Entropic Scalar EFT: From Entanglement Microstructure to Gravity and Cosmic Structure

Contents

Part I. Physical Idea and Foundations — 5

1. Introduction: The Physical Claim — 5 - 1.1 Primitive Inputs and Closure Status — 6 - 1.2 Physical Motivation for the Primitives — 9

2. Canonical Field Content and Definitions — 11

3. The Three Postulates — 13 - 3.1 Information–Geometry Equivalence — 13 - 3.2 Mass–Entropy Equivalence — 13 - 3.3 Many-Pasts Hypothesis — 14

4. Relativistic Continuum Structure — 15 - 4.1 Capacity budget and continuum symmetry — 15 - 4.2 Dependency Map of the Theory — 16

Part II. UV Coefficient Chain — 17

5. Why a Tetrahedral Boundary Ensemble — 17

6. Admissibility Closure — 19 - 6.1 Minimal isotropic kernel — 19 - 6.2 Closure condition and uniqueness — 19 - 6.3 Effective sharing entropy — 20

7. Edge Kernel and Tree-Level Coupling — 20

8. Finite-Loop Renormalization — 21

9. Continuum Stiffness and SI Normalization — 22

Part III. The Closed Static Branch: Einstein Gravity in the Capacity Variable — 24

10. Einstein Parent Action and the Reduced Capacity Frame — 24

11. Capacity Variable, Bridge Law, and Variational Status — 26

12. Newtonian Gravity and the Point-Source Limit — 27

13. Electron Anchor: One-Bit Mass and Seven-Sector Length — 28 - 13.1 Why the electron is the anchor — 28 - 13.2 One-bit mass anchor — 28 - 13.3 Seven-sector length anchor — 29 - 13.4 Decorated marked-transfer vertex — 29 - 13.5 Consistency checks — 31 - 13.6 Composite sectors — 32

14. Baseline Metric Closure: No Slip and PPN — 32

Part IV. The Carrier-Resolved Galactic Branch — 32

15. Carrier-Resolved Galactic Dynamics — 32

16. Galactic Metric, Lensing, and Local Tests — 37

Part V. Transport, Clusters, and Cosmology — 42

17. Causal Transport and Telegrapher Dynamics — 42

18. Cluster Source Projection and the Diffuse–Decoupled Channel Split — 43

19. Cosmology and the Hubble-Tension Sector — 48

20. The Saturated Phase and the Cosmic Microwave Background — 49

Part VI. Strong Fields, Many-Pasts, and Microstructure — 54

21. Strong-Field Action: Spherical Closure and Its Boundary — 54

22. Many-Pasts: The History-Space Ontology — 56

23. Microstructure Hamiltonian and Underlying Dynamics — 59

Part VII. The Substrate on a Dynamical Lattice — 61

24. Lattice Tests: Compatibility, Defect Response, and Transport — 61 - 24.1 The host geometry and the cell identification — 61 - 24.2 The coupled ensemble and its controls — 62 - 24.3 What the closure sector cannot supply: vacuum stiffness — 63 - 24.4 The externally hosted vacuum and the role of conditioning — 63 - 24.5 Compatibility: the weighting on dynamical geometry — 64 - 24.6 The defect experiment: geometry responds to the theory's mass — 64 - 24.7 Reaching Newtonian range: the conservation requirement — 65 - 24.8 What the finite-lattice results establish — 67 - 24.9 The capacity-decorated continuum target — 67

25. Equilibrium Vacuum and Cosmological Term — 69

Part VIII. Closure Status, Falsifiability, and Comparisons — 73

26. Closure-Status Table — 73

27. Falsifiability and Observational Tests — 83 - 27.1 Static weak-field falsifiers — 83 - 27.2 Dynamical falsifiers — 84 - 27.3 Cosmological falsifiers — 85 - 27.4 Correlated-constant falsifiers — 86 - 27.5 Many-Pasts status — 86

28. What the Theory Would Have to Get Wrong to Fail — 86

29. Comparison with Other Approaches — 87 - 29.1 Relative to ΛCDM — 87 - 29.2 Relative to MOND-like interpolation programs — 87 - 29.3 Relative to Verlinde-style emergent gravity — 88 - 29.4 Relative to TeVeS and other multi-field modified gravities — 88 - 29.5 Relative to AeST — 88 - 29.6 Relative to scalar-tensor gravity — 88 - 29.7 Relative to quantum-mechanical interpretations — 88 - 29.8 Relative to CDT, spin foams, and group field theory — 89 - 29.9 Relative to algebraic and information-geometric gravity — 89

30. Conclusion — 90

Part I. Physical Idea and Foundations

1. Introduction: The Physical Claim

Space, in this proposal, is a finite medium of entanglement capacity. The particles we call matter are stable defects that lock away part of that capacity, and the surrounding medium responds to the commitment. Seen at large scales, that response is gravity:

• Matter is a localized capacity defect of the substrate.
• Mass is the entanglement that defect commits, read in mass units.
• Gravity is the extended capacity strain — the fractional capacity deficit — the medium carries around the defect.
• Dark-matter phenomenology comes from two further regimes of the same medium: the long-range capacity strain on galactic scales and the saturated phase in the early universe.
• General relativity is the low-energy geometry of this capacity medium.

Written as a continuum theory, this becomes a scalar EFT for a vacuum-relative entanglement field \(S_{\text{ent}}(x)\) and its deficit \(\delta S\) relative to the background capacity. The defect sector is written at continuum scale in ordinary stress-energy variables, but its ontology is unchanged: inertial mass enters through the mass-per-entropy map \(\kappa_m\), and the weak-field potential is the fractional deficit \(\delta S/S_\infty\).

The gravitational response normally attributed to a dark halo is assigned here to the capacity structure of the vacuum itself. Ordinary weak gravity, the galactic excess, and the homogeneous cosmological mode are different regimes of one medium: general relativity supplies its low-energy geometry, and the capacity variable tracks how localized defects deplete and redistribute the available entanglement. Section 29.9 compares this proposal with recent information-theoretic constructions of gravity, after the paper's own development is complete.

The finite-capacity substrate, three postulates, and tetrahedral ensemble define the framework. The selected realization specifies its auxiliary response coordinates, marked routing, additive charged generator, and geometry–capacity matching. Finite enumeration fixes the admissibility entropy and marked weights. The charged transfer and electron anchor then calibrate \(L_*\), while source projection and physical matching give the ordinary gravitational normalization.

The capacity/record action splits a primitive shared face into a seven-state matched channel and a nine-state retained mismatch record. Appendix H.11 constructs its finite complex state geometry, composition, and record probabilities within the specified quadratic and operational completion. The ordinary geometric branch uses a selected fixed-facet host. Maximal fusion supplies the canonical \(V_3^{\otimes N} \to V_{3N}\) blocking map, and the metric-Regge branch supplies the controlled two-helicity TT limit and exact primitive pullback. The capacity premises permit different native geometric tensors and measures, so selection of a nonperturbative Einstein phase remains a geometric input and phase problem.

The cosmological release model uses further infrared prescriptions: a post-refresh energy ledger, a homogeneous projection on the capacity clock carried by source-associated flow, and the initial condition \(\rho_{\Lambda,i} = 0\). Its finite volume projection and renewal identities are exact. A constrained covariant clock action reproduces the background exchange and fixes its lapse, clock, and velocity perturbations for any specified release history. For a closed progenitor cohort under common advection–diffusion, the fully resolved position-information loss is bounded and monotone. The coarse observable used by the physical release law requires its own monotonicity check in the self-gravitating progenitor-tagged zoom before the full likelihood is evaluated. Progenitor records taken to mix on infall into the turbulent cores of first-galaxy halos, in about one halo dynamical time, give \(\Omega_{\Lambda,0} = 0.68^{+0.03}_{-0.04}\); that form and the order-unity range of its coefficient were adopted with the observed abundance known, and the same zoom must measure the mixing time.

The most controlled branch is the ordinary static weak-field action and source map

$$\text{microstructure} \longrightarrow \text{coefficient chain} \longrightarrow \text{static capacity EFT} \longrightarrow \{G, \text{baseline metric}\}.$$

It recovers the Newtonian point-source limit, the leading no-slip metric, and the parametrized post-Newtonian values of general relativity through its Einstein parent. A specified carrier-resolved transverse branch produces the galactic acceleration scale \(a_0\), the radial-acceleration relation, its conservative nonspherical extension, and the leading lensing metric without per-system tuning, subject to the microscopic matching conditions stated in Sections 15–16 and Appendix N. Evaluated on the joint coarse source of all active carriers, the branch conserves momentum between a galaxy and its satellites, and the response budget set by each carrier's Lagrangian baryons passes the isolated-lens weak-lensing test. The electron anchor, memoryless dressing, and marked vertex fix the substrate length inside the stated support-to-rate branch, and the resulting Newton normalization and corrected charged-lepton ratios agree with current measurements within one standard deviation. The comparison also runs backward: with the marked weight held at its action value, the measured Newton constant selects the vertex's routing integer, and the unique survivor is the same seven fixed by the tetrahedral alphabet (Section 13.5). Appendix L separates this action-level closure from the historical fact that the residuals were already known.

Later parts treat time-dependent transport, galaxy clusters, cosmology, the saturated early phase, strong fields, and particle and gauge extensions. Their derivational status is listed in Part VIII.

Many-Pasts, the third postulate, already does work in this chain: faithful sector resolution selects the memoryless electron-dressing kernel, and Many-Pasts supplies the history space in which that kernel operates. Its consequences for quantum probability, branch realization, and the arrow of time are developed in Section 22 and Appendix G.

1.1 Primitive Inputs and Closure Status

The word "closure" is used here in a specific sense. The paper does not derive the existence of a finite entanglement substrate or the tetrahedral boundary architecture from a deeper microscopic Hamiltonian; those are theory-defining inputs, and the closure claim begins only after they are fixed. There are five such inputs: finite local entanglement capacity; geometry–capacity equivalence; mass–entropy equivalence, including matter as localized defects of committed capacity; the Many-Pasts ontology of Postulate III, with ordinary probability normalization and finite additivity for mutually exclusive complete physical records; and the tetrahedral ultraviolet architecture, including positive oriented matching of the two primitive descriptions of a shared face. Appendix B writes that matching rule as a primitive pair operator. Within the selected sharp reversible vertex, unitarity proves that the retained carrier is its orthogonal fusion complement; the hard-core realization, flags, relative phases, and routing remain selected microscopic details. This refines the existing ultraviolet input rather than adding a fourth postulate. Maximum caliber is likewise not a sixth input: it restates, for histories, the same faithful full-support condition already used to select the admissibility ensemble — every pass carries the largest path entropy compatible with the same fixed marginal. The equivalence has to be stated explicitly because finite capacity by itself does not imply renewal.

Given those inputs, the chain closes step by step. Finite counting fixes the admissibility weighting and the effective entropy. Faithful full-support resolution fixes the memoryless replacement kernel, and the decorated native-cell vertex realizes its reversible update. The state-weighted determinant supplies the baseline seven-channel recurrence, and the same closure amplitude, projected through two directed singlet returns and canonically dilated, supplies the marked correction and its 21-edge determinant. Finite-state recurrence fixes the shell support factor, while a common scalar fluctuation gives a positive realization of its coherent quadratic overlap. The electron anchor fixes the proper-time cadence. The continuum speed identification defines its causal length as \(L_* = c\tau_*\); the rest-gap calculation alone does not fix a microscopic spatial hopping coefficient. Edge transport and source projection then determine the ordinary static response. The longitudinal functional turns out to be the Einstein scalar-constraint sector in a different variable, so it yields the Newtonian limit, the baseline no-slip metric, and the PPN values of general relativity. Controlled Regge results and the exact primitive pullback establish the Einstein transverse-traceless limit on the selected metric branch. On that branch, one covariant Einstein–Dirac action carries the shell mass operator, relativistic matter propagation, and metric sourcing together. The joint perfect-action pullback and renewal identity preserve this coupled system exactly at finite regulator. The empty-capacity lift is exact. The host-underdetermination theorem shows that the present capacity premises do not determine the geometric gluing tensor or measure. The galactic branch is closed as a specified leading EFT; its carrier projector, thermal cell Hamiltonian, and metric contact still require a microscopic derivation. In the cosmological branch, a constrained capacity-clock action fixes the covariant exchange once the release history is supplied. Common advection–diffusion proves monotonicity for fully resolved position information; the coarse physical mixing observable remains an output of the cosmological zoom.

The five commitments define the physical framework. Its quantitative realization also specifies the marked field content, routing, auxiliary response coordinates, and charged generator. The internal trace determines the motif weights; the additive-generator rule of Appendix H.9 determines how those weights enter the energy. The normalized geometry–capacity matching of Appendix C.5 sets \(G = G_*\). These prescriptions are part of the selected action and matching data. Their numerical consequences can be tested even though the broad commitments admit other realizations.

In compressed form, the central claim is

$$\text{primitive UV capacity hypothesis} \to \text{finite counting + admissibility} \to L_*$$ $$\to \gamma, \kappa/\gamma \to \delta S \leftrightarrow \Phi \to G, \text{baseline metric},$$ $$a_0, \text{RAR, galactic lensing in the carrier-resolved transverse EFT}.$$

The microstructure is an input: a finite ultraviolet counting problem from which the weak-field sector is derived.

The absolute scale calibration uses the electron, the lightest elementary charged defect, as the dimensional anchor. Faithful full-support resolution forces the replacement kernel to be memoryless: a dressing pass can carry the full admissibility entropy only if it retains no memory of the endpoint it replaces (the one-line proof appears in Section 1.2). The lightest-defect functional selects the fermionic ceiling \(k = 7\) with \(\Delta_7 = 0\), and the state-weighted determinant of the seven renewed clouds is \(r = e^{-7g_{\text{share,eff}}}\). The decorated vertex fixes the residual marked-fiber factor and its electron routing \(Z_e\). The additive-generator prescription assigns these weights to the charged energy. Positivity gives the raw survival energy \(E_{\text{raw}} = -(\hbar/\tau_*) \ln(1 - r)\), and the dressed electron identification \(m_e c^2 = (3/2)Z_e E_{\text{raw}}\) fixes \(\tau_*\). The associated causal length is defined by \(L_* = c\tau_*\). This conversion introduces no geometric diameter, but a microscopic matter action must still reproduce the same invariant speed in its spatial dispersion. Appendix H gives the finite action, the adversarial audits, and an exact benchmark for that remaining test.

Appendix I separates the arithmetic recurrence statistic that supplies the shell support from the geometric typical recurrence measured by entropy and gives an explicit positive realization of the coherent \(N^2\) overlap.

The reduced capacity functional can superficially resemble the scalar sector of a Brans–Dicke theory [78], but the resemblance is misleading. In the ordinary static branch it is the Einstein constraint action rewritten through \(\delta S = -2S_\infty\Phi/c^2\), with no second scalar–tensor action. The open action questions are narrower: the microscopic origin and nonlinear covariant variation of the carrier-resolved transverse contact, the generic covariant capacity observable outside controlled reductions, and the saturation-boundary functional.

Several tasks remain: derive the ensemble from a deeper Hamiltonian, complete and audit the coupled geometry–capacity interaction beyond the factorized branch, derive a charged spatial transfer with the same infrared speed as the geometric sector, audit the separate finite-loop stiffness return operator, derive the carrier projector and transverse metric contact microscopically, and construct the strong-field boundary action. The selected metric-Regge GFT branch already has the massless two-helicity TT limit and an exact fixed-\(j = 3\) primitive realization. The remaining geometric problem is to find a principle that selects a unique native gluing tensor and measure, then prove that its nonperturbative measure lies in the Einstein phase. The independent capacity-decorated CDT route must exhibit an extended four-dimensional phase with a massless transverse–traceless transfer sector; a CDT critical surface is the stronger optional cutoff-removal test. Each unresolved step is listed explicitly in the closure table.

The construction rests on five commitments.

First, the vacuum is a medium with a bounded local capacity for entanglement.

Second, spacetime geometry and that capacity structure are the same substrate seen at different scales, so that in the weak field gravity is the fractional deficit of locally available capacity.

Third, matter is localized committed capacity: a particle is a stable defect of the medium, and its inertial mass is the entanglement content of that defect read in mass units.

Fourth, a recorded present is supported by many compatible microscopic pasts. The operational branch assigns probabilities only to decoherent record histories through the history Gram kernel derived in Appendix H.11, then conditions them on the realized present. Its finite retained sector gives Born statistics and no-signaling. The reversible renewal dilation supplies a concrete microscopic role for the history degrees of freedom: they receive the previous local state while the present register is renewed.

Fifth, the ultraviolet cell has tetrahedral boundary architecture. Each shared face begins with two fermionic primitive slots and positive oriented matching. The matching operator transmits the maximal coupled multiplet. In the selected sharp reversible vertex, unitarity identifies the complete nine-state marked carrier with the discarded complement. The three-dimensional closure vector then fixes \(j_0 = \frac{3}{2}\) and the seven-state alphabet internally. For a different microscopic vertex, \(j_0 = \frac{3}{2}\) remains the surviving branch of the discrete audit in Appendix B. Four relational ports, single-copy channel capacity, and the two orientations then give the 1680-state ensemble. The same faithful-resolution standard is applied to its histories: the local process carries the full available path entropy and retains no endpoint memory. "Maximum caliber" names this temporal application of the same standard; it adds no independent commitment to the five listed here.

The finite-capacity substrate is the ultraviolet premise; geometry–capacity equivalence, mass–entropy equivalence, and Many-Pasts are the three postulates; and the tetrahedral ensemble is the ultraviolet architecture. Faithful full-support resolution acts on both states and histories. From these inputs the paper derives the admissibility weighting, \(g_{\text{share,eff}}\), the replacement kernel, the one-layer native-vertex update, the marked transfer, the tree edge factor, the weak-field bridge, and the Newtonian metric branch, all inside the displayed decorated action. The separate loop-dressed stiffness, the galactic acceleration scale, and the radial-acceleration relation carry the conditional grades recorded in the closure table.

The architecture therefore contains three postulates, one finite-capacity substrate premise, one minimal ultraviolet ensemble, and a small number of explicitly labeled conditional readings. Applied to the same ensemble, faithful resolution fixes both the state entropy and the maximum-caliber history process.

1.2 Physical Motivation for the Primitives

These commitments are premises, and the derivation begins only after they are fixed. Each addresses a specific open problem in black-hole thermodynamics, quantum information, the equality of inertial and gravitational mass, or the quantum-mechanical role of records. Their motivations do not prove them; their numerical and structural consequences provide the tests.

Three established results motivate a finite-capacity substrate. A black hole's entropy scales with horizon area; the Bekenstein bound limits the information a bounded region can hold; and entanglement-based reconstructions relate spatial geometry to entanglement structure. The present model treats the vacuum as a medium with a finite local entanglement budget and geometry as its large-scale description. In this picture, a black hole exhausts the interior capacity, leaving the active bookkeeping at the boundary between exhausted and available capacity. A finite cell state space also supplies an ultraviolet cutoff because the continuum description ends below the cell scale. The assumption that the empty state maximizes capacity agrees with the thermodynamic direction suggested by gravitational entropy. Three recent continuum results provide compatible limits: the observer-dressed de Sitter algebra makes empty de Sitter the maximum-entropy gravitational state [27], a horizon modular calculation recovers Einstein curvature from the relative information carried by an excitation [26], and a geometric-relative-entropy action yields local bulk information dynamics with an Einstein limit [28]. None supplies the finite cell or its coefficients; Section 29.9 states their precise role in the present construction.

The mass–entropy identification addresses a coincidence that general relativity encodes but does not explain. General relativity builds in the equality of inertial and gravitational mass geometrically, through the equivalence principle, but gives no microphysical account of why the mass that resists acceleration and the mass that sources attraction should be one and the same. Here both are readings of a defect's committed entanglement, so the equality follows from the construction instead of being imposed by hand. The same identification bears on why gravity is so weak. The induced gravitational scale contains the exact factor \(Z_e^2 \ln^2(1 - e^{-7g_{\text{share,eff}}})\), whose dominant hierarchy is \(e^{-14g_{\text{share,eff}}}\): a large sharing entropy makes gravity exponentially feeble, and the gap between gravity and the other forces becomes a matter of arithmetic rather than fine-tuning. The galactic extension carries a lower closure grade. In its carrier-resolved transverse branch, \(a_0 = \epsilon cH_0\) with \(\epsilon \equiv g_{\text{share,eff}}/(4\pi^2)\) ties the onset of anomalous rotation to the cosmic horizon. The thermal contact fixes the response of each retained carrier, and the auxiliary action relates that response to the resolved baryonic source. The phase-cell loading, carrier projector, and microscopic metric vertex are the tests of that connection.

The scale-setting proposal separates two questions that sound alike: how hard the medium works to keep a defect bound, and how often the defect fully re-forms. Because the medium never holds still — every instant it is re-drawn from all the ways it could be — a particle is a pattern the medium must continually re-form, and the entanglement it commits is a maintained quantity, re-established at every update. Keeping the knot bound means holding a grip on every strand at once, and separate holds add up, so the maintenance is large. Full re-completion, with every strand falling into alignment at the same instant, is a simultaneous coincidence, and coincidences multiply, so it is exponentially rare. The particle's mass follows the second question, not the first: it is fixed by how rarely the binding fully re-closes. This rare recurrence allows the electron to lie far below the natural substrate scale without an inserted small number: deep binding makes full coherent recurrence exponentially unlikely. Quantitatively, seven channel entropies add, so their effective support multiplies; the state-weighted determinant turns that support into a record-conditioned transfer rate; the positive survival operator gives the rest-energy gap; and the decorated marked vertex fixes the finite closure-response correction while the electron anchor fixes the clock. Section 23 and Appendix H give this its full form.

A memoryless update rule posits no hidden machinery carrying information forward from tick to tick, and in this construction it is also forced. For any stationary per-channel kernel with marginal \(p_{\eta_*}\),

$$H(B_{t+1} \mid B_t) = g_{\text{share,eff}} - I(B_t; B_{t+1}) \leq g_{\text{share,eff}},$$

so a dressing pass carries the full admissibility entropy if and only if the endpoint mutual information vanishes, which fixes the refresh kernel \(K(b, b') = p_{\eta_*}(b')\). The allowed local single-label dynamics cannot realize this requirement: they freeze into disconnected sectors that never explore the full space (Appendix D.4), so the refresh must act nonlocally on the native cell. The decorated vertex prepares the diagonal fresh amplitude and realizes the charged recurrence; its geometric condensate embedding and durable history capacity remain open. Compatibility with Lorentz tests requires the retained matter sector and the geometric modes to share the same invariant infrared speed and requires substrate loops to suppress dimension-four Lorentz-violating operators. Renewal fixes neither result. Section 4 and Appendix H.11 state these as tests of a microscopic completion.

Many-Pasts holds that the present configuration of the entanglement network is supported not by one definite microscopic past but by a conditional ensemble of compatible decoherent histories. It provides an ontological account of interference, entanglement correlations, and measurement while preserving ordinary quantum dynamics. The interference in a double-slit experiment is the persistence of the unrecorded alternative histories in that ensemble; a durable which-path record conditions the ensemble, and the interference goes away. The correlations of an entangled pair come from weighting the histories of the whole joint system, which reproduces the nonclassical statistics with no signal passing between the two wings. Measurement adds no separate collapse law; it lays down a durable record, after which the relevant histories are the ones compatible with it. The operational construction is therefore ordinary quantum mechanics equipped with a history-space ontology, and it leaves laboratory predictions intact: Born-rule statistics and no-signaling both hold. Its arrow-of-time extension still needs a substrate typicality theorem. The postulate also constrains the construction: faithful sector resolution independently selects the memoryless dressing, and Many-Pasts supplies the history-space setting in which that process operates and in which its exported registers live.

Tetrahedra are standard building blocks in several approaches to quantum geometry. The specific addition here is a primitive matching rule. After the two descriptions of a shared face are transported into one orientation, its positive mismatch operator has the maximal-spin sector as its unique null space. For the physical branch, the independently constructed nine-state marked register matches the complete nonmaximal fusion sector only at \(j_0 = \frac{3}{2}\). Conditional on identifying those two objects dynamically, three-dimensional closure fixes seven face labels; four relational ports, single-copy channel capacity, and two orientations then give exactly 1680 boundary states. Without that identification the same result is the unique surviving point of a stated discrete branch audit, not a consequence of minimality. The closure structure also permits only three nondegenerate charged-lepton shells, a candidate answer to why the Standard Model contains three charged-lepton generations. Appendix L preserves the historical provenance of the construction. Once the branch is fixed, its entropy contains no adjustable continuous parameter.

Motivation alone carries little evidential weight, because a proposal of this scope can almost always assemble a list of mysteries explained after the fact. The relevant questions are how much freedom remained when the construction was chosen and whether one fixed construction survives measurements it did not anticipate. Appendix L audits the first question. The following sections derive the weak-field consequences and compare them with observation.

2. Canonical Field Content and Definitions

Before the symbols, four plain words recur throughout. Capacity is the entanglement support locally available in the medium. A defect is a stable, localized commitment of that capacity — what we coarse-grain into a particle. A deficit is capacity no longer freely available to the surrounding vacuum because a defect has committed it. Strain is the extended profile of that deficit reaching out into the medium, whose fractional size the weak-field potential tracks. The field variables below are the precise versions of these words.

We define the fundamental continuum variable as the vacuum-relative coarse-grained entanglement assigned to a UV probe cell of size \(L_*\) centered at \(x\):

$$S_{\text{ent}}(x) \in \mathbb{R},$$

measured in nats and therefore dimensionless. This is not a literal microscopic entropy density at a mathematical point. It is the leading scalar order parameter associated with a vacuum-relative entanglement defect after coarse-graining over a UV cell.

This definition keeps the microscopic and continuum pictures tied together. At continuum level, \(S_{\text{ent}}(x)\) is the field that appears in the action and field equations. At the microscopic level it is the coarse variable recording how much local entanglement capacity remains available in the underlying medium after averaging over a UV cell.

The asymptotic vacuum-capacity baseline is denoted \(S_\infty\), and the deficit field is

$$\delta S(x) \equiv S_\infty - S_{\text{ent}}(x).$$

Positive \(\delta S\) denotes reduced available vacuum entanglement capacity in the neighborhood of a localized defect or defect distribution. It is the extended capacity-strain field sourced by the defect sector, not an independent medium acted on by matter from outside. For nonlinear work it is useful to define the bounded occupancy fraction

$$q(x) \equiv \frac{S_{\text{ent}}(x)}{S_\infty} = 1 - \frac{\delta S}{S_\infty} \in [0, 1].$$

The variables \(S_{\text{ent}}\), \(\delta S\), and \(q\) therefore describe the same local physics in three closely related ways: available capacity, missing capacity relative to vacuum, and surviving-capacity fraction. Each is used where it is most transparent: \(\delta S\) for the weak-field theory, because it maps directly onto the Newtonian potential; \(q\) for the nonlinear and strong-field completion, because boundedness is built in from the start; and \(S_{\text{ent}}\) itself for the covariant EFT, because it is the field that appears in the action. The operational meanings are:

• \(q = 1\): vacuum capacity fully available in the absence of local defect-induced capacity strain;
• \(0 < q < 1\): partial local capacity reduction around a defect configuration;
• \(q = 0\): complete local exhaustion of available capacity on the physical branch.

Fixed-epoch normalization. The entropy unit is conventional at a fixed epoch. The physical ratio \(\kappa/(\gamma S_\infty)\) is fixed separately by the geometry–capacity matching in Appendix C.5. Under a constant rescaling

$$S_{\text{ent}} \mapsto K S_{\text{ent}}, \quad S_\infty \mapsto K S_\infty, \quad \delta S \mapsto K \delta S,$$

the observable bridge

$$\frac{\Phi}{c^2} = -\frac{\delta S}{2S_\infty}$$

is unchanged. The source equation is invariant in the same sense: rescaling the entropy field rescales the source coefficient with it, so the observable Newtonian normalization depends on the gauge-invariant combination \(\kappa/(\gamma S_\infty)\) rather than on \(S_\infty\) alone. A cell-normalized description and a horizon-normalized description can therefore assign different numerical values to \(S_\infty\) without changing \(\Phi\), \(G\), or the PPN limit. This is not a time-dependent gauge symmetry; it is a fixed-epoch entropy-unit convention. Gravity sees fractional capacity depletion.

Substrate length scale. The canonical UV cell length is not taken to be the conventional Planck length as an input. Faithful full-support resolution fixes the renewal kernel and its history-space factorization. The state-weighted determinant gives the seven-channel recurrence

$$r = e^{-7g_{\text{share,eff}}}, \quad L_*^{(0)} = -\frac{3}{2}\lambda_e \ln(1 - r), \quad \lambda_e = \frac{\hbar}{m_e c}.$$

The decorated marked-transfer vertex derived in Appendix H fixes

$$\zeta_* = 9e^{-g_{\text{share,eff}}}\left(1 - \frac{8\eta_*}{49}\right)^{21/2}, \quad Z_e = (1 + \zeta_*)(1 + 7\zeta_*^2).$$

The physical electron-anchored scale is

$$\boxed{L_* = Z_e L_*^{(0)}} = 1.6162537024 \times 10^{-35}\ \text{m}.$$

The corresponding induced gravitational scale is

$$G_* := \frac{c^3 L_*^2}{\hbar} = \frac{9}{4}\frac{\hbar c}{m_e^2}Z_e^2 \ln^2\left(1 - e^{-7g_{\text{share,eff}}}\right) = 6.6742890813 \times 10^{-11}\ \text{m}^3\text{kg}^{-1}\text{s}^{-2}.$$

The lightest one-bit fermionic defect resolves the seven face sectors once and exports the transverse \(2/3\) share of that dressing block. The marked vertex accounts for its finite closure-response fiber and label-return loop. The conventional Planck length \(L_P = \sqrt{\hbar G/c^3}\) remains useful for comparison and for standard black-hole thermodynamic notation, but it is not the primitive scale-setting input here.

The principal coefficients and derived quantities used throughout are:

$$\gamma : \text{entanglement-field stiffness}, \tag{1}$$ $$\kappa : \text{defect–entropy coupling}, \tag{2}$$ $$\kappa_m(\ell) : \text{mass-per-entropy map at scale } \ell, \tag{3}$$ $$L_* : \text{substrate cell length in the electron-anchored support-to-rate map}, \tag{4}$$ $$G_* : \text{gravitational scale induced by } L_*, \tag{5}$$ $$g_{\text{share,max}} = \ln(1680), \tag{6}$$ $$g_{\text{share,eff}} : \text{admissibility-weighted effective sharing entropy}, \tag{7}$$ $$J_{\text{bare}}, J^{\text{tree}}_{\text{eff}}, J^{(\text{ren})}_{\text{eff}} : \text{UV edge-kernel couplings}, \tag{8}$$ $$a_0 = \frac{cH_0 g_{\text{share,eff}}}{4\pi^2} \text{ in the conditional compact two-phase normalization}. \tag{9}$$

The gravitational potentials are denoted \(\Phi\) and \(\Psi\), and the canonical weak-field bridge will be written as

$$\frac{\Phi}{c^2} = -\frac{\delta S}{2S_\infty}.$$

These same symbols reappear in the ultraviolet closure chain, the continuum action, and the phenomenology sections. From this point onward each one keeps the same meaning, so the later derivations build on a single notation rather than shifting between parallel conventions.

3. The Three Postulates

The three postulates answer distinct physical questions. Postulate I identifies spacetime geometry with the long-wavelength expression of the capacity substrate. Postulate II identifies matter with localized committed capacity and mass with its inertial reading. Postulate III describes the realized present through a history-space ontology over the compatible decoherent pasts recorded in it. Appendix H.11 derives the finite retained-sector probability and history machinery; its global continuum/Fock and cosmological realization remains open. The proposed arrow of time separately requires an additional typicality result. Faithful sector resolution, applied to this history space, selects the memoryless dressing kernel used by the decorated scale-setting action.

3.1 Information–Geometry Equivalence

The first postulate states that spacetime geometry is the continuum expression of the capacity substrate. This is stronger than saying entanglement contributes an additional piece of stress-energy inside otherwise standard general relativity: the metric and the scalar capacity sector are two projections of one finite medium, and \(S_{\text{ent}}\) is not appended to an independent background geometry. In the weak field, gravitational potential is the fractional deficit of available capacity.

Two consequences of this reading should be kept distinct from the start. First, absolute \(S_{\text{ent}}\) is not itself "the gravitational potential"; the observable weak-field potential comes from the fractional deficit \(\delta S/S_\infty\), which is why a fixed-epoch rescaling of entropy units leaves gravity unchanged (Section 2). Second, because geometry and capacity are two descriptions of one response, the deficit is not an extra force appended to an independently existing metric. Section 10 proves that the ordinary reduced capacity functional is the Einstein constraint action in the capacity coordinate. The remaining common-parent problem is the microscopic embedding of the carrier-resolved transverse contact and the saturation boundary.

3.2 Mass–Entropy Equivalence

The second postulate identifies mass as the inertial reading of localized capacity commitment. At scale \(\ell\),

$$m(\ell) = \kappa_m(\ell) \Delta S.$$

A particle is already a localized defect of the entanglement substrate, so \(m = \kappa_m \Delta S\) does not assert an analogy between two independent things; it asserts that the inertial content of the defect is its entanglement content, read in mass units.

For elementary fermionic sectors the canonical defect increment is

$$\Delta S_f = \ln 2.$$

The one bit here is not arbitrary. An elementary fermionic exclusion is binary — the face is occupied or unoccupied — and a binary distinction carries exactly \(\ln 2\) of missing entanglement. This is the simplest possible defect increment, which is why the lightest such defect, the electron, becomes the cleanest anchor for the mass–entropy map (Section 13). Composite sectors instead require their fully dressed bound-state entanglement budgets.

Two corollaries are used later. First, because mass and entanglement budget are two descriptions of the same defect, and the masses of separated defects add, capacity committed in service of one defect cannot simultaneously serve another: shared service would make the joint budget, and with it the joint mass, sub-additive. Commitment is therefore per-defect — each committed unit carries the label of the defect it serves. Second, the same bookkeeping makes the saturated early phase countable: its abundance follows from the number of committed units, with no double-counting across separated defects (Section 20).

3.3 Many-Pasts Hypothesis

The third postulate concerns the microscopic support of the present entanglement network. A recorded present can be compatible with many coarse-grained histories of the substrate. Many-Pasts takes those alternative pasts seriously while retaining one realized macroscopic present. Its probability theory must distinguish amplitudes for alternatives that still interfere, probabilities for recorded presents, and conditional probabilities for decoherent histories compatible with a given record.

The operational construction takes the decoherent-histories form [93]. Appendix H.11 derives that form for every finite retained sector from its real dynamics, record composition, and norm. A coarse history \(h = (\alpha_1, \ldots, \alpha_n)\) has class operator

$$C_h = \Pi^{(n)}_{\alpha_n} U_{n,n-1} \cdots \Pi^{(1)}_{\alpha_1} U_{1,0},$$

and decoherence functional

$$\mathcal{D}(h, h') = \text{Tr}\left(C_h \rho_0 C^\dagger_{h'}\right).$$

When a family decoheres, \(\mathcal{D}(h, h') \simeq 0\) for \(h \neq h'\), its diagonal entries obey the ordinary probability sum rules. If \(P\) denotes a final macroscopic record and \(\mathcal{H}_P\) is a decoherent refinement of the histories ending in that record, then

$$p(P) = \sum_{h\in\mathcal{H}_P} \mathcal{D}(h, h) = \text{Tr}(\Pi_P \rho_{\text{now}}), \quad p(h \mid P) = \frac{\mathcal{D}(h, h)}{p(P)}.$$

The probability of the present is obtained from the common normalized measure over all records before the history distribution is conditioned on \(P\). Unresolved alternatives remain combined at amplitude level; no positive probability is assigned to individual fine-grained paths that have not decohered.

This construction makes the finite operational branch standard quantum mechanics. Reversible coupling to a retained record gives the Kraus operators, POVMs, conditional update, and trace-preserving reduced channel. The norm rule gives their Born probabilities, and locality plus trace preservation gives no-signaling. Many-Pasts changes the history-space interpretation without adding a collapse term or a signaling bias.

The ontological reading is record-retentive. The realized present includes both its current macroscopic configuration and the physical records that encode earlier events. Compatible pasts are distinguished to the resolution carried by those present records; they are not additional coexisting spacetimes. When a physical process erases a record distinction, the conditional history measure coarsens by summing the histories that the surviving record can no longer separate. Conditioning on a future record is defined only after that record belongs to a present configuration, so the construction supplies no future-to-past force or retrocausal update law.

For the retained one-mark sector, Appendix H.11 derives rather than assumes \(\mathcal{H}_{\text{mark}} \simeq \mathbb{C}^9\) and the ray space \(\mathbb{CP}^8\). Reversible internal equivalence makes a branch weight a function only of its squared norm. Additivity under coarse-graining of orthogonal durable records makes that function additive; positivity and normalization then fix

$$\boxed{p(\Pi \mid \psi) = \|\Pi\psi\|^2 = \langle\psi|\Pi|\psi\rangle.}$$

This proof uses record completeness, nonnegativity, normalization, and finite additivity for mutually exclusive physical records. Gleason's theorem [87] remains an independent consistency check rather than the step that carries the Born derivation. The same norm makes the finite decoherence functional the Gram kernel of history branch vectors. The full continuum/Fock-space kinematics, the cosmological initial state, the histories that actually decohere, and durable global record capacity remain open.

The postulate and the renewal theorem have separate logical roles. Many-Pasts supplies the ontology and the record-conditioned history space. The already-stated faithful full-support condition selects the memoryless dressing kernel when applied to paths at fixed admissibility marginal; maximum caliber names this history-space form and is independent of the Born measure on laboratory records. The lightest-defect functional selects the same absence of temporal memory and, within the support-to-length map, selects seven occupied channels with vanishing inter-channel correlation. A reversible dilation exports the old replaceable closure register to history. The determinant-survival action and additive-generator prescription combine the finite marked response with the Compton phase readout.

The renewal statement applies to the replaceable closure register, not to every degree of freedom carried by a charged defect. The marked internal fiber and its position degree of freedom belong to the retained system. Appendix H.11 proves the corresponding coherence criterion: after a dilation, a spatial off-diagonal is multiplied by the overlap of the discarded records produced by its two branches. Free marked transport must therefore export no cell address or other which-path label. This is a consistency condition on the microscopic realization of the existing operational quantum postulate, not an additional founding premise.

4. Relativistic Continuum Structure

4.1 Capacity budget and continuum symmetry

The continuum description is expected to be covariant because the substrate itself is finite-capacity, isotropic, and relational — covariance is read off from the substrate's own properties rather than added as a geometric axiom at the outset.

The first ingredient is a finite maximal update rate, denoted by the same constant \(c\) that later appears in the transport relation \(D/\tau_0 = c^2\). In the present interpretation, \(c\) measures the largest rate at which the substrate can propagate and reorganize information. A defect at rest spends that budget entirely on local temporal evolution. A defect in motion must spend part of the same budget on spatial transport within the surrounding network. Because the substrate is isotropic, the cost of motion depends only on the rotational scalar \(v^2\) at leading order, with the temporal rate maximal at \(v = 0\) and vanishing when the budget is exhausted at \(v = c\). These endpoint conditions alone admit many interpolating functions and so do not fix the form of the time-dilation relation. The form is fixed once the finite update speed is treated as invariant across inertial coarse descriptions: homogeneity, isotropy, and the relativity principle then select the Lorentz group rather than the Galilean one, giving the invariant interval

$$c^2 d\tau^2 = c^2 dt^2 - d\mathbf{x}^2,$$

and hence

$$\frac{d\tau}{dt} = \sqrt{1 - \frac{v^2}{c^2}}.$$

The capacity-budget picture supplies the substrate interpretation of this Lorentzian kinematics: motion allocates part of the finite update budget to spatial transport, leaving the remaining fraction as proper-time evolution.

This argument fixes the continuum kinematics once a common invariant speed exists. It does not derive the spatial transfer coefficient from the renewal gap. The charged transfer fixes a rest-frame cadence \(\tau_*\), while a spatial matter operator must determine its own small-momentum coefficient and reproduce \(c\). The definition \(L_* = c\tau_*\) is therefore a causal-length conversion. It does not identify \(L_*\) with a bond length, a tetrahedron diameter, or a face area. Appendix H.11 gives an exact fixed-graph benchmark that separates these quantities.

The same capacity language also unifies motion-induced and gravity-induced clock slowing. In the nonlinear branch the surviving-capacity fraction is

$$q = \frac{S_{\text{ent}}}{S_\infty},$$

so smaller \(q\) means that less local update capacity remains available. Motion reduces the temporal share of the budget by consuming part of it in spatial transport; a nearby defect reduces the local budget by depleting available capacity. The two familiar time-dilation effects are therefore interpreted as two regimes of one mechanism.

The second ingredient is the relational character of the substrate. The network is not embedded in a prior physical manifold whose coordinate labels carry independent meaning; its physical content is the pattern of local capacities, defects, and neighborhood relations. Continuum coordinates are descriptive labels imposed on that relational structure, and smooth coordinate changes relabel the same underlying configuration. This is precisely why the low-energy theory must be written in generally covariant form.

The metric sector, then, is not introduced from outside. Lorentzian geometry is the natural coarse description of a finite-capacity, isotropic, relational substrate, and the Einstein sector is its lowest-order continuum gravitational expression, with the entanglement scalar tracking how localized defects redistribute the same capacity geometry. As with any discrete substrate, this continuum claim faces a sharp known obstacle: a discrete structure with a preferred rest frame feeds dimension-four Lorentz-violating operators into the infrared with order-unity coefficients through loops [35], against laboratory bounds many orders of magnitude below unity. The protection here is structural. The tetrahedral ensemble is combinatorial and pre-geometric: it lives in the state counting from which the continuum is constructed, defines no embedding lattice in the emergent spacetime, and imprints on the EFT only through the frame-independent scalars \(L_*\), \(g_{\text{share,eff}}\), and \(\eta_*\). Discreteness of this class is compatible with exact low-energy Lorentz symmetry, as causal-set sprinkling demonstrates by construction [36, 37]. The cosmological bath does select a frame, but only in the environmental sense the CMB does: a state rather than an operator, while the laboratory bounds constrain operators. The supporting calculation this argument calls for — that substrate loops generate no dimension-four Lorentz-violating operators — is still required and is listed in the closure table.

4.2 Dependency Map of the Theory

The logical flow begins with the three foundational postulates — Information–Geometry, Mass–Entropy, and Many-Pasts — with faithful full-support resolution applied to both states and histories, and runs through the static weak-field chain before reaching the conditional sectors:

$$\{\text{three postulates}\}$$ $$\to \text{finite-capacity substrate ontology} \to \text{tetrahedral boundary ensemble}$$ $$\to \text{faithful resolution of states and histories} \to \text{local replacement / history export}$$ $$\to \text{edge transport / loop dressing / source map}$$ $$\to \text{capacity form of the Einstein constraint} \to \text{Newton / baseline no-slip / GR PPN},$$ $$\text{with carrier projection + thermal contact + horizon matching}$$ $$\to \{a_0, \text{RAR, nonspherical fields, lensing}\} \text{ in the leading EFT}.$$

Only then come the conditional and frontier sectors — transport, clusters, cosmology, strong field, and the particle/gauge extensions — each developed as a consequence or completion of the same framework.

Two features of this map matter. First, it is a dependency graph, not an equality of closure status: the ordinary static branch is closed more tightly than the transverse, cosmological, or strong-field sectors, and Part VIII makes that difference explicit in a closure-status table. Second, Many-Pasts appears at the top of the map because the scale-setting chain uses it: faithful full-support resolution selects the local renewal process on the history space that Many-Pasts supplies. Their combination gives a reversible present/history exchange; neither the Born history measure nor record conditioning alone selects memorylessness.

Part II. UV Coefficient Chain

Part I fixed what the theory is about. The question now is whether the local capacity-sharing structure can actually be counted. If the substrate has finite local capacity, the coefficients that appear in the continuum weak-field theory should not be free continuum parameters; they should descend from a finite local boundary problem. The next five sections follow that problem through: the smallest boundary cell that can carry capacity and close isotropically, the weighting that selects well-closed configurations, the cost of neighboring cells disagreeing, the local returns that dress that cost, and the continuum coefficient they leave behind. The baseline calculation is internal to the microscopic construction. Appendix B separately propagates nearby discrete branches to the Newton scale, and Appendix L records that this comparison is postdictive rather than historically blind.

Several of the ultraviolet choices below may look at first like independent tunings: the tetrahedral cell, the seven labels, the injective assignment, the parity doubling, the admissibility kernel, the transverse export, and the electron anchor. None is a phenomenological knob, and none varies from galaxy to galaxy. Section 5 and Appendix B derive the finite ensemble and its entropy; Appendix C derives the edge projection; Appendices D and H derive the replacement process, the reversible history export, the factorized lightest branch, and the decorated marked-transfer action; Appendix L records the historical fork accounting. The remaining microscopic task is to embed that finite transfer vertex in a stable geometric continuum, through a specified GFT action or a capacity-decorated CDT transfer matrix on a critical trajectory.

5. Why a Tetrahedral Boundary Ensemble

The problem is to find a finite boundary cell that can carry channel entropy, close isotropically, and hand a scalar response to the continuum. In three spatial dimensions the minimal volumetric simplex is a tetrahedron. The construction uses five ingredients:

• a tetrahedral volumetric cell;
• half-integer primitive data on the two sides of each shared face;
• positive matching after the two sides are placed in one orientation;
• four distinguishable relational ports with single-copy channel capacity;
• binary cell orientation.

This package is not presented as the only possible ultraviolet completion. Postulate II assigns a half-integer primitive spin

$$j_0 = \frac{1}{2}, \frac{3}{2}, \frac{5}{2}, \ldots$$

to each side of a shared face. Before gluing, the pair spans

$$V_{j_0} \otimes V_{j_0} = \bigoplus_{J=0}^{2j_0} V_J.$$

After orientation transport, positive coherent matching is generated by a nonnegative mismatch operator whose unique null space is the maximal coupled sector. Its sharp limit is therefore

$$\mathcal{G}_{\text{sharp}} = P_{2j_0},$$

and the transmitted face alphabet has

$$|M| = \dim V_{2j_0} = 4j_0 + 1$$

states. Appendix B derives this operator on the full tensor product, gives its finite-width spectrum, and states the Hessian tests that would falsify the primitive matching rule.

The marked sector provides a second, more selective relation. The independently constructed present/history response is a product of two spatial vectors,

$$\mathcal{H}_{\text{mark}} = V_1^P \otimes V_1^H = V_0 \oplus V_1 \oplus V_2.$$

The complete nonmaximal information left by maximal fusion is

$$Q(j) = (V_j \otimes V_j) \ominus V_{2j} = \bigoplus_{J=0}^{2j-1} V_J.$$

Because both sums are multiplicity-free,

$$\mathcal{H}_{\text{mark}}(s) \cong Q(j) \iff j = s + \frac{1}{2}.$$

The closure response is a three-component spatial vector, so in \(d = 3\) it carries \(s = 1\). In the selected sharp reversible vertex, unitarity makes the complete marked carrier the retained nonmaximal fusion output, and therefore

$$j_0 = \frac{3}{2}, \quad V_{2j_0} = V_3, \quad |M| = 7.$$

This is a derivation within the selected sharp reversible incidence vertex, not a new minimality axiom. Unitarity fixes the retained output to the orthogonal fusion complement, and completeness of the nine-state H.9 carrier then makes the complement map an isometric identification. The result fixes the carrier space; the hard-core realization, flags, phases, and detailed routing remain selected structure of the displayed microscopic vertex. If that vertex is replaced, \(j_0 = \frac{3}{2}\) remains the branch selected by the discrete audit of Appendix B.5.

The four faces are relationally distinct ports. A displayed letter is the occupation of one mode in a single cell-level channel resource; the fermionic single-copy ceiling therefore forbids the same channel from being routed through two ports at once. This gives an injective assignment without antisymmetrizing away the port labels. The two global orientations remain distinct microscopic states. The resulting count is

$$\Omega_{\text{tet}} = 2P(7, 4) = 1680,$$

and the combinatorial sharing ceiling is

$$g_{\text{share,max}} = \ln(1680) = 7.42654907240.$$

The exact equality

$$16 = 7 + 9$$

has a direct meaning: in the sharp primitive pair map, seven states form the geometric link and the other nine can be retained by the marked record. Within the physical three-dimensional rotation algebra the conditional chain is

$$d = 3 \implies V_1^P \otimes V_1^H \implies j_0 = \frac{3}{2} \implies V_3 \implies 7.$$

It is not asserted as a theorem under dimensional continuation to arbitrary \(\text{Spin}(d)\).

The exact \(K^2\) spectrum and branch audits are given in Appendix B. The \(j\)-labeled tetrahedron used here coincides with the quantum tetrahedron of simplicial spin networks [84, 85], whose discrete geometric spectra [86] arise from the same \(SU(2)\) representation theory. The present construction differs in weighting these states by admissibility closure rather than by a spin-foam amplitude, and in routing them to a capacity entropy rather than to area and volume operators.

6. Admissibility Closure

6.1 Minimal isotropic kernel

Not every boundary configuration should count equally. The raw combinatorial ensemble is too permissive to be the complete ultraviolet input: some configurations sit close to the regular closure pattern expected of a smooth local cell, while others are badly distorted. Admissibility closure is the statement, in its mildest form, that more poorly closed configurations contribute less to the coarse ensemble. The minimal rotationally invariant measure of that distortion is a single quadratic closure-defect scalar \(K^2\), and the weighting it induces is

$$p_\eta(b) \propto e^{-\eta K^2(b)}.$$

Normalization, isotropy, and a fixed quadratic closure moment select this maximum-entropy kernel. Higher invariants such as \(K^4\) carry additional ultraviolet information and enter as subleading refinements. One distinction is fixed here and holds throughout: this capacity (admissibility) closure is soft, and its invariant \(K^2\) is strictly positive on the ensemble, while the exact Gauss closure of local gauge redundancy is a different operator on a different tensor factor. The two cannot be one operator — Appendix Q proves the impossibility and locates every quantity of the numerical spine on the gauge-inert factor.

6.2 Closure condition and uniqueness

The admissibility precision \(\eta\) is not chosen externally; it is fixed by maximizing the normalized closure evidence. Tetrahedral closure is the vanishing of the three-component oriented-face sum, so the closure-defect space is three-dimensional, and the quadratic family on it carries a determinant weight \(\eta^{3/2}\). The closure-evidence functional is therefore

$$\mathcal{F}(\eta) = \ln Z(\eta) + \frac{3}{2}\ln\eta,$$

and its stationary point gives the closure condition

$$\langle K^2\rangle_\eta = \frac{3}{2\eta},$$

in which the factor \(3/2\) is the determinant weight of the three independent closure components, while the discreteness and multiplicities of the spectrum stay inside the exact sum \(Z(\eta)\). This is the stationary normalized-evidence point of the exact closure spectrum, and it is a maximum rather than a bare root (Appendix B.2).

On that spectrum it is unique,

$$\eta_* = 0.0298668443935.$$

The closed branch is locally stiff: small fractional changes in \(\eta\) produce only small fractional changes in the downstream effective sharing entropy.

6.3 Effective sharing entropy

The admissibility-weighted effective sharing entropy is

$$g_{\text{share,eff}} = 7.41980002357.$$

The gap between \(g_{\text{share,max}}\) and \(g_{\text{share,eff}}\) is therefore not loss imposed by hand. It is the difference between the raw combinatorial ceiling and the admissibility-closed effective boundary entropy that actually propagates into observable couplings.

The continuum description does not inherit the naive channel-counting ceiling; it inherits the portion of the channel space that survives after closure is imposed. The downstream couplings should therefore be read as consequences of admissibility-closed sharing, not of raw combinatorics alone.

With \(\eta_*\) fixed, the effective sharing entropy carries no remaining freedom; the exact spectrum, multiplicities, and uniqueness proof are given in Appendix B.2.

7. Edge Kernel and Tree-Level Coupling

Admissibility determines the capacity of one cell. The edge kernel determines the cost when neighboring cells differ, and that cost becomes the continuum stiffness: stronger resistance to local disagreement makes capacity deficits spread less readily. The same ultraviolet closure data fix both quantities. The geometric bridge is the tetrahedral identity

$$\sum_{i=1}^4 \hat{n}_i\hat{n}_i^T = \frac{4}{3}I_3,$$

which implies a channel-averaged transverse fraction of \(2/3\) and gives the bare edge smoothness coupling

$$J_{\text{bare}} = \frac{2}{3}\eta_*.$$

If adjacent cells disagree strongly the edge pays a larger penalty; if they agree, the penalty is small. The factor \(2/3\) is the geometric fraction that survives after averaging the four tetrahedral channel directions into the isotropic continuum limit — the part of the disagreement that the scalar sharing channel actually carries.

For a \(z = 4\) regular coarse adjacency graph, the tree-to-lattice reduction then yields

$$J^{\text{tree}}_{\text{eff}} = \frac{J_{\text{bare}}}{3} = \frac{2\eta_*}{9}.$$

The division by 3 comes from the branching geometry of the rooted \(z = 4\) graph. One neighboring link points back toward the source, while the remaining \(z - 1 = 3\) links carry forward transport into the tree. The net long-range transport \(J^{\text{tree}}_{\text{eff}}\) is therefore the portion of the microscopic edge penalty that survives this local branching.

Origin of the horizon target. The horizon target

$$\sigma_* = \frac{\pi}{g_{\text{share,eff}}}$$

is the closure-consistency value required by the horizon-normalized field convention. In the admissibility-closed boundary ensemble, one active microscopic sharing unit carries effective entropy \(g_{\text{share,eff}}\). In the continuum normalization used for the weak-field scalar, the occupancy variable is normalized by

$$S = \pi Q_{\text{occ}},$$

so a coarse horizon-normalized channel with occupancy \(Q_{\text{occ}} = 1\) carries entropy \(\pi\) in the \(S\)-field convention. If \(\sigma_*\) denotes the asymptotic conditional-independence weight seen by the rooted shell hierarchy (Appendix B.3), consistency between the boundary entropy count and the horizon-normalized continuum field requires

$$\sigma_* g_{\text{share,eff}} = \pi,$$

and therefore

$$\sigma_* = \frac{\pi}{g_{\text{share,eff}}} = 0.42340665\ldots.$$

The factor \(\sigma_*\) matches the admissibility-closed microscopic entropy normalization to the horizon-normalized scalar-field convention. The rooted shell observable converges rapidly to this closure target, constraining the nonlocal correction at small shell depth. The four tetrahedral channel directions average to the isotropic tensor structure in the continuum limit, so the combinatorial data that fix admissibility also fix tree-level transport. Appendix C gives the shell hierarchy and phase-selection checks.

8. Finite-Loop Renormalization

Tree level is not the end of the ultraviolet chain. The full lattice admits local closed-return motifs that recycle part of the transmitted information before it contributes to net coarse transport. The leading correction is organized as a local Dyson self-energy dressing,

$$J^{(\text{ren})}_{\text{eff}} = \frac{J^{\text{tree}}_{\text{eff}}}{1 + J^{\text{tree}}_{\text{eff}}\Sigma_{\text{ret}}}.$$

A purely tree-like transmission rule would let the relevant amplitude move outward once and never locally return; a real coarse graph is not that simple. Some of the transmitted information cycles back through short closed motifs before contributing to long-distance transport. The renormalized coupling is therefore the true stiffness felt by the coarse field after these local returns have been resummed.

The structure of that self-energy is not a generic loop number. The returns split into seven sector-diagonal channels and one collective mode. The seven are the face-label channels, each returning independently without mixing. The one is the permutation-symmetric combination across channels, which returns as a shared closure-singlet rather than as a channel-specific loop, and it is weighted by the same transverse projection and branch-dilution factors that define the tree edge map,

$$\left(\frac{2}{3}\right)\left(\frac{1}{3}\right) = \frac{2}{9}.$$

The leading local self-energy is therefore

$$\Sigma_{\text{ret}} = 7 + \frac{2}{9} = \frac{65}{9}.$$

Equivalently, on the seven-channel scalar return space,

$$R_{\text{ret}} = I_7 + \frac{2}{9}P_{\text{sing}}, \quad P_{\text{sing}} = |u\rangle\langle u|, \quad u = \frac{1}{\sqrt{7}}(1, \ldots, 1),$$

with \(\Sigma_{\text{ret}} = \text{Tr}(R_{\text{ret}})\). The orthogonal six-dimensional sum-zero sector carries no net scalar charge in the coarse branch and so adds no separate scalar return. The singlet weight is fixed by the tree map, not introduced here, so no new loop parameter appears. Permutation symmetry fixes the existence of the singlet but not the placement of the suppression factors on it alone; that placement is the minimal-return-operator reading whose graph-level derivation Appendix C.3 records as the outstanding audit. The induced uncertainty is bounded: replacing \(\Sigma_{\text{ret}} = 65/9\) by 7, \(7 + 2/3\), or 8 shifts \(J^{(\text{ren})}_{\text{eff}}\) and \(\gamma\) by at most 0.5% and leaves \(G\), \(a_0\), and \(\kappa/\gamma\) exactly unchanged, since \(J^{(\text{ren})}_{\text{eff}}\) cancels in the source-to-stiffness ratio (Appendix C.5).

$$c^{(\text{ren})}_{\text{loop}} \equiv \frac{J^{(\text{ren})}_{\text{eff}}}{J^{\text{tree}}_{\text{eff}}} = \frac{1}{1 + J^{\text{tree}}_{\text{eff}}\Sigma_{\text{ret}}} \approx 0.95426,$$

and

$$J^{(\text{ren})}_{\text{eff}} \approx 0.00633348.$$

This reproduces the shell-target crossing near \(J_{\text{bare,cross}} \sim 0.019\) at the 0.05% level.

The finite renormalization is written as an explicit local self-energy. The remaining audit task is the independent graph-level derivation of the relative diagonal and singlet weights of the same scalar-return operator, not the introduction of any new loop parameter.

9. Continuum Stiffness and SI Normalization

The last UV step reads the lattice weighting as a quantum action rather than a thermal one: the lattice quadratic form is interpreted as a Euclidean action weight,

$$\frac{I_E}{\hbar} = \frac{J^{(\text{ren})}_{\text{eff}}}{2}\sum_{a,i}(Q_a - Q_{a+L_*\hat{n}_i})^2,$$

where the sum runs over one sublattice representative \(a\) of each bipartite primitive cell and its four outgoing bonds \(\hat{n}_i\), so each undirected nearest-neighbor edge is counted once (the convention of Appendix C.4). The microscopic four-cell is assigned the volume

$$\Delta V_4 = \frac{L_*^4}{c}$$

as a coarse-graining convention: the abstract tetrahedral cell complex has no space-filling regular-tetrahedron Euclidean embedding (Appendix C.5), so cell volumes and face areas enter as normalization conventions of the coarse map, not as geometry supplied by the graph. Up to this point the derivation has determined a dimensionless lattice weighting. The continuum EFT, however, needs a dimensionful coefficient multiplying derivatives of a field in spacetime. The Euclidean-action interpretation upgrades the lattice closure data into a continuum action density with the right units and the right covariant target.

The same tetrahedral identity used in the edge-kernel reduction then yields the continuum coefficient for the occupancy field \(Q_{\text{occ}}\),

$$\gamma_Q = \frac{4\hbar c}{3L_*^2}J^{(\text{ren})}_{\text{eff}}.$$

Here \(L_*\) is the canonical tetrahedral spacing, with one coarse cell carrying volume \(L_*^3\) up to the fixed cell-shape convention, and \(J^{(\text{ren})}_{\text{eff}}\) is the loop-dressed edge coupling. The numerical factor \(4/3\) is the isotropic projection

$$\sum_i \hat{n}_i\hat{n}_i^T = \frac{4}{3}I_3$$

that turns the tetrahedral edge directions into the continuum gradient tensor.

The field normalization is fixed by horizon capacity:

$$S = \pi Q_{\text{occ}}.$$

Therefore the canonical EFT coefficient in the \(\frac{\gamma}{2}(\partial S)^2\) convention is

$$\gamma = \frac{4\hbar c}{3\pi^2 L_*^2}J^{(\text{ren})}_{\text{eff}}.$$

Physically, \(\gamma\) is the continuum stiffness of the entanglement-capacity field. A larger \(\gamma\) makes spatial gradients more costly and suppresses the capacity-deficit response to a given source; a smaller \(\gamma\) allows larger variations of the field. The faithful sector-resolution principle fixes \(L_*\) without using \(G\). It is nevertheless useful to define the gravitational scale induced by this length,

$$G_* := \frac{c^3 L_*^2}{\hbar}.$$

Then the stiffness may be written in Einstein-normalized form as

$$\gamma = \frac{4J^{(\text{ren})}_{\text{eff}}}{3\pi^2}\frac{c^4}{G_*}.$$

This is the same algebra as the familiar Planck-cell rewrite, but read in the opposite direction: the substrate cell length induces the gravitational scale rather than being chosen by first inserting the measured value of \(G\). Within the Euclidean-action and cell-volume conventions stated above, the SI-normalized stiffness coefficient is fixed; because the absolute normalization of an isolated scalar functional is conventional, the invariant content of this step is the ratio \(\kappa/(\gamma S_\infty)\) that the weak-field matching of Section 11 consumes. In that ratio \(J^{(\text{ren})}_{\text{eff}}\) cancels (Appendix C.5), so the loop-dressed coupling carries no content for the Newton normalization; its nontrivial input enters the stiffness itself and the dynamical and galactic sectors.

This completes the micro-to-continuum coefficient chain. The tetrahedral ensemble determines the effective sharing entropy; the edge kernel and loop dressing turn that entropy into a discrete stiffness; and the Euclidean matching turns the discrete stiffness into the continuum coefficient \(\gamma\) of the weak-field EFT.

Closed UV-to-IR chain. The UV coefficient chain can now be summarized as

$$\{\Omega_{\text{tet}}, K^2, \eta_*, g_{\text{share,eff}}, L_*, J_{\text{bare}}, J^{\text{tree}}_{\text{eff}}, \Sigma_{\text{ret}}, J^{(\text{ren})}_{\text{eff}}, \gamma\} \longrightarrow \{\kappa, G, a_0, g_{\text{obs}}(g_{\text{bar}})\}.$$

The first bracket is the micro-to-continuum closure chain; the second collects the weak-field observables it feeds. All later weak-field coefficients come from this chain.

The ordinary branch uses the Euclidean action-kernel interpretation and the physical matching of Appendix C.5. Fuller inhomogeneous dynamics must reproduce both prescriptions.

Part III. The Closed Static Branch: Einstein Gravity in the Capacity Variable

10. Einstein Parent Action and the Reduced Capacity Frame

The ordinary longitudinal capacity branch has the ordinary metric action as its covariant parent; no independently varied capacity scalar is added to it:

$$I_0[g, \psi] = \frac{c^3}{16\pi G}\int_\mathcal{M} d^4x \sqrt{-g}(R - 2\Lambda) + I_{\text{GHY}}[g] + I_{\text{matter}}[g, \psi],$$

where covariant coordinates use \(x^0 = ct\). When the time integral is instead written in seconds, \(dx^0 = c\,dt\) and the ADM prefactor is correspondingly \(c^4/(16\pi G)\). Matter is coupled once, to one physical metric. The capacity functional is the static scalar-constraint reduction of \(I_0\), expressed in a different field coordinate. This statement can be proved without appealing to the final Poisson equation.

Static scalar reduction of Einstein–Hilbert gravity. Use Newtonian gauge,

$$ds^2 = -\left(1 + \frac{2\Phi}{c^2}\right)(dx^0)^2 + \left(1 - \frac{2\Psi}{c^2}\right)\delta_{ij}dx^i dx^j, \quad x^0 = ct,$$

and retain the static scalar sector through quadratic order. In ADM variables [11] the shift and extrinsic curvature vanish, so the Einstein–Hilbert plus Gibbons–Hawking–York action is

$$I^{\text{static}}_{\text{ADM}} = \frac{c^4}{16\pi G}\int dt\, d^3x\, N\sqrt{h}\, ^{(3)}R + I^{\text{static}}_{\text{matter}} + I_\infty.$$

Expanding \(N = 1 + \Phi/c^2\) and \(h_{ij} = (1 - 2\Psi/c^2)\delta_{ij}\), cancelling the reference boundary term at infinity, and using \(I^{(1)}_{\text{matter}} = -\int dt\, d^3x\, \rho\Phi\), gives

$$I^{(2)}_{0,\text{scal}}[\Phi, \Psi] = \int dt\, d^3x\left[\frac{1}{8\pi G}\left((\nabla\Psi)^2 - 2\nabla\Phi\cdot\nabla\Psi\right) - \rho\Phi\right].$$

The lapse perturbation remains a constraint variable. Its variation and the spatial-scalar variation give, respectively,

$$\nabla^2\Psi = 4\pi G\rho, \quad \nabla^2(\Phi - \Psi) = 0.$$

For the isolated solution with regular interior matching and no independent harmonic boundary data, asymptotic flatness gives \(\Phi = \Psi\). Eliminating \(\Psi\) therefore produces

$$\boxed{I_{\text{Newton}}[\Phi] = \int dt\, d^3x\left[-\frac{(\nabla\Phi)^2}{8\pi G} - \rho\Phi\right].}$$

Thus no independent scalar stress tensor is needed to create the linear potential, and the equality \(\Phi = \Psi\) in the baseline branch is a metric constraint equation rather than an anisotropic-stress assumption. Appendix N gives the expansion, boundary bookkeeping, and degree-of-freedom audit in full.

Exact reduced-action identity. On the renormalized static branch, write \(S_{\text{ent}} = S_\infty - \delta S\). Source-independent extensive terms are removed by the vacuum normalization proved for the regulated joint measure in Section 25. The capacity functional is then

$$I^{\text{static}}_{\text{cap}}[\delta S; \rho] = \int dt\, d^3x\left[-\frac{\gamma}{2}(\nabla\delta S)^2 + \kappa\rho\,\delta S\right].$$

The field redefinition

$$\delta S = -\frac{2S_\infty}{c^2}\Phi$$

turns it into

$$I^{\text{static}}_{\text{cap}} = \int dt\, d^3x\left[-\frac{2\gamma S_\infty^2}{c^4}(\nabla\Phi)^2 - \frac{2\kappa S_\infty}{c^2}\rho\Phi\right].$$

Using

$$G = \frac{c^2\kappa}{8\pi\gamma S_\infty}$$

gives the action-level equality

$$\boxed{I^{\text{static}}_{\text{cap}} = Z_S I_{\text{Newton}}, \quad Z_S \equiv \frac{2\kappa S_\infty}{c^2}.}$$

The field-independent factor \(Z_S\) cannot affect the classical reduced equations. It can matter when the UV construction is asked to normalize fluctuations or correlation functions, but it does not represent a second determination of \(G\) and it does not license adding \(I_{\text{cap}}\) to \(I_0\). The equality assumes the same asymptotically flat or Dirichlet boundary data on both sides. At a finite boundary the Newton surface term and its image under \(\delta S = -2S_\infty\Phi/c^2\) must be included as well.

Scope of the background-covariant notation. For transport calculations the same reduced equation is packaged as

$$I_{\text{cap}}[S_{\text{ent}}; \chi \mid g_{\text{ref}}] = \int d^4x \sqrt{-g_{\text{ref}}}\left[-\frac{\gamma}{2}g^{\mu\nu}_{\text{ref}}\partial_\mu S_{\text{ent}}\partial_\nu S_{\text{ent}} - \lambda S_{\text{ent}} - \kappa\chi S_{\text{ent}}\right].$$

The vertical bar is essential: \(g_{\text{ref}}\) and the reduced source projection \(\chi\) are held fixed while \(S_{\text{ent}}\) is varied. This notation is useful for extending the reduced response in time, but it is not a covariant scalar–tensor parent action. In the static nonrelativistic sector \(\chi \simeq \rho\); covariantly the source is the full stress tensor through \(I_{\text{matter}}[g, \psi]\).

Capacity coefficients and the source theorem. The UV calculation still fixes how the geometric constraint is coordinatized by the substrate variable. With

$$\sigma_{\text{def}} = \frac{\rho}{\kappa_m(L_*)},$$

the Green-matched projection is

$$\nabla^2\delta S = -\frac{3L_*}{4G_{\text{tet}}(0)}\sigma_{\text{def}}, \quad \frac{\kappa}{\gamma} = \frac{3L_*}{4G_{\text{tet}}(0)\kappa_m(L_*)}.$$

This is the microscopic map between defect density and the capacity coordinate on the Einstein constraint surface. It is not an extra matter coupling in the covariant parent theory.

The length backbone of Newton's constant. The weak-field normalization is

$$G = \frac{c^2\kappa}{8\pi\gamma S_\infty}.$$

Faithful full-support resolution and the positive marked-transfer spectrum fix

$$L_* = -\frac{3}{2}Z_e\lambda_e\ln\left(1 - e^{-7g_{\text{share,eff}}}\right), \quad \lambda_e = \frac{\hbar}{m_e c},$$

and therefore induces

$$G_* = \frac{c^3 L_*^2}{\hbar} = \frac{9}{4}\frac{\hbar c}{m_e^2}Z_e^2 \ln^2\left(1 - e^{-7g_{\text{share,eff}}}\right).$$

Substituting the source-map identities

$$\frac{\kappa}{\gamma} = \frac{3L_*}{4G_{\text{tet}}(0)\kappa_m(L_*)}, \quad \kappa_m(L_*) = \frac{\hbar}{cL_* \ln 2}, \quad S^{\text{cell}}_\infty = \frac{3\ln 2}{32\pi G_{\text{tet}}(0)},$$

into the weak-field expression gives identically

$$G = \frac{c^3 L_*^2}{\hbar} = G_*.$$

Thus the electron recurrence fixes \(L_*\) and the candidate scale \(G_* = c^3 L_*^2/\hbar\). Physical \(G = G_*\) follows only after the selected Einstein/horizon normalization \(S_\infty = S_0\) is imposed. The stiffness and source coefficient give a consistent static-EFT representation of that matched length backbone rather than an independent numerical determination. The comparison with the measured Newton constant is therefore a test of the electron-anchored candidate together with this physical matching.

The decorated scale gives \(G_* = 6.6742890813 \times 10^{-11}\ \text{m}^3\text{kg}^{-1}\text{s}^{-2}\), or \(-0.073\sigma\) relative to CODATA. Because both the early entropy construction and the later residual-closing vertex were developed with the discrepancy known, Appendix L treats this as a high-precision postdiction. The nontrivial content is the shared action that also fixes the two charged-lepton corrections.

11. Capacity Variable, Bridge Law, and Variational Status

Varying the reduced capacity-frame functional with respect to \(S_{\text{ent}}\) gives

$$\gamma\Box S_{\text{ent}} = \lambda + \kappa\chi.$$

On the renormalized static, nonrelativistic branch this becomes

$$\nabla^2\delta S = -\frac{\kappa}{\gamma}\rho.$$

Define the surviving fractional capacity

$$q(x) \equiv \frac{S_{\text{ent}}(x)}{S_\infty} = 1 - \frac{\delta S(x)}{S_\infty}.$$

The action reduction above fixes the weak-field bridge directly,

$$\frac{\Phi}{c^2} = -\frac{\delta S}{2S_\infty}.$$

Equivalently,

$$q = 1 + \frac{2\Phi}{c^2} + O(c^{-4}).$$

The bounded nonlinear rule

$$N^2 = q$$

is the continuous multiplicative completion selected by the capacity-composition rule. Its status must now be stated more precisely. In a static spherical exterior, \(q\) is the invariant geometric scalar

$$q = h^{ab}\partial_a R\partial_b R = 1 - \frac{2GM_{\text{MS}}}{c^2 R},$$

and in Schwarzschild coordinates it equals \(N^2\). In a generic spacetime the lapse is foliation dependent, so \(N^2 = q\) by itself is not a covariant constraint. The metric-only parent therefore treats the weak-field \(\delta S\) and the spherical \(q\) as reduced or composite geometric variables; it does not promote either to an unconstrained second gravitational field. Appendix N.6 formulates the remaining target as a diffeomorphism-invariant, potentially quasilocal and state-dependent functional, rather than presuming that an additional fundamental scalar is needed.

Combining the static source equation with the weak-field bridge gives

$$G = \frac{c^2\kappa}{8\pi\gamma S_\infty}.$$

This relation uses only the invariant combination \(\kappa/(\gamma S_\infty)\): a fixed-epoch rescaling of the entropy units changes \(S_\infty\) and \(\kappa\) together and leaves the observable potential unchanged.

Two coordinates on one response. The metric parent solves the Hamiltonian and spatial constraints for \(\Phi\) and \(\Psi\); the capacity frame uses \(\delta S\) as a field coordinate on that reduced solution. Around a constant background, a canonical scalar stress would begin as \((\partial\delta S)^2 = O(\rho^2)\) and could not be the source of the observed \(O(\rho)\) potential. The action identity removes that mismatch: the linear capacity response is the reduced metric constraint itself.

Matter enters once. For the nonrelativistic static branch, the microscopic source theorem reduces the full metric source to the defect density \(\rho\). Covariantly, matter enters only through \(I_{\text{matter}}[g, \psi]\), so the full stress tensor gravitates, including trace-free radiation, and the Bianchi identity enforces the usual conservation law. The notation \(\chi \simeq \rho\) belongs only to the reduced nonrelativistic source map; an explicit universal term \(S_{\text{ent}}T^\mu_\mu\) is neither required nor adopted.

Parent-action decision. The preferred minimal construction for the ordinary branch is therefore metric-only:

$$I^{\text{long}}_{\text{parent}} = I_0[g, \psi], \quad \delta S = \delta S[g, \psi] \text{ after constraint reduction}.$$

It propagates the two tensor polarizations of general relativity and no extra scalar. Constrained-clock and scalar–tensor alternatives remain useful control cases, but both add structure and generically add a mode; Appendix N records why neither is selected. The galactic excess belongs to the separate carrier-resolved contact of Sections 15–16; its leading EFT is explicit, while its microscopic metric vertex remains conditional.

12. Newtonian Gravity and the Point-Source Limit

In the renormalized static weak-field sector the scalar equation reduces to

$$\nabla^2\delta S = -\frac{\kappa}{\gamma}\rho.$$

After the background is renormalized away and the source is taken to be nonrelativistic, the deficit field obeys an ordinary Poisson equation. Its mathematical structure is the one used in standard weak-field gravity, with \(\delta S\) as the field variable.

For a point source \(M\),

$$\delta S(r) = \frac{\kappa M}{4\pi\gamma r}.$$

Using the bridge law,

$$\frac{\Phi}{c^2} = -\frac{\delta S}{2S_\infty},$$

the gravitational acceleration becomes

$$g(r) = \frac{c^2\kappa}{8\pi\gamma S_\infty}\frac{M}{r^2} = \frac{GM}{r^2}.$$

Thus Newtonian gravity is recovered as the weak-field response of the entanglement-capacity medium: the sourced scalar equation and the bridge law together imply the familiar point-mass force law, with nothing further assumed.

Interpretation. A point defect produces a \(1/r\) capacity deficit, and the bridge maps its gradient to the Newtonian inverse-square force. Ordinary gravity is the small-deficit, weak-curvature limit of the extended capacity strain around localized defects. Section 24 exhibits the same structure on the discrete substrate: cell-by-cell re-equilibration screens at sub-cell range, whereas a conserved capacity current with maintenance sinks obeys the massless graph-Poisson equation and produces the \(1/r\) deficit. The Newtonian form therefore diagnoses the conservation law behind it.

13. Electron Anchor: One-Bit Mass and Seven-Sector Length

The electron supplies the elementary mass anchor \(m_e/\ln 2\). Its Compton scale also assigns the length associated with the seven-sector effective support. The decorated marked-transfer action closes the finite correction to that scale and routes the same correction through the heavier charged-lepton shells. These are two readings of one dimensionful datum, not independent measurements, so the weak-field normalization remains a single calibration.

13.1 Why the electron is the anchor

The mass–entropy map needs an elementary anchor because the elementary matter sector is the localized defect sector. The electron is the lightest simple charged fermionic defect, and its mass is not obscured by hadronic or QCD dressing. A single fermionic face-exclusion defect carries the canonical increment

$$\Delta S_f = \ln 2,$$

one bit of missing entanglement, because an excluded face is a binary occupied/unoccupied defect of the local network.

13.2 One-bit mass anchor

At the electron Compton scale \(\ell = \lambda_e\) the mass–entropy map reads

$$\kappa_m(\lambda_e) = \frac{m_e}{\ln 2}.$$

Dividing the electron mass by the fixed one-bit increment fixes the mass-per-entropy conversion at the electron's own scale. Run back to the cutoff cell, the conversion is

$$\kappa_{m,\text{UV}} = \frac{\hbar}{cL_*}\frac{1}{\ln 2},$$

with canonical running law

$$\kappa_m(\ell) = \kappa_{m,\text{UV}}\left(\frac{L_*}{\ell}\right)^{1+\alpha_{\text{cl}}}, \quad \alpha_{\text{cl}} = 0$$

in the closed branch. One bit fixes the electron-scale conversion; the running law carries it to the UV scale; the same conversion then feeds the weak-field source map. This is the only point at which the mass anchor enters gravity.

13.3 Seven-sector length anchor

The same electron anchors the cell length through the marked support-to-rate map. Faithful full-support resolution uniquely selects the memoryless kernel. Within the stated recurrence mass functional, fermionic exclusion and electron lightness jointly select the maximum channel count \(k = 7\) and vanishing inter-channel correlation \(\Delta_7 = 0\). On the commutative renewed-state algebra, the state-weighted determinant of the likelihood multiplication operator is

$$\Delta_{\tau_p}(R) = \exp[\tau_p(\ln R)] = e^{-g_{\text{share,eff}}}.$$

Multiplicativity across the seven factorized clouds gives

$$r = \Delta_{\tau_p}^{\otimes 7}(R^{\otimes 7}) = e^{-7g_{\text{share,eff}}}.$$

This is the record-conditioned geometric transfer rate, not the collision probability of two independently sampled blocks. The positive survival spectrum gives the baseline scale

$$L_*^{(0)} = -\frac{3}{2}\lambda_e\ln(1 - r),$$

with \(3/2\) the transverse export factor of Appendix C.5.

Appendix H realizes this determinant transfer in a finite charged action. The construction separates the baseline recurrence from the defect-bound closure response, so the one-channel mixing projector, seven-channel survival operator, and marked-fiber determinant remain distinct. The finite occupation bound in H.2 excludes fewer-channel competitors in the specified positive-generator class; H.5 establishes the unique product minimum over seven-channel readout laws at fixed marginals and marked vertex.

This does not mean an electron is a single tetrahedral cell carrying seven simultaneous labels. The local ensemble supplies seven distinguishable dressing layers. Each layer has fermionic occupation at most one, and the recurrence mass minimum occupies all seven once. The electron is the lightest coherent one-bit defect on that selected branch and exports the transverse share of the resulting support. The Compton scale calibrates the substrate length hierarchy; the one-bit mass calibrates the source map. The two uses impose a nontrivial joint requirement on the electron's role in the gravitational normalization without duplicating a single input.

13.4 Decorated marked-transfer vertex

The original Newton normalization was low by about 1.05%. Because \(G_* \propto L_*^2\), the missing amplitude in the substrate length was 0.5309%. The baseline muon ratio required a 0.5124% uplift. At the PDG tau mass the tau ratio required 0.6601%, which separates into the same universal uplift and a smaller 0.1470% second-shell factor. Their common sign and scale motivated one small charged response with shell-dependent routing. Appendix L records that this was an action-level postdiction.

That response also had to be additive. The tetrahedral ensemble already fixed the entropy, closure spectrum, edge projection, source map, and the micro-to-macro coefficient chain. Changing its label count, admissibility weight, or transverse export to repair the residuals would move results that did not share the discrepancy. The admissible repair was consequently required to vanish in the unmarked vacuum, reuse the established closure incidence, and act only on the charged transfer graph. This criterion motivates the marked vertex below; it does not determine the answer numerically. The field content, determinant power, and routing still have to follow from the displayed action and survive the alternative-kernel audits.

The renewed closure amplitude has the exact Gaussian representation

$$e^{-\eta_* C^2/2} = \int\frac{d^3\xi}{(2\pi)^{3/2}}\exp\left[-\frac{1}{2}\xi^2 + i\sqrt{\eta_*}\,\xi_a C_a\right].$$

The amplitude-level closure incidence is therefore \(\sqrt{\eta_*}\). For each occupied unordered channel pair \(e = (m, m')\), the two directed scalar returns have row operator

$$R_e = \frac{2}{7}\left(\langle m\to m'| + \langle m'\to m|\right).$$

The contraction \(B_e = \sqrt{\eta_*}R_e\) obeys

$$B_e B^\dagger_e = 2\eta_*\left(\frac{2}{7}\right)^2 = \frac{8\eta_*}{49} \equiv u.$$

Its canonical unitary dilation has no-event amplitude \(\sqrt{1 - u}\). The seven-channel lightest branch activates all \(\binom{7}{2} = 21\) pair records, giving

$$Z_{\text{edge}} = \left(1 - \frac{8\eta_*}{49}\right)^{21/2}.$$

The present and history Gaussian strands each have three normalized first excitations. Their ordered products form nine orthonormal marked states, so the internal trace of one hard-core marked fiber gives

$$\boxed{\zeta_* = 9e^{-g_{\text{share,eff}}}\left(1 - \frac{8\eta_*}{49}\right)^{21/2}} = 0.005123584484947.$$

This internal trace computes a scalar vertex weight at fixed path branch; it is not a measurement that discards the defect's position label. Free transport applies the same internal contraction on every position branch, as required by Appendix H.11. The finite transfer graph has a universal marked alternative, one two-vertex label return, and one second-shell singlet passage. Its exact factors are

$$Z_\mu = 1 + \zeta_*, \quad Z_e = (1 + \zeta_*)(1 + 7\zeta_*^2), \quad Z_{\tau,2} = 1 + \frac{2}{7}\zeta_*.$$

With the additive-generator prescription, the dressed scale is

$$\boxed{L_* = Z_e L_*^{(0)}.}$$

Appendix H writes the controlled vertex, decomposes its complete edge Hessian, and enumerates all 56,800 states of the one-step routing block. H.9 also constructs a finite recurrence contraction with \(E^{\text{evt}}_e = -(3\hbar/2\tau_*)\ln(1 - rZ_e)\); its relative energy difference is \(7.37 \times 10^{-26}\). The additive-generator prescription supplies the exact formulas used here.

The anchor is a gauge choice. Because counting fixes only dimensionless quantities, exactly one dimensionful measurement must be supplied, and which one is a convention. The physical content of the framework is carried by anchor-invariant statements: the lepton ratios \(m_\mu/m_e\) and \(m_\tau/m_e\); the hierarchy

$$\frac{m_e}{m_P} = -\frac{3}{2}Z_e\ln(1 - e^{-7g_{\text{share,eff}}});$$

the horizon normalization identity \(1/4\) within the stated cell convention; the abundance ratio \(\Omega_c/\Omega_b\); and, within its branch assignment, \(a_0/cH_0\). The electron remains the anchor because its mass is measured most precisely. Appendix L records that the high-precision marked correction was constructed after the residuals were known, so anchor invariance does not turn the agreement into a blind prediction.

13.5 Consistency checks

The marked action uses no continuously fitted coefficient. Substitution gives

$$G_* = 6.6742890813 \times 10^{-11}\ \text{m}^3\text{kg}^{-1}\text{s}^{-2}\quad (-0.073\sigma),$$ $$\frac{m_\mu}{m_e} = 720\frac{2}{7}Z_\mu = 206.768280237\quad (-0.535\sigma),$$ $$\frac{m_\tau}{m_e} = 720^2\left(\frac{2}{7}\right)^4 Z_\mu Z_{\tau,2} = 3477.343310\quad (-0.125\sigma).$$

These three comparisons probe one marked-fiber weight through different graph polynomials. The scalar stiffness, source map, and weak-field bridge reproduce the same dressed \(G_*\) after substitution, so they remain one consistency chain rather than an independent determination.

Routing-integer audit. The displayed action makes the coefficient of \(\zeta_*^2\) in \(Z_e\) equal to the number of persistent charged labels: the second marked return sums over the alphabet, so the factor is \(1 + 7\zeta_*^2\) (Appendix H.9). Holding \(\zeta_*\) fixed and temporarily treating that integer as unknown, each unit step in \(n\) moves the induced \(G_*\) by 2.34 CODATA standard deviations: \(n = 6\) gives \(-2.41\sigma\), \(n = 7\) gives \(-0.073\sigma\), and \(n = 8\) gives \(+2.26\sigma\). Within \(-200 \leq n \leq 200\), seven is the only integer within two standard deviations. Dropping the second-order term (\(n = 0\)) gives \(-16.4\sigma\). Appendix B now supplies two readings of this result. In the selected sharp reversible vertex, the carrier-incidence theorem and the three-dimensional closure vector fix the same seven internally, and the Newton comparison checks its routing. For a different microscopic vertex, the comparison selects the seven-state member of the stated discrete family. Neither reading is historically blind: the target and the residual were already known, as Appendix L records.

Two remarks keep this honest. The resolvability is partly fortunate: \(\zeta_*^2 \simeq 2.6 \times 10^{-5}\) places the rung spacing just above the CODATA uncertainty, so the integer is measurable at all; a smaller \(\zeta_*\) would bury the ladder inside the error bar, and a larger one would leave \(n = 7\) one near-miss among many. And since the alphabet is symmetric about zero, one might expect the neutral label \(m = 0\) to drop out of a return sum, giving coefficient six; \(n = 6\) lies outside the band, so the neutral label demonstrably participates in the second marked return. With the current PDG tau mass, \(2/7\) is the only member of the forty-three distinct rationals in \([0, 1]\) with denominator at most eleven that lies within one experimental standard deviation. Its residual is \(-0.125\sigma\); \(3/11\) gives \(-1.436\sigma\). This finite candidate test does not assign probabilities to microscopic models.

The same comparison has a dimensionless form. Squaring the hierarchy gives the electron's gravitational coupling,

$$\frac{G_* m_e^2}{\hbar c} = \frac{9}{4}Z_e^2 \ln^2\left(1 - e^{-7g_{\text{share,eff}}}\right),$$

with \(9/4\) the square of the transverse export factor and \(Z_e^2\) the marked-transfer dressing. The relation also has an area reading. A horizon stores one bit in area \(4\ln 2\, L_P^2\), while the electron spreads its single bit over the Compton area \(\lambda_e^2\). The marked factor changes the finite packing correction without altering the dominant hierarchy \(e^{-14g_{\text{share,eff}}}\).

13.6 Composite sectors

For composite hadrons the claim is weaker and different in kind. The relevant quantity is the dressed, vacuum-subtracted bound-state entropy,

$$m_{\text{hadron}} = \kappa_m(\ell_H)S^{\text{dressed}}_{\text{ent},H},$$

with the dressed budget generated by confinement, gluonic structure, trace-anomaly dynamics, and chiral vacuum reorganization. A lattice derivation of that dressed entropy is not yet available. The present claim is limited to structural compatibility between the mass–entropy map and the standard QCD mass budget. The elementary-fermion anchor is settled in the simple sectors; the hadronic coefficients remain open.

14. Baseline Metric Closure: No Slip and PPN

For the branch that reproduces Newtonian gravity, the physical metric is already fixed by the Einstein parent action. The scalar constraint reduction of Section 10 gives

$$\nabla^2(\Phi - \Psi) = 0,$$

so asymptotic flatness implies

$$\Phi = \Psi.$$

This is an action-level result. It is not inferred from the canonical stress tensor of \(\delta S\), because no independently gravitating capacity scalar is present. Since the parent action of this branch is exactly Einstein–Hilbert plus minimally coupled matter, its vacuum post-Newtonian solution has

$$\gamma_{\text{PPN}} = \beta_{\text{PPN}} = 1$$

and the remaining standard PPN parameters vanish, subject to the usual assumptions on the matter sector and boundary conditions. The capacity redefinition leaves these values unchanged. The carrier rule introduced below also removes localized Solar-System sources from the transverse influence functional. The leading Solar-System metric is therefore the Einstein solution in the specified carrier-resolved EFT. A nonlinear covariant audit of the record projector and the cosmic-state vector remains required for time-dependent preferred-frame terms.

Part IV. The Carrier-Resolved Galactic Branch

15. Carrier-Resolved Galactic Dynamics

The Einstein branch responds to the full stress tensor and supplies ordinary gravity at every scale. The galactic excess is a second, coarse-grained response associated with retained source records. Its leading effective completion consists of a carrier projector, a thermal contact law, and a conservative auxiliary-field action. These ingredients fix the static equations without introducing an independently coupled matter charge. Their microscopic origin in the GFT state and metric vertex remains conditional.

Carrier nesting. An active carrier \(P\) is a retained, dynamically coherent source record at the resolution relevant to the long-range response. Let \(\chi_P\) be its support and \(E_P\) a conditional expectation on that support. It preserves support, is idempotent, and preserves the integrated source. Define

$$\boxed{\Pi_P X = E_P(\chi_P X).}$$

The active set \(\mathfrak{A}\) contains disjoint carriers covering the modeled source; a parent and descendant are never both active. Therefore

$$\Pi_P^2 = \Pi_P, \quad \Pi_P \Pi_Q = 0 \quad (P \neq Q,\ P, Q \in \mathfrak{A}).$$

For nested record algebras on the same parent support, \(E_P E_A = E_P\) when the parent algebra is contained in the finer algebra. A spatially restricted child operator instead obeys source assembly:

$$\Pi_P X = E_P\left(\sum_{A\prec P}\chi_A X\right), \quad \sum_{A\prec P}\chi_A = \chi_P.$$

This counts all descendants once before applying the parent response.

The transverse functional consequently has the carrier sum

$$\Gamma^{\text{nest}}_\perp = \sum_{P\in\mathfrak{A}}w_P\Gamma_{\perp,w_P}[g_\pm; T^\mathfrak{A}_\pm, \chi_P, \mathcal{R}_P], \quad T^\mathfrak{A} = \sum_{Q\in\mathfrak{A}}\Pi_Q T.$$

Each carrier's share of the response is evaluated on the joint coarse source \(T^\mathfrak{A}\) and weighted by its record support; for a single active carrier \(T^\mathfrak{A} = \Pi_P T\). The Einstein action continues to use the unprojected \(T_{\mu\nu}\). A star contributes once to its galaxy's coarse transverse source and retains its resolved stellar field in the Einstein branch. It does not also carry an independent transverse halo. This source-space nesting removes parent–descendant cross-terms before the nonlinear response is evaluated. The joint source makes the forces between active carriers reciprocal. A response computed from each carrier's own field alone would not be: the excess force of \(P\) on \(Q\) scales as \(M_Q\sqrt{M_P}\) and that of \(Q\) on \(P\) as \(M_P\sqrt{M_Q}\), and a Milky-Way–LMC pair would self-accelerate at about \(10^2\ \text{km s}^{-1}\ \text{Gyr}^{-1}\). On the joint source the excess forces cancel identically (paragraph "Several active carriers" below).

Thermal contact and the acceleration scale. The transverse doublet has two compact phase angles, with Haar volume \((2\pi)^2\) for one phase cell. Loading one sharing entropy into that cell and coupling it reversibly to the apparent-horizon state gives

$$a_0 = \frac{g_{\text{share,eff}}}{4\pi^2}cH_0.$$

The phase-cell loading and reversible horizon coupling are physical matching premises. With the Planck value of \(H_0\), they give \(a_0 = 1.231 \times 10^{-10}\ \text{m s}^{-2}\), compared with the fitted RAR scale \(1.20 \times 10^{-10}\ \text{m s}^{-2}\) [3, 73]. The same premise gives \(a_0(z) \propto H(z)\); that extrapolation is tested separately in Section 27.3.

For an active carrier \(P\), the effective occupation cell is specified by

$$H_{\text{cell},P} = k_B T_H\, x_P : N_P :, \quad x_P = \sqrt{\frac{|\nabla\Phi_{b,P}|}{a_0}}.$$

Its equilibrium free energy and response are

$$F_{\text{th},P} = k_B T_H\ln(1 - e^{-x_P}), \quad \frac{\partial F_{\text{th},P}}{\partial E_P} = n_B(x_P) = \frac{1}{e^{x_P} - 1}.$$

The dimensionless matching \(E_P/(k_B T_H) = x_P\) is part of the leading EFT and awaits a microscopic derivation. The rank-one capacity Hessian of Appendix N remains a thermodynamic consistency condition for one capacity variable. Its massless tangent and massive eigenvalue \(A_L + A_T\) do not supply the thermal energy or an additional propagating exchange mode.

Conservative static action. Define the carrier density and its Newtonian auxiliary potential by

$$\rho_P = \frac{u_\mu u_\nu}{c^2}\Pi_P T^{\mu\nu}, \quad \nabla^2\Phi_{b,P} = 4\pi G\rho_P,$$

and set

$$z_P = \frac{|\nabla\Phi_{b,P}|^2}{a_0^2}, \quad x_P = z_P^{1/4}.$$

The required excess function is

$$\mathcal{Q}_\perp(z) = \int_0^z\frac{ds}{e^{s^{1/4}} - 1}, \quad \mathcal{Q}'_\perp(z) = n_B(z^{1/4}),$$

so the total auxiliary function \(\mathcal{Q}(z) = z + \mathcal{Q}_\perp(z)\) satisfies

$$\mathcal{Q}'(z) = 1 + n_B(z^{1/4}) = \frac{1}{1 - e^{-z^{1/4}}}.$$

For each active carrier, introduce a total carrier potential \(\Phi_P\) and the QUMOND-type action [22]

$$I_{Q,P} = -\int dt\, d^3x\left[\frac{1}{8\pi G}\left(2\nabla\Phi_P\cdot\nabla\Phi_{b,P} - a_0^2\mathcal{Q}(z_P)\right) + \rho_P\Phi_P\right].$$

The functional added to the Einstein branch is the excess \(I_{\perp,P} = I_{Q,P} - I_{N,P}\), where \(I_{N,P}\) is the same auxiliary action with \(\mathcal{Q}(z) = z\). Variation gives

$$\nabla^2\Phi_P = \nabla\cdot\left(\mathcal{Q}'(z_P)\nabla\Phi_{b,P}\right).$$

Writing \(\phi_{\perp,P} = \Phi_P - \Phi_{b,P}\) isolates the transverse excess:

$$\boxed{\nabla^2\phi_{\perp,P} = \nabla\cdot[n_B(x_P)\nabla\Phi_{b,P}].}$$

This equation is conservative and applies to nonspherical sources. When active carriers overlap, \(\Phi_{b,P}\) in the flux is replaced by the joint potential \(\Psi = \sum_{Q\in\mathfrak{A}}\Phi_{b,Q}\) and carrier \(P\)'s flux is weighted by \(\chi_P\); an isolated carrier is unchanged. For spherical systems it reduces to

$$g_{\text{obs}} = g_{\text{bar}}\left[1 + n_B\left(\sqrt{g_{\text{bar}}/a_0}\right)\right] = \frac{g_{\text{bar}}}{1 - \exp[-\sqrt{g_{\text{bar}}/a_0}]}.$$

The high-acceleration limit is Newtonian, and the deep limit gives \(g_{\text{obs}} \simeq \sqrt{a_0 g_{\text{bar}}}\) and \(v^4 \simeq GM_b a_0\) [71, 72]. Disks and other nonspherical sources generally carry geometry-dependent curl corrections. Their predictions require direct Poisson solves using resolved baryonic maps; the pointwise spherical formula is not imposed on each pixel.

Direction-stiffness completion for resolved disks. The preceding local contact slaves the transverse frame to the source direction \(e_{L,P} = \nabla\Phi_{b,P}/|\nabla\Phi_{b,P}|\). At a thin-disk midplane that direction reverses across roughly two scale heights, and the local divergence acquires a narrow phantom sheet with surface density of order \(n_B\Sigma_b\). The sheet multiplies the vertical self-gravity of the disk by \(1+n_B\), with \(n_B \simeq 0.5\) at the solar radius, and produces the failed local Milky-Way benchmark below.

The selected resummation promotes the source direction to a nematic unit director \(\hat{e}_P \equiv -\hat{e}_P\) and defines

$$\tilde{z}_P = \frac{(\hat{e}_P\cdot\nabla\Phi_{b,P})^2}{a_0^2}, \quad \tilde{x}_P = \tilde{z}_P^{1/4}, \quad \mathbf{J}_{\perp,P} = n_B(\tilde{x}_P)(\hat{e}_P\cdot\nabla\Phi_{b,P})\hat{e}_P,$$

with

$$\boxed{\nabla^2\phi_{\perp,P} = \nabla\cdot\mathbf{J}_{\perp,P}.}$$

The tetrahedral kinetic term contains \(Z_0 X^2(\partial\hat{e})^2\) (Appendix N.8). Identifying its longitudinal amplitude with the carrier record potential \(X_P\) gives the stiffness \(Z_{\theta,P} = 2c X_P^2\). The record potential is the potential of the carrier's response stock, \(X_P \leftrightarrow \Phi_{R,P}\) (Appendix O.3), which vanishes at the carrier vacuum. For an isolated carrier in equilibrium \(\Phi_{R,P}\) is a fixed functional of \(\Phi_{b,P}\). The baryonic potential then serves as a proxy, \(Z_{\theta,P} = 2c\Phi_{b,P}^2\), with the same stiff-limit solutions. Overlapping carriers require the stock potential itself (paragraph "Several active carriers" below). The tetrahedral term does not fix the normalization \(c\). The Milky Way and satellite dispersions place it near \(c \simeq 0.03\) for the stock amplitude, and the resolved solutions are insensitive to it within that range. At constant stiffness the one-constant density \(|\partial\hat{e}|^2 \equiv \partial_i\hat{e}^a\partial_i\hat{e}^a\) cannot be distinguished from \(|\partial\hat{e}|^2 + \kappa\, s[\hat{e}]\), where

$$s[\hat{e}] = \partial_i\hat{e}^a\partial_a\hat{e}^i - (\nabla\cdot\hat{e})^2 = \partial_i\left[(\hat{e}\cdot\nabla)\hat{e}^i - \hat{e}^i\nabla\cdot\hat{e}\right].$$

Once \(Z_{\theta,P}\) follows the record potential, the \(\kappa\) term integrates to the connection cross term \(-\frac{1}{2}\kappa\,\partial_i Z_{\theta,P}[(\hat{e}\cdot\nabla)\hat{e}^i - \hat{e}^i\nabla\cdot\hat{e}]\), one of the completion data named in Appendix N.8. Requiring the spherical law to hold exactly fixes \(\kappa\). The radial field has \(|\partial\hat{r}|^2 = 2/r^2\) and \(s[\hat{r}] = -2/r^2\), so \(|\partial\hat{e}|^2 + \kappa\, s[\hat{e}]\) vanishes on every radial director only for \(\kappa = 1\):

$$\mathcal{S}[\hat{e}] = |\partial\hat{e}|^2 + s[\hat{e}] = (\lambda_1 - \lambda_2)^2 + |(\hat{e}\cdot\nabla)\hat{e}|^2 \geq 0,$$

where \(\lambda_{1,2}\) are the principal splays of the director congruence. The density penalizes anisotropic splay and bend and vanishes for radial and uniform directors. Because \(s\) has zero principal symbol, the director equation keeps the ellipticity of the one-constant term.

Expanding \(-a_0^2\mathcal{Q}_\perp(\tilde{z}_P)\) to second order in the tilt \(\delta = \alpha - \alpha_L\) from the source direction gives the restoring density \(n_B|\nabla\Phi_{b,P}|^2\delta^2\). Against the stiffness \(c X_P^2|\nabla\delta|^2\) this sets the coherence length

$$\boxed{\xi^2_{\theta,P} = \frac{c X_P^2}{n_B(\tilde{x}_P)|\nabla\Phi_{b,P}|^2}.}$$

For the Milky Way \(\Phi_R/\Phi_b \simeq 2.8\) at the solar radius. The stock amplitude at \(c = 0.03\) therefore gives \(\xi_\theta(R_0, 0) \simeq 6\ \text{kpc}\), twenty times the stellar scale height, and the baryonic proxy at \(c = 1\) gives 12 kpc. Either way the director cannot follow the source reversal across the disk thickness. Spherical sources carry the radial director, \(\mathcal{S}\) vanishes, and the local law holds exactly. In a disk the near-radial director passes through the midplane without reversal. Near \(z = 0\), \(\hat{e} \simeq \hat{R} + (z/R)\hat{z}\) and \(\partial_z J_{\perp,z} \simeq n_B g_R/R\), where the local contact has \(\partial_z(n_B\partial_z\Phi_b) \simeq 4\pi G n_B\rho_b\). The ratio \(g_R/(4\pi G\rho_b R)\) is 0.03–0.1 in stellar disks, so the excess is a smooth, nearly round component without a phantom sheet. An isotropic scalar smoothing kernel wide enough to remove the Milky-Way sheet shifts dwarf and inner-disk SPARC accelerations by 0.03–0.10 dex, which the RAR data exclude. One auxiliary action generates the Newtonian Poisson equation, the excess flux, the director equation, and a stiffness source \(c X_P\mathcal{S}[\hat{e}_P]\) in the excess equation (Appendix N.8). The source vanishes for every spherical carrier. The director completion was formulated after the failure of the local-contact Milky-Way vertical benchmark was known. The comparisons of Section 16 are therefore development-set postdictions unless they are identified as subsequently frozen tests; the old-disk dispersions and excess shapes registered in Section 27.1 are such tests.

Several active carriers. Let the active set contain carriers whose fields overlap, such as a galaxy and a satellite that keeps an independent record. With \(\Psi = \sum_P\Phi_{b,P}\) and the local contact \(\mathbf{J} = n_B\left(\sqrt{|\nabla\Psi|/a_0}\right)\nabla\Psi\), two integrations by parts with \(\rho_i = \nabla^2\Phi_{b,i}/4\pi G\) give the excess force on carrier \(i\),

$$F^\perp_i = -\int\rho_i\,\partial_z\phi_\perp\, d^3x = -\frac{1}{4\pi G}\int\mathbf{J}\cdot\nabla\partial_z\Phi_{b,i}\, d^3x.$$

Summed over carriers the integrand becomes \(n_B\,\partial_z\frac{1}{2}|\nabla\Psi|^2\), a total derivative, so

$$\boxed{\sum_{i\in\mathfrak{A}}F^\perp_i = 0.}$$

For a Milky-Way-like Plummer carrier (\(6\times10^{10}M_\odot\), \(b = 3\ \text{kpc}\)) and an LMC-like one (\(3\times10^9M_\odot\), \(b = 1\ \text{kpc}\)) 50 kpc apart, the joint source gives excess forces \(\pm1.471\times10^{12}\, M_\odot(\text{km s}^{-1})^2\text{kpc}^{-1}\) with zero net. Per-carrier responses give \(+8.14\) and \(-1.72\times10^{12}\), a centre-of-mass self-acceleration of \(10^4\ \text{km s}^{-1}\ \text{Gyr}^{-1}\). An independent implementation reproduced both results.

The director completion carries over with one director per carrier, a record support \(\chi_P\) (\(\sum_P\chi_P = 1\)), and the joint field in \(\tilde{z}_P\). Translating each carrier with its support, stiffness amplitude, and director core leaves the static energy unchanged, so the forces balance on-shell. For the Milky-Way–LMC pair, the on-shell solution balances to \(2\times10^{-6}\) of the mutual force. With the baryonic amplitude \(Z_{\theta,P} = 2c\Phi_{b,P}^2\) it has two defects:

• a negative excess-density layer of \(-7.5\times10^9M_\odot\), two and a half times the LMC's baryons, on the support edge;
• accelerations of 0.4–2.8% that act on LMC-labelled but not MW-labelled matter at the same point.

A single continuous director removes the layer only by locking the satellite or by creating a compensating defect. Appendix N.8 shows that two changes remove both defects:

• the nematic order melts where carrier records overlap;
• the stiffness amplitude becomes the potential of each carrier's response stock (Appendix O.3).

With an overlap zone half as wide as the support and normalization \(c \simeq 0.03\), the combined model gives the following:

• an interface layer of \(-0.85 \times 10^9M_\odot\), against \(-0.33 \times 10^9\) for a single joint field;
• an intact satellite interior;
• a mutual force within the finite-difference systematic of the joint-field value;
• no label-dependent force on baryons;
• a Milky-Way vertical potential of 1416 and 1315 \(\text{km}^2\text{s}^{-2}\) for the two baryonic models of Section 16.

The joint source fixes momentum balance but not inertia. The Milky Way's reflex velocity toward the LMC, integrated along a radial infall from 250 to 50 kpc at \(300\ \text{km s}^{-1}\), is \(8.7\ \text{km s}^{-1}\) when the response has no inertia of its own. The measured travel velocity is \(32 \pm 4\ \text{km s}^{-1}\) at 40–120 kpc [62], and about 40 near 100 kpc [63]. The response must therefore be carried by an inertial stock that moves with its carrier. Appendix O.3 identifies that stock with the reallocated capacity and fixes its budget.

Empirical domain. The exponential law is specified for equilibrium active carriers in the regime established by rotationally supported galaxies and isolated-galaxy lensing. Dispersion-supported low-mass dwarfs now provide a distinct boundary test. A GravSphere analysis of 12 nearby dwarfs with \(10^4 < M_{\text{bar}}/M_\odot < 10^{7.5}\) finds that most lie above the extrapolated RAR, trace multivalued loci, and show substantial galaxy-to-galaxy scatter; the reported tidal and nonequilibrium checks do not account for the result [54]. The carrier projector could address this evidence only through a derived phase-selection prediction. The present EFT therefore makes no universal RAR claim for ultrafaint dwarfs.

Satellites and tidal dwarfs test the carrier rules directly. On the joint source a satellite's response carries its host's external field.

• Classical satellites. Nonlinear virial dispersions in the Milky-Way field are 11.0, 7.29 and \(1.25\ \text{km s}^{-1}\) for Fornax, Sculptor and Crater II. The measured values are 11.7, 9.2 and \(2.7 \pm 0.3\) [64, 65]. A response computed from Crater II's own field alone gives \(4.0\ \text{km s}^{-1}\). Neither reproduces Crater II, which remains a boundary case of the equilibrium phase.
• Carrier-local stiffness. A stiffness built from the joint potential would lock satellite directors to the host. It would reduce their dynamical masses by factors of 2.4–3.9, and Fornax would fall to \(7.2\ \text{km s}^{-1}\). The stiffness amplitude is therefore carrier-local.
• Tidal dwarfs. Descendants that inherit their parent's record sit under the parent's director and are locked. For the NGC 5291 tidal dwarfs, with the parent at \(1\text{–}2 \times 10^{11}M_\odot\) and 60 kpc, the locked response gives dynamical-mass boosts of 1.5–2.0, against the measured 1.2–1.5 [66]. Treating each dwarf as its own carrier gives 3.6–4.9. For a \(2\times10^{11}M_\odot\) parent, \(\chi^2\) over the three dwarfs is 0.5 for inheritance and 48 for separate carriers.

Status. The thermal contact free energy, local auxiliary action, and carrier nesting rule close the leading slowly varying static EFT once its matching data are supplied. Evaluated on the joint coarse source, the carrier sum conserves momentum between active carriers. The direction-resummed response follows from the same auxiliary action with the record-potential stiffness. Several inputs of its multi-carrier completion remain conditional: the overlap width, the normalization \(c\), the tie of supports and director cores to the response stocks, and the stock dynamics. The amplitude identification \(X_P \leftrightarrow \Phi_{R,P}\), the saddle-splay coefficient set by spherical exactness, the one-entropy horizon loading, \(E_P/(k_B T_H) = x_P\), active-carrier projector, and potential-zero prescription remain conditional. A microscopic GFT derivation must produce the cell Hamiltonian, projector, metric vertex, and director stiffness. The ultrafaint-dwarf phase boundary remains empirical work.

16. Galactic Metric, Lensing, and Local Tests

No-slip metric contact. The leading stationary metric response of carrier \(P\) is

$$\boxed{h^{\perp,P}_{\mu\nu} = -\frac{2\phi_{\perp,P}}{c^2}(\bar{g}_{\mu\nu} + 2u_\mu u_\nu).}$$

In Newtonian gauge this gives \(\Delta\Phi_{\perp,P} = \Delta\Psi_{\perp,P} = \phi_{\perp,P}\). A leading closed-time-path contact that produces this mean response is

$$\boxed{\Gamma^{\text{QS}}_\perp = -\frac{1}{2}\sum_{P\in\mathfrak{A}}w_P\left\langle h_\Delta, \mathcal{E}h_{\perp,P}[T^\mathfrak{A}_c]\right\rangle,}$$

where \(\mathcal{E}\) is the linearized Einstein operator and the bracket includes the invariant spacetime integral. This contact satisfies \(\Gamma_\perp[g, g] = 0\) and is real in the static mean. Its induced stationary source has \(T^{00}_{\perp,P} = \rho_{\perp,P}c^2\) and \(T^{ij}_{\perp,P} = 0\), so it carries no scalar anisotropic stress. The quasistatic contact introduces no independent scalar Cauchy data and adds no pole to the vacuum graviton propagator. A canonical local scalar-gradient stress would produce anisotropic stress and is not the metric realization used here.

The linear Ward identity follows from the Bianchi identity for \(\mathcal{E}\). Nonlinear conservation requires the carrier and record variables to be varied with the metric. In covariant form the projector must obey

$$\Pi_P[\varphi^* g, \varphi^*\psi, \varphi^*\mathcal{R}] = \varphi^*\Pi_P[g, \psi, \mathcal{R}].$$

Freezing \(\Pi_P\) during metric variation would leave the nonlinear Ward identity unproved. The full covariant completion retains a conditional grade for this reason.

Lensing response. No additional lensing coefficient appears. The effective density inferred by dynamics and lensing is

$$\rho_{\text{lens},P} = \rho_P + \frac{1}{4\pi G}\nabla\cdot\mathbf{J}_{\perp,P}.$$

For a deep-regime point mass, define \(A = \sqrt{GMa_0}\). Then

$$g = \frac{A}{r}, \quad \rho_\perp(r) = \frac{A}{4\pi Gr^2},$$

and the projected profiles and asymptotic deflection are

$$\Sigma_\perp(R) = \Delta\Sigma_\perp(R) = \frac{A}{4GR}, \quad \alpha_\infty = \frac{2\pi\sqrt{GMa_0}}{c^2} = \frac{2\pi v_\infty^2}{c^2}.$$

The response of an isolated carrier is bounded by its stock (Appendix O.3). The committed capacity of the carrier's Lagrangian baryons, less the released share, gives

$$M_\perp \leq C_R M_b, \quad C_R = (1 - \eta_*)(1 - f_b)\frac{M_h}{M_b},$$

and the cap is reached at \(r_C = r_M/\ln(1 + 1/C_R)\). The isolated-lens KiDS-1000 RAR [68] tests this bound. For a point baryonic mass the prediction at fixed \(g_{\text{bar}}\) is independent of \(M_b\), and the model surface density converts to \(g_{\text{obs}} = 4G\Delta\Sigma\) as for the data.

• Cap per present baryon. \(C_R = (1-\eta_*)/\epsilon = 5.16\) puts the cap at the innermost lensing point. It gives \(\chi^2 = 326\) on the seven isolation-safe points (\(g_{\text{bar}} \geq 10^{-13}\ \text{m s}^{-2}\)), against 40 uncapped, and is excluded. An independent implementation reproduced both values.
• Lagrangian cap. With \(M_h\) from the stellar-to-halo relation of Moster et al. [61] at \(z = 0.2\), the four stellar-mass bins give \(C_R = 20\text{–}49\) and \(r_C = 21\text{–}50\, r_M\), near the virial radius. Over 31 mass-binned isolation-safe points this gives \(\chi^2 = 86\), against 75 uncapped and 69 for halo masses 0.2 dex higher.

Beyond 0.3 Mpc every capped model falls below the data, so the signal there must come from neighbouring structures, as it does in ΛCDM.

Joint kinematic and weak-lensing data extend the same RAR by about 2.5 decades in acceleration. With strict isolation and consistent stellar and gas masses, early- and late-type galaxies follow the same relation [68, 69]. This supports the leading no-slip choice. Direct solutions for resolved nonspherical lenses remain necessary.

Resolved Milky-Way test. For the flat-inner-disk baryonic model of Wang et al. [41], two independent solvers give the local-contact value \(\Delta\Phi(8.17, 1.055) = 1792.2\ \text{km}^2\text{s}^{-2}\) against \(1294.8 \pm 54.0\). That baryonic model also gives only \(v_c(R_0) = 199\ \text{km s}^{-1}\) under the local response, against \(237\ \text{km s}^{-1}\). The director response changes the circular speed at \(R_0\) by less than 0.5%, so it cannot supply the missing enclosed mass. The comparison below therefore continues the inner thin disk exponentially, with \(h_{\text{in}} = 2.2\text{–}2.5\ \text{kpc}\) and \(M_b \simeq (7\text{–}8) \times 10^{10}M_\odot\), a baryonic premise that RAR-based fits share.

With those baryons the responses compare as follows, at Wang's local stellar surface density and at the McKee et al. [42] level. The rotation-curve \(\chi^2_v\) uses 34 points and the vertical-potential \(\chi^2_\Phi\) uses 43.

The director completion uses the stock amplitude at \(c = 0.03\) (Appendix N.8), with the Milky Way's stock the \(C_R = 14\) capped response of Appendix O.3. It fits the vertical potential as well as the NFW controls, both with \(\chi^2_\Phi/N \simeq 3.2\), and leaves the rotation-curve fit essentially unchanged. The solution sits in the stiff limit:

• at \((R_0, 1\ \text{kpc})\) the director lies \(1.0°\) from radial while the source field is tilted by \(14.3°\);
• varying \(0.03 \leq c \leq 1\) spans \(\Delta\Phi = 1396\text{–}1416\ \text{km}^2\text{s}^{-2}\), and \(c = 0.01\) gives 1445;
• the stiffness source is \(-0.25\%\) of \(\nabla\cdot\mathbf{J}_\perp\) at \(R_0\) and \(-0.03\%\) of the excess mass within 30 kpc.

The excess density at \(R_0\) is 0.37, 0.35, and \(0.30\ \text{GeV cm}^{-3}\) at \(|z| = 0, 1,\) and 3 kpc.

The baryonic proxy at \(c = 1\) gives 1401 and 1302 \(\text{km}^2\text{s}^{-2}\) and 0.35, 0.34, and \(0.29\ \text{GeV cm}^{-3}\). With the proxy, rescaling \(1/4 \leq c \leq 4\) spans 1397–1414. For that solution two independently written solvers agree to \(0.1\ \text{km}^2\text{s}^{-2}\), find the same minimizer from local and radial starting fields with no negative Hessian pivots, and are unchanged when the director grid is halved. The two amplitudes differ by about 1% in \(\Delta\Phi\) and by \(0.02\ \text{GeV cm}^{-3}\) in the local excess density.

Resolved disk galaxies. A director solve with the stiffness source for each of the 131 SPARC galaxies passing the quality cut (\(Q \neq 3\), \(i \geq 30°\), \(\delta V/V < 0.1\), \(R < 2.5\, r_M\); 1683 points) reproduces the algebraic RAR to 0.012 dex in every \(r/r_M\) bin, with rms residual 0.147 dex for both. It stays within 0.011 dex of the rigid radial director at every point, and the stiffness source moves no point by more than 0.001 dex. The local two-dimensional solve departs from the algebraic relation by up to 0.025 dex in the same bins. These solves use the baryonic proxy at \(c = 1\). They stay within 0.011 dex of the rigid radial director, which is the stiff limit shared by both amplitudes.

Vertical dynamics of face-on disks. Removing the phantom sheet changes vertical dynamics, and this separates the completion from QUMOND. The DiskMass and NGC 6946 solutions below use the baryonic proxy at \(c = 1\), and the predictions registered in Section 27.1 keep that basis. For a face-on disk the mass-weighted dispersion \(\langle\sigma_z^2\rangle = \int z\nu\,\partial_z\Phi\, dz/\int\nu\, dz\) follows from the solved potential at the rotation-curve mass-to-light ratio. For the DiskMass galaxies [48, 49], with \(\Upsilon_K\) from MOND rotation-curve fits [50], the director excess raises \(\sigma_z^2\) above the Newtonian value by a median factor 1.20 at the DiskMass scale heights, against 1.73 for QUMOND. Matching the measured integrated-light dispersions of the 27 unflagged galaxies then requires stellar scale heights \(h^*_z/h_{z,\text{DMS}} = 0.55\) (16–84%: 0.39–0.77) for the completion, 0.37 for QUMOND, and 0.61 for Newtonian baryons alone. As a pipeline check, Newton at the DiskMass mass-to-light ratios reproduces the measured dispersions to 3%.

The shortfall common to every baryons-only model traces to the integrated-light dispersion. It mixes a cold young layer with the old disk, whose scale height enters the vertical Jeans relation [51]. A forward model of the measurement uses isothermal populations with a Milky-Way-shaped age–velocity relation, \(V\)-band, \(K\)-band, and Mg b light weights, the \(17\ \text{km s}^{-1}\) PPak instrumental profile, a single-Gaussian fit, and the \(K\)-band scale height. Relative to a single-population disk it gives \(\Sigma_{\text{true}}/\Sigma_{\text{DMS}} = 1.25\) (16–84% over 66 population variants: 1.11–1.49). A DiskMass quality cut keeps the 22 galaxies whose DiskMass Newtonian model reproduces its own dispersion fit within 25% and whose dispersions stay above \(12\ \text{km s}^{-1}\). With the correction these require \(h^*_z/h_{z,\text{DMS}} = 0.76\) (0.66–0.90) for the completion and 0.55 (0.46–0.66) for QUMOND. The completion-to-QUMOND ratio stays within 1.32–1.45 in every variant.

NGC 6946 supplies the old-disk dispersion directly, from two-component fits to high-resolution integrated light in the inner disk and planetary nebulae farther out: \(\sigma_{\text{old}} = 65, 65, 32, 21,\) and \(15\ \text{km s}^{-1}\) at 1.5–9 kpc [52]. With \(h_z = 346\ \text{pc}\) from the DiskMass scaling relation, an isothermal old disk, a 15% cold layer, and the measured gas, one stellar mass-to-light ratio must fit both the SPARC rotation curve and \(\sigma_{\text{old}}\). The completion gives \(\Upsilon = 0.31^{+0.09}_{-0.08}\) from \(\sigma_{\text{old}}\) against 0.39 from the rotation curve, a \(1.0\sigma\) difference, with \(\chi^2 = 5.9\) for five points at the rotation-curve value. QUMOND gives \(0.19^{+0.08}_{-0.04}\), a \(2.6\sigma\) difference, and \(\chi^2 = 13.7\). The completion remains acceptable for an exponential vertical profile, the TRGB distance, and \(h_z\) changed by \(\pm20\%\), with \(\chi^2 = 4.9, 4.9, 4.5,\) and 10.1 (\(p \geq 0.07\)). The single-component dispersions of the same galaxy give \(\chi^2 = 154\) for the completion and 345 for QUMOND. The DiskMass-type tension therefore disappears in this galaxy once the old-disk dispersion is used.

Shape of the excess. The same solution fixes the shape of the excess. For the Milky Way the total potential has \(q_\Phi = 0.87\text{–}0.88, 0.95, 0.98,\) and 0.99 at 10, 20, 30, and 50 kpc. The baryonic proxy gives 0.88–0.89, 0.96, 0.98, and 0.99, with excess-density axis ratio \(q_\rho = 0.91\text{–}0.92, 0.94, 0.95,\) and 0.98. For the 53 SPARC disks with \(V_{\text{flat}} > 150\ \text{km s}^{-1}\) the radial-director limit gives \(q_\Phi = 0.95 \pm 0.05\) at 20 kpc and \(0.98 \pm 0.02\) at 30 kpc. Disks with \(V_{\text{flat}} = 120\text{–}150\ \text{km s}^{-1}\) are rounder, with \(q_\Phi = 0.99\) at 20 kpc. Hydrodynamic ΛCDM halos of Milky-Way mass have inertia-tensor density axis ratios \(c/a \simeq 0.67\text{–}0.72\) with halo-to-halo scatter 0.1 at these radii [46, 47]. An oblate NFW homoeoid with \(c/a = 0.67\text{–}0.72\) has potential axis ratio 0.85–0.87 at 20 kpc and 0.87–0.89 at 50 kpc. In potential flattening the two predictions therefore differ by about 0.1.

Milky-Way tracers with six-dimensional data give \(q_\rho = 1.00 \pm 0.09\) from RR Lyrae at 5–20 kpc [45] and \(1.06 \pm 0.06\) from three streams near 20 kpc [43]. Their weighted mean, \(1.04 \pm 0.05\), lies \(1.9\sigma\) above the predicted 0.94. It lies 2.9–3.3\(\sigma\) above the ΛCDM halos, including their scatter, and the likelihood ratio on statistical errors is 20–90 in favor of the completion. A single galaxy caps this ratio: for a prediction uncertainty of 0.02, even a perfect measurement at the predicted value would give only 60–190, because the ΛCDM prediction for one halo carries the population scatter.

A population measurement has no such cap, and Chemaly et al. [44] provide the first extragalactic one. They fit projected tracks of nearby extragalactic streams with a halo-only axisymmetric NFW potential of free orientation and potential flattening \(q\), and infer a truncated-Gaussian population. Their gold subsample of 17 streams gives \(\mu_q = 0.72^{+0.16}_{-0.14}\) and \(\sigma_q = 0.34^{+0.18}_{-0.19}\); the bronze subsample of 15 gives \(0.97^{+0.19}_{-0.17}\) and \(0.30^{+0.20}_{-0.16}\). Synthetic orbit-segment tracks with 1 kpc noise, viewed at random orientations and fitted the same way with halo mass free and the disk omitted, return \(\hat{\mu}_q = 1.02\text{–}1.03\) for a round halo with a Milky-Way disk and 0.86–0.91 for \(q_\Phi = 0.87\). Both inputs have zero intrinsic scatter and still return \(\hat{\sigma}_q = 0.3\text{–}0.5\), so the inferred scatter carries no shape information. Bootstrapped through the same hierarchical model, the round input yields a gold-like \(\hat{\mu}_q \leq 0.72\) in at most 3% of 17-stream samples, and the ΛCDM-like input in 1–16%. The bronze result distinguishes neither. The combined likelihood ratio is about 4 in favor of a ΛCDM-like halo, with a range of 3–14 over the assumed per-stream likelihood width, and comes almost entirely from the gold subsample. Read as a potential flattening, \(\mu_q = 0.72\) corresponds to a density axis ratio near 0.4, flatter than the simulated halos. With per-stream noise of 0.3 in \(q\), separating 0.97 from 0.87 at \(3\sigma\) requires about 80 streams of gold quality. The Milky Way and the present stream population therefore point in opposite directions, and a disk-modeled population of order \(10^2\) streams can separate the two predictions.

Solar-System quadrupole and wide binaries. In the Solar System the Milky Way is the active carrier. The Sun and planets remain resolved descendants in the Einstein branch and do not re-enter \(\Gamma_\perp\) as separate transverse sources. The Sun-generated anomalous quadrupole is therefore

$$\boxed{Q^{\odot,\perp}_2 = 0}$$

at leading order. The remaining smooth Galactic contribution is a tide of order

$$\frac{g_{\text{MW}}}{R_0} \simeq 9.2 \times 10^{-31}\ \text{s}^{-2},$$

about two thousand times smaller than the current Cassini \(1\sigma\) uncertainty. Cassini tracking gives \(Q_2 = (1.6\pm1.8)\times10^{-27}\ \text{s}^{-2}\) and places universal nonlinear functions of the total local field in 3–15\(\sigma\) tension with galaxy fits; it also limits their Milky-Way boost at the Sun to two percent at 95% confidence [53]. Carrier nesting avoids that total-field cross-term.

The avoidance fixes a minimum resolution for the parent's conditional expectation. A descendant of mass \(M\) enters the parent's coarse source smoothed over a width \(L\). The linear response to the Galactic field then gives, at its centre,

$$Q_2 = \frac{4}{15\sqrt{2\pi}}(n_\| - n_\perp)\frac{GM}{L^3},$$

with \(n_\perp = n_B(x_e) = 0.645\) and \(n_\| = d(n_B g)/dg|_{g_e} = 0.148\) at the Sun. The coefficient was verified analytically and numerically by an independent calculation. Cassini's \(2\sigma\) interval, \(-2.0\) to \(+5.2 \times 10^{-27}\ \text{s}^{-2}\), therefore requires:

• \(L \geq 0.049\ \text{pc}\) (10 200 AU) if the Galactic director follows the local field, where the quadrupole is negative;
• \(L \geq 0.024\ \text{pc}\) (4900 AU) if the director is locked (\(n_\perp \to 0\)), where the quadrupole is positive.

Both lengths are comparable to the Sun's MOND radius, 0.034 pc. Globular clusters bound \(L\) more strongly. Palomar 14 resolved in the joint source would have \(\sigma = 1.49\ \text{km s}^{-1}\), against the measured \(0.38\pm0.12\) [67], and 0.54 when smoothed into its parent. If Palomar 14 is a descendant of the Milky-Way carrier, \(L \gtrsim 25\text{–}50\ \text{pc}\).

Wide binaries inside the Milky-Way carrier are likewise Newtonian apart from the smooth Galactic tide. For a \(1.5M_\odot\) pair on the flat Galactic rotation curve the Jacobi radius \(r_J = [GM/(4\Omega^2 - \kappa^2)]^{1/3}\) is about 1.6 pc, and the fractional speed corrections from the tide satisfy

$$\frac{\Delta v}{v_N} \simeq \frac{1}{2}\left(\frac{s}{r_J}\right)^3 \lesssim 2 \times 10^{-5},\ 1 \times 10^{-4},\ 4 \times 10^{-4}$$

at separations of 10, 20, and 30 kAU, respectively. Current Gaia analyses remain divided: one recent treatment favors Newtonian dynamics after modeling unresolved triples, while two analyses report a low-acceleration anomaly [38, 39, 40]. The carrier-resolved branch predicts the Newtonian outcome.

The background vector \(u^\mu\) enters the parent carrier response, but its local spatial variation is limited by the Galactic tide. Leading Solar-System PPN values and vacuum gravitational-wave propagation remain those of the Einstein branch. Time-dependent vector contractions and nonlinear preferred-frame terms remain part of the covariant projector audit.

Fluctuations. For \(N_{\text{eff}}\) independent occupation cells,

$$\text{Var}(\bar{n}) = \frac{n_B(1 + n_B)}{N_{\text{eff}}}, \quad \frac{\sigma_g}{g} = \sqrt{\frac{n_B}{N_{\text{eff}}(1 + n_B)}} \leq \frac{1}{\sqrt{N_{\text{eff}}}}.$$

A stochastic contribution below ten percent requires \(N_{\text{eff}} \gtrsim 100\); three percent requires \(N_{\text{eff}} \gtrsim 10^3\). The symmetrized occupation covariance gives a positive noise kernel. A dynamical completion must also satisfy the KMS fluctuation–dissipation relation. The bath coupling, equilibration rate, and correlation volume have not yet been derived.

Part V. Transport, Clusters, and Cosmology

The microstructure-to-static-weak-field chain is the most directly constrained part of the theory. The sectors of this part ask how the same finite-capacity substrate behaves when sources move. Transport asks how the capacity-strain field propagates after a source changes; the cluster sector asks how it couples to matter in distinct dynamical phases; cosmology treats the homogeneous mode; and the saturated phase treats the proposed committed carrier. These sectors share the static weak-field ontology but not its evidential status: the Einstein/capacity action equivalence, ordinary source map, and finite marked-transfer scale are the controlled baseline inside their displayed actions, while the transport, cluster, and cosmological additions carry their stated conditional or open grades.

17. Causal Transport and Telegrapher Dynamics

The static equations give settled fields. Moving sources require a causal response with the same zero-frequency limit. Relative to the substrate four-velocity \(u^\mu\), the ordinary capacity deficit retains the covariant telegrapher completion

$$\tau_0(u^\mu\nabla_\mu)^2\delta S + u^\mu\nabla_\mu\delta S = Dh^{\mu\nu}\nabla_\mu\nabla_\nu\delta S + A\chi, \quad h^{\mu\nu} = g^{\mu\nu} + u^\mu u^\nu,$$

with \(A/D = \kappa/\gamma\). The carrier-resolved transverse excess has the corresponding local-frame equation

$$\boxed{\tau_0\ddot{\phi}_{\perp,P} + \dot{\phi}_{\perp,P} - D\nabla^2\phi_{\perp,P} = -4\pi GD\,\rho_{\perp,P}[\Pi_P T],}$$

where

$$\rho_{\perp,P} = \frac{1}{4\pi G}\nabla\cdot\mathbf{J}_{\perp,P}.$$

Its static limit is exactly the Poisson equation of Section 15. No separate static form factor is introduced.

Causal propagation and the minimal infrared scale choice give

$$\frac{D}{\tau_0} = c^2, \quad \tau_0^{-1} = H_0, \quad D = \frac{c^2}{H_0}.$$

For a Fourier mode, the response relative to the static solution is

$$F(\omega, k) = \frac{Dk^2}{Dk^2 - \tau_0\omega^2 - i\omega}, \quad F(0, k) = 1.$$

The Laplace poles are

$$s_\pm = \frac{-1 \pm \sqrt{1 - 4\tau_0 Dk^2}}{2\tau_0}.$$

Both have negative real part, and the characteristic propagation speed is \(\sqrt{D/\tau_0} = c\). For galactic source frequencies \(\omega \sim vk\), departures from the static branch begin at \(O(v^2/c^2) \sim 10^{-6}\); the linear damping correction is smaller on kiloparsec scales. The quasistatic RAR and lensing equations therefore survive without spectral suppression.

This transport law governs propagation delay, relaxation, and merger-era lag. Higher-derivative corrections at the block scale remain ultraviolet data. The cluster and merger phenomenology, which supplies the source weights evolved by this equation, is developed in Section 18.

18. Cluster Source Projection and the Diffuse–Decoupled Channel Split

Clusters require one account of both the hook-shaped residuals of relaxed systems and the collisionless-galaxy lensing peaks of mergers, without hidden baryonic galaxy mass or a modified galaxy law.

The proposed long-range entropic-excess channel couples differently to the three baryonic phases. A diffuse, phase-averaged medium carries the transverse projection \(\epsilon\) inherited from the conditional galactic normalization; a dynamically decoupled collisionless component recovers the full projection; and a virialized coherent bath can be lifted above it. The projection value is fixed once that transverse branch is adopted. The phase-selection rule, lift profile, and resolved-map test remain open.

The cluster residual varies with acceleration. Lensing and kinematic analyses of relaxed clusters [6, 19, 20] find the ratio of observed to galaxy-RAR-predicted acceleration near unity in stellar-dominated centers, rising to roughly 3–5 at intermediate accelerations (\(g_{\text{bar}} \sim 10^{-11}\text{–}10^{-10}\ \text{m s}^{-2}\)), and apparently returning toward the galaxy relation at the lowest probed accelerations, subject to gas-extrapolation caveats [20]. The older integrated value of 1.5–2 from higher-acceleration hydrostatic analyses samples the high-\(g_{\text{bar}}\) edge of this hook-shaped profile. Merging clusters add a spatial constraint. In Bullet-type systems, ram pressure displaces the intracluster plasma from the collisionless galaxies while the lensing peaks remain with the outgoing collisionless components. A viable cluster sector must explain both observations without hidden baryonic mass in galaxies or a change to the galactic mass anchor.

Transport lag and extra galaxy mass do not supply the required offset. In the canonical \(\tau_0^{-1} = H_0\) branch, disturbances propagate at \(c\) and cross a megaparsec-scale configuration in roughly 3 Myr, about two and a half orders below a gigayear merger timescale. The field therefore tracks the moving source; an underdamped configuration retains memory of the pre-merger centroid rather than the outgoing collisionless component. Extra galaxy mass is excluded because the galactic acceleration scale and gas-dominated-dwarf RAR fix the deficit per unit baryonic mass.

The projection coefficient as the galactic reduction factor. Section 15 fixes the galactic acceleration scale as the transverse reduction of the horizon thermal scale,

$$a_0 = \frac{g_{\text{share,eff}}}{4\pi^2}a_H, \quad a_H = cH_0,$$

where \((2\pi)^2\) is the angular Haar volume of one compact transverse phase cell and \(g_{\text{share,eff}}\) is the admissibility-sharing content conditionally loaded into it. The same loading and horizon-coupling construction gives

$$\epsilon \equiv \frac{a_0}{a_H} = \frac{g_{\text{share,eff}}}{4\pi^2} \simeq 0.188.$$

This introduces no cluster-specific coefficient. It imports the reduction factor of the galactic branch and reads it as a source projection: a source restricted to the transverse static sector couples at strength \(\epsilon\) relative to a source accessing the full horizon projection.

The diffuse and decoupled source classes. The assignment follows from coherence under coarse-graining. Diffuse matter is a continuum of locally uncorrelated, thermalized source elements; under coarse-graining its off-diagonal source cross-terms average away and only the diagonal transverse static projection survives. It therefore couples at \(\epsilon\). This includes shocked or unvirialized intracluster plasma, the warm–hot intergalactic medium, and cold but diffuse galactic H i when the galaxy is treated as a single smooth source—which is why gas-dominated dwarfs and low-surface-brightness galaxies remain on the galaxy RAR. The suppressed projection is thus a coherence effect, not a temperature effect; a hot phase that has virialized into a coherent bath is the exception, taken up below.

The unsuppressed projection is accessed by matter that is not part of the phase-averaged continuum: a collisionless overdensity that is spatially separated from, and dynamically decoupled from, a surrounding diffuse medium. The criterion is relational, not intrinsic compactness. Compactness alone would misclassify: stars in ordinary galaxies, isolated ellipticals, and globular clusters are compact and bound yet must remain on the galaxy RAR, and they do, because none is a collisionless node decoupled from a distinct diffuse continuum. A galaxy in a cluster is different only because it is embedded in, and decoupled from, the intracluster medium. The suppression is the property of participating in the continuum; matter that has decoupled from the continuum escapes it. So decoupled collisionless matter sits at the baseline weight, and incoherent diffuse gas is suppressed to \(\epsilon\), with \(\epsilon = g_{\text{share,eff}}/4\pi^2\).

A virialized bath is the third state. Once the diffuse atmosphere relaxes into a coherent, extended phase it may open a collective response above the decoupled baseline. The source weight includes the ordinary unit response plus the maximally recruited excess. The cap reading is therefore

$$\boxed{W^{\text{max}}_{\text{bath}} = 1 + \frac{1}{\epsilon} \simeq 6.32,}$$

the same total-to-baryon factor that appears in the pinned abundance, \(\Omega_m/\Omega_b = 1 + 1/\epsilon\). The earlier \(1/\epsilon\) value counts only the excess/committed component and is not the ceiling on the total bath source weight. The lift profile between \(W_{\text{bath}} = 1\) and \(1 + 1/\epsilon\) is evaluated against cluster data below.

The effective entropic-channel source \(\chi_{\text{ent}}\) is therefore regime-dependent. In a relaxed cluster the gas is a virialized bath,

$$\boxed{\chi_{\text{rel}} = \rho_{\text{dec}} + W_{\text{bath}}\rho_{\text{bath}}}$$

with \(\rho_{\text{dec}}\) the decoupled collisionless substructure and \(\rho_{\text{bath}}\) the virialized continuum; in a non-equilibrium merger the central gas is shocked and incoherent while only a residual atmosphere stays virialized,

$$\boxed{\chi_{\text{merge}} = \rho_{\text{dec}} + \epsilon\rho_{\text{shock}} + W_{\text{bath}}\rho_{\text{vir}}}$$

which in the Bullet limit, where the displaced gas is shocked and little virialized bath remains on the cores, reduces to \(\chi_{\text{Bullet}} \simeq \rho_{\text{dec}} + \epsilon\rho_{\text{shock}}\). Ordinary matter continues to gravitate through the usual metric coupling; the projection rule concerns only the long-range entropic-excess channel.

The measured hot-atmosphere factor. A component is decoupled only relative to a surrounding medium, so the relevant factor is tied to a measured property of that medium: the fraction of the halo's cosmic baryon allotment that has become an extended virialized hot phase,

$$\boxed{B_{\text{bath}} = \text{clip}_{[0,1]}\left[\frac{M_{\text{hot,vir}}(< r_{500})}{f_{b,\text{cos}}M_{500}}\right]}\quad f_{b,\text{cos}} = \frac{\Omega_b}{\Omega_m} \simeq 0.156,$$

with \(f_{b,\text{cos}}\) fixed by Planck values [73], within \(0.7\sigma\) of the branch value \(\epsilon/(1 + \epsilon) = 0.158\) and \(M_{\text{hot,vir}}, M_{500}\) read from X-ray/SZ and total-mass estimates. Once the transverse branch is adopted, \(\epsilon\) is shared across all systems, the baryon fraction is fixed cosmologically, and the per-system quantities are measured. The undetermined object is the coherence-growth profile \(W_{\text{bath}}(B_{\text{bath}})\). In merging systems \(M_{\text{hot,vir}}\) refers to the pre-merger virialized atmosphere.

Relaxed-cluster residual. For a relaxed cluster the diffuse gas is a virialized continuum, and the residual is carried by that continuum, not by the decoupled stellar component. Let

$$f_{\text{cont}} = \frac{M_{\text{hot,vir}}}{M_{\text{baryon}}}$$

be the fraction of observed baryons in the virialized diffuse continuum. The residual relative to a galaxy-RAR extrapolation is then

$$\boxed{\mathcal{R}_{\text{rel}} = 1 + (W_{\text{bath}} - 1)f_{\text{cont}}}$$

The minimal linear lift is

$$W_{\text{bath}} = 1 + (1 - \epsilon)B_{\text{bath}}, \quad 1 - \epsilon = 1 - \frac{g_{\text{share,eff}}}{4\pi^2} \simeq 0.812,$$

where \((1 - \epsilon)\) is the part of the full horizon channel missing from the suppressed transverse branch, rather than a new coefficient. This form has the correct mass trend because \(B_{\text{bath}}\) and \(f_{\text{cont}}\) increase from groups to massive clusters [18, 17], but its amplitude is bounded by \(\mathcal{R}_{\text{rel}} < 1 + (1 - \epsilon) \simeq 1.81\). CLASH-scale values \(B_{\text{bath}} \simeq 0.83\) and \(f_{\text{cont}} \simeq 0.9\) give \(\mathcal{R}_{\text{rel}} \simeq 1.6\), below the measured peak band 3–5 [6, 19]. The alternative \(\mathcal{R} = 1 + 4.32 B_{\text{bath}}f_{\text{dec}}\), with \(f_{\text{dec}} = f_\star/(f_\star + f_{\text{gas}})\), can reach that amplitude but predicts the wrong mass trend because \(f_{\text{dec}}\) falls with mass. These exclusions assign the residual to the virialized continuum while leaving its nonlinear coherence-growth profile to be derived.

The hook morphology. Independently of the lift amplitude, the channel split predicts the radial shape of the cluster residual. The decoupled BCG stellar component dominates cluster centers and carries weight 1, so the local residual starts near unity. Farther out, the virialized gas continuum dominates and raises the residual toward the bath-weighted value. At the lowest accelerations, the deep branch compresses a bounded source weight \(W\) toward \(\sqrt{W}\) in acceleration terms and lowers the residual again. The resulting profile is a hook: near unity in the stellar-dominated center, maximal where the bath dominates at intermediate acceleration, and closer to the galaxy relation in the deep outskirts. Current measurements report this morphology [19, 20]; neither a total-baryon modified-gravity law with no relaxed-cluster excess nor a constant offset gives the same shape.

At the cap \(W_{\text{bath}} = 1 + 1/\epsilon\), developed-bath parameters give

$$\boxed{\mathcal{R}_{\text{cap}} = f_{\text{dec}} + \left(1 + \frac{1}{\epsilon}\right)f_{\text{cont}} = 1 + \frac{f_{\text{cont}}}{\epsilon} \simeq 5.7\text{–}5.9.}$$

This is an upper bound, not a prediction that relaxed clusters sit at saturation. The ceiling 5.7–5.9 lies above the measured 3–5 mass-residual band, while the acceleration-resolved hook peak \(\mathcal{S} \simeq 3.7\text{–}4.9\) remains inside that band. Developed relaxed clusters therefore reach roughly three quarters of the available cap rather than saturating it. Applying the cap uniformly at group scale would overshoot strongly, so the attained fraction must depend on bath development; X-ray-faint groups remain far below it, with the group-scale data requiring \(W_{\text{bath}} \lesssim 1.4\) there.

One caveat accompanies the bounded profile: the predicted central residual depends on the stellar/gas decomposition and on excluding multiphase cool-core gas from the coherent bath. The deep-outskirt question—power-law continuation [6] versus convergence toward the galaxy relation [20]—is adjudicated directly below.

The ceiling against cluster data. Inverting the X-COP hydrostatic measurements [94] gives twenty-four source-weight tests. For each point, \(\mathcal{S}\) solves

$$\frac{\mathcal{S}g_{\text{bar}}}{1 - \exp[-\sqrt{\mathcal{S}g_{\text{bar}}/a_0}]} = g_{\text{obs}}.$$

Every point respects \(\mathcal{S} \leq 1 + 1/\epsilon\); the maximum is \(\mathcal{S} = 3.96\), and the relaxed systems span \(\mathcal{S} \simeq 1.8\text{–}2.7\) at \(R_{500}\) and 1.4–2.2 at \(R_{200}\).

The same inversion adjudicates the deep end: the power-law continuation requires \(\mathcal{S} \simeq 6\text{–}8\) at \(g_{\text{bar}} \simeq 1\text{–}2 \times 10^{-11}\ \text{m s}^{-2}\), precisely the accelerations of the \(R_{500}\)–\(R_{200}\) points, which sit at \(\mathcal{S} \simeq 1.5\text{–}2.7\); within this sample the deep end converges rather than continuing, and the strongest published challenge to the bound is not borne out.

The methodological caveat is that the continuation was fit to lensing-based masses of higher-redshift systems while the inversion here uses local hydrostatic masses; breaching the bound at \(R_{500}\) would require the non-thermal-corrected masses to be low by a factor of \(\simeq 2.3\), well beyond any claimed hydrostatic bias. The measured radial run of the source weight—\(\mathcal{S} \simeq 3.7\text{–}4.9\) in the hook-peak window, where the observed peak band is reached, declining to \(\simeq 2.2\) at \(R_{500}\) and \(\simeq 1.5\) at \(R_{200}\)—is the quantitative target the coherence-growth profile must reproduce.

Direct and fluctuation-based turbulence measurements find low non-thermal support in relaxed systems at all probed radii [95, 96], so the decoherence agent gating the lift cannot be the cluster-to-cluster turbulence level: it must grow with radius even in fully relaxed atmospheres. The coherence-growth profile between the fixed endpoints, so constrained, is the object the channel-selection theorem must deliver, with the group end requiring \(W_{\text{bath}} \lesssim 1.4\).

Bullet-type mergers. The relaxed residual and Bullet morphology use the same branch coefficient \(\epsilon\) with different source expressions because the gas occupies different states. A relaxed atmosphere enters through the bath-lift term; shocked displaced gas carries the suppressed weight while collisionless cores remain decoupled. The two regimes share one conditional coefficient, not one universal scalar law.

In the merger, then, the decoupled galaxies and subcluster cores retain the unsuppressed projection and the shocked diffuse gas couples at \(\epsilon\). With a gas/galaxy baryon ratio near 5.7, the gas contributes \(\epsilon\times5.7 \simeq 1.07\) in the entropic channel against the galaxy contribution of 1.0: the projection brings the two components to near-parity, removing the factor \(\sim5.7\) by which the gas would otherwise dominate, but it does not by itself invert them. The inversion is completed by projected compactness. For two roughly symmetric outgoing components the ratio of one edge peak to the central gas contribution scales as

$$\frac{\Sigma_{\text{edge}}}{\Sigma_{\text{gas}}} \sim \frac{f_{\text{dec}}}{2\epsilon f_{\text{gas}}}\frac{A_{\text{gas}}}{A_{\text{edge}}}, \quad \frac{f_{\text{dec}}}{2\epsilon f_{\text{gas}}} \simeq 0.47,$$

so an edge peak dominates the projected map once the shocked gas is spread over more than about twice the projected area of a compact outgoing core—a condition the observed morphology satisfies by a wide margin. The projection rule and this geometry therefore produce the observed gas/lensing inversion—by projection and geometry together, not by projection alone and not by transport lag—as a spatial surface-density prediction rather than an integrated-mass argument. This is consistency, not yet a test of the coefficient. In standard flexible lens reconstructions the gas weight is degenerate with free halo and substructure components, so a model that fits comparably well with or without the fixed X-ray gas map constrains \(\epsilon\) only weakly; Bullet-type mergers are thus consistent with the projection rule but do not yet measure \(\epsilon\). Peak location alone is in any case insensitive to the coefficient, since sufficiently broad shocked gas yields clump-centered peaks across a wide range of gas weights; the coefficient is tested only by the resolved amplitude fit below.

Relation to the transport sector. The telegrapher sector of Section 17 is not the cluster mechanism; it governs how the field propagates and relaxes once the source weights are set. The source-projection rule supplies the static weights \(\chi_{\text{ent}}\); the transport sector then evolves them. This division avoids the failure mode of a transport-only account, in which a field sourced equally by all baryons cannot hold a lensing peak on the outgoing collisionless component. The full merger observable is obtained by evolving \(\chi_{\text{ent}}(x, t) = \rho_{\text{dec}}(x, t) + [\cdots]\) through the causal equation with the observed geometry as input.

Falsifiers and open status. The rule makes quantitative predictions beyond the relaxed normalization. (i) The channel split predicts a hook: a residual near unity in BCG-dominated centers, one peak where the virialized bath dominates at intermediate acceleration, and convergence toward the galaxy relation in the deep outskirts. A profile monotonic in acceleration, or one that peaks in the stellar-dominated center, would falsify the channel split. The total-source cap supplies the ceiling \(\mathcal{R}_{\text{rel}} \leq 1 + f_{\text{cont}}/\epsilon \simeq 5.7\text{–}5.9\) for developed-bath parameters. All twenty-four X-COP source-weight inversions respect it. A confirmed relaxed-cluster residual above this bound would falsify the cap reading.

(ii) At fixed mass, X-ray-bright bath-developed systems (larger \(B_{\text{bath}}\), larger \(f_{\text{cont}}\)) should deviate more from the galaxy RAR than X-ray-faint systems; the residual turns on with the developed diffuse atmosphere, not with mass alone, so two systems of equal mass but different bath development should separate.

(iii) Resolved lensing maps test the three source components directly. Because the relaxed and merger regimes use different source expressions, the convergence is a three-component channel-weighted map—a relaxed virialized-bath component, a shocked non-equilibrium continuum at the suppressed weight, and a decoupled collisionless component,

$$\boxed{\kappa_{\text{obs}}(x, y) = A\left[W_{\text{bath}}\Sigma_{\text{bath}}(x, y) + \epsilon\Sigma_{\text{shock}}(x, y) + \Sigma_{\text{dec}}(x, y)\right] + b}$$

with the branch value \(\epsilon = g_{\text{share,eff}}/4\pi^2 \simeq 0.188\) or with \(\epsilon\) floated as a test, and \(W_{\text{bath}}\) fit within \([1, 1 + 1/\epsilon]\). Recovering \(\epsilon \simeq 0.19\) across relaxed and merging systems would support both the cluster source rule and the inherited transverse normalization. A best fit near 1 or 0 would falsify the cluster branch. The fit must use the channel-weighted baryonic maps without free dark haloes, which would otherwise absorb the gas weight.

Three open items remain at the theory level. First, the relaxed residual and Bullet morphology use one inherited coefficient with two regime-specific source expressions. The boundary between the virialized and shocked regimes is not derived.

Second, suppression of a phase-averaged continuum and collective lift of a virialized bath are independent premises. Neither follows from the current microscopic source map. Their derivation must also reproduce the measured radial decline and the proposed branch endpoints.

Third, identifying the decoupled component with stellar or galaxy mass and the continuum with gas is a coarse split; intracluster light and tidally stripped stars blur it at a level the resolved-map fit would expose. Abell 520, whose reported gas-coincident dark core is disputed, is a phase-state stress case rather than a direct test: a re-cohering or quasi-bound central component would raise its effective \(B_{\text{bath}}\) and return lensing toward the gas. A confirmed young merger with a statistically secure gas-centered, galaxy-free lensing peak would leave no time for that re-coherence and would challenge the model.

This sector is a structured, falsifiable proposal. The branch value of \(\epsilon\) is inherited rather than re-fitted, \(B_{\text{bath}}\) is measured from the hot-atmosphere fraction, and current data support the trend and hook morphology while excluding the linear lift candidate. The microscopic transverse normalization, coherence-growth profile, resolved-map test, and channel-selection theorem remain open.

19. Cosmology and the Hubble-Tension Sector

The scalar capacity field has two cosmological roles, and the sector works only if they stay separate. Its homogeneous mode \(\bar{S}(t)\) affects the background expansion and the sound horizon; its inhomogeneous fluctuations \(s(x, t)\) still govern local weak-field gravity. The cosmological sector is the homogeneous continuation of the same medium, not an unrelated dark-energy component appended to the weak-field theory: what changes is the kinematic regime, not the ontology, as the background mode becomes dynamically relevant on horizon scales while the local branch stays encoded in the fluctuations.

The cosmological sector uses the same field split,

$$S(x, t) = \bar{S}(t) + s(x, t),$$

where \(\bar{S}(t)\) is the homogeneous mode and \(s(x, t)\) the inhomogeneous sector responsible for local weak-field dynamics. The vacuum baseline is fixed by apparent-horizon capacity,

$$S_\infty(t) = \pi\frac{R_A(t)^2}{L_*^2}.$$

This is the horizon-normalized representation of the same entropy field used locally. It is compatible with the cell-normalized source theorem because local observables depend on \(\delta S/S_\infty\) and \(\kappa/(\gamma S_\infty)\) rather than on an absolute entropy unit.

Two independent results clarify what this baseline means. First, the type II\(_1\) static-patch construction makes empty de Sitter the maximum-entropy gravitational state and places excited semiclassical states below it [27]. Second, Bianconi obtains the same de Sitter area scaling from a bulk rather than a horizon entropy. In the \(c = 1\) convention of her low-curvature Friedmann approximation, the local geometric-relative-entropy density and the causal-diamond four-volume scale as

$$\frac{\delta s}{\delta v} \simeq \bar{\omega}_{[1]}H^2, \quad V_{\text{dS}} \sim H^{-4},$$

so their product gives

$$S_{\text{GfE}} \simeq \frac{\bar{\omega}_{[1]}}{\ell_P^4 H^2} \propto H^{-2} \propto A_{\text{dS}}.$$

Integrating Bianconi's local volumetric information density over the observer's causal diamond gives an area-sized finite capacity [28]. Her result does not fix the coefficient used here, replace \(\ell_P\) by the independently calibrated \(L_*\), or derive the homogeneous evolution. Her intrinsic geometric \(k\)-temperatures scale as \(H^2\), whereas the Gibbons–Hawking temperature used in Section 15 scales as \(H\).

Because the entanglement field couples to the trace of the stress-energy tensor, the homogeneous mode is largely dormant during radiation domination but can become active near matter–radiation equality. The conditional proposal would reduce the sound horizon and shift the CMB-inferred Hubble constant upward: the mechanism has the required sign and turns on at the required epoch. Appendix O.3 fixes a covariant perturbation system for any supplied source-clock release history. Its joint Boltzmann evolution with radiation, baryons, and the committed component has not been computed, and the full likelihood remains open.

The local weak-field predictions are protected by the separation between \(\bar{S}(t)\) and \(s(x, t)\). This is the role of the shear-lock logic: changing the homogeneous background mode does not rewrite the local static Poisson branch that governs galactic dynamics and lensing. The two regimes therefore use one scalar medium without changing the galactic coefficients. The covariant capacity-clock equations make the corresponding dynamical separation test well posed for the full Boltzmann calculation.

20. The Saturated Phase and the Cosmic Microwave Background

The conditional transverse normalization of Section 15 is now applied to the early universe. The source-allocation cap is applied to the estimated recombination-era demand through the recruitment prescription below. The galactic force law itself has an unbounded low-acceleration boost. The resulting saturated phase contains a conserved count of committed channels that redshifts as \(a^{-3}\). The zero-pressure constraint theorem below is conditional on the pinned realization of that phase, and the numerical abundance inherits the transverse coefficient \(\epsilon\).

Saturation requirement at recombination. Use \(y \equiv g_{\text{pert}}/a_0(z)\) for the acceleration ratio and \(x \equiv \sqrt{y}\) for the thermal argument of Section 15. The linear-domain estimate \(y \lesssim 2 \times 10^{-3}\) today and \(y \propto \sqrt{a}\) gives \(y \lesssim 6 \times 10^{-5}\) at \(z \simeq 1100\), hence \(x \lesssim 8 \times 10^{-3}\) and \(\nu(x) \gtrsim 10^2\). This exceeds the total source-allocation cap \(1 + 1/\epsilon = 6.32\). Under the adopted demand-to-allocation mapping, the estimate requires saturation for those perturbation modes. The relation \(a_0(z) = \epsilon cH(z)\) keeps the excess ratio \(cH/a_0 = 1/\epsilon\) fixed at every redshift; adding the ordinary unit response gives the total cap.

The RAR response \(\nu(y) = [1 - e^{-\sqrt{y}}]^{-1}\) exceeds \(1 + 1/\epsilon\) whenever

$$y < [\ln(1 + \epsilon)]^2 = 0.02966\ldots.$$

The number \(1 + 1/\epsilon\) therefore bounds the adopted committed allocation per baryonic source; it cannot be a universal ceiling on the galactic acceleration ratio. The recombination estimate assumes a mapping from the unsaturated response demand to that allocation. A cross-phase microscopic calculation must establish this mapping and the recruitment dynamics.

Absolute capacity, source-normalized recruitment, and coherence. Three quantities that answer different questions must be kept separate. The availability field \(q\) measures the absolute local capacity remaining for ordinary transactions; in the static normalization it obeys \(q = 1 + 2\Phi/c^2 + O(c^{-4})\). It is not a universal identity with \(-g_{00}\) in arbitrary FLRW coordinates. Recruitment saturation is instead defined relative to each source's cap. For a Lagrangian source element \(A\) of baryonic mass \(M_A\), let \(M_{c,A}\) be the mass-equivalent committed allocation assigned to that source. Per-defect bookkeeping gives the cap and its normalized occupancy,

$$M^{\text{max}}_{c,A} = \frac{M_A}{\epsilon}, \quad \boxed{\sigma_A = \frac{M_{c,A}}{M^{\text{max}}_{c,A}} = \epsilon\frac{M_{c,A}}{M_A} \in [0, 1].}$$

The label \(A\) is physical bookkeeping: allocations belonging to distinct defects may overlap spatially, but they cannot be counted twice. In the continuum \(A\) becomes a Lagrangian source label, so \(\sigma_A\) is source-supported rather than a local projector attached independently to every cell.

Coherence is likewise a property of the committed flow, not a microscopic Boolean register. Write \(\mathcal{C} = 1\) when the source-labelled currents combine into one hypersurface-orthogonal, single-stream flow with a single-valued renewal phase; locally this requires vanishing curl of the normalized flow, and globally it requires vanishing phase holonomy. At first shell crossing the Lagrangian map loses invertibility and several velocities occupy one Eulerian point, so \(\mathcal{C}\) falls to zero. Loss of single-stream coherence ends the single-potential pinned description. It does not erase the Lagrangian source label or reroute the committed allocation: source-labelled multistream stock remains conserved until the separate carrier-loss condition \(\mathsf{E}_A = 1\) is met. The physical pinned branch is therefore

$$\boxed{\sigma_A = 1 \text{ for every recruited source}, \quad \mathcal{C} = 1,}$$

and the subsequent sequence is

$$\boxed{\begin{array}{c}\text{coherent pinned committed capacity}\\\downarrow\\\text{source-labelled multistream committed stock}\\\downarrow\\\text{carrier loss and rerouting}\end{array}}$$

generally with the absolute availability still close to its vacuum value. Terminal strong-field saturation is the different condition \(q_{\text{geo}} = 0\). Cosmological pinning saturates a small active subsector; it does not exhaust the vacuum budget.

The pinned reading and the zero-pressure constraint theorem. The pinned reading is the statement that the saturated bath holds exactly at the source-normalized ceiling and stays there. The transfer law quantizes channel transport at one bit per tick as an upper bound; saturation is the attainment of that bound, so on the pinned reading every committed channel advances by exactly one unit per tick, and while the phase remains pinned no channel can decommit, since there is no slack for it to relax into. The committed count is then conserved and dilutes only with the expanding volume.

The unit-timelike normalization is not a further independent condition once saturation and coherence are granted. Let \(\theta\) be the coherent renewal phase, increasing by one on each electron-calibrated interval \(\tau_*\). The dimensionful clock \(\phi = \tau_*\theta\) therefore satisfies \(d\phi/d\tau = 1\) along a committed worldline. Coherence identifies the flow covector as \(u_\mu = -\nabla_\mu\phi\). Hence

$$1 = \frac{d\phi}{d\tau} = u^\mu\nabla_\mu\phi = -u^\mu u_\mu, \quad \boxed{X = -\frac{1}{2}\nabla_\mu\phi\nabla^\mu\phi = \frac{1}{2}.}$$

Thus the spectral tick fixes the normalization of the coherent clock; no cosmological coefficient is inserted. This proves the constraint within the pinned coherent branch. It does not derive the transition that makes \(\sigma_A \to 1\) and \(\mathcal{C} \to 1\).

The coarse-grained field \(\phi\) consequently carries a constraint rather than a generic kinetic term. The leading action consistent with it is

$$S_{\text{comm}} = \int d^4x \sqrt{-g}\,\lambda\left(X - \frac{1}{2}\right),$$

whose variation gives \(\nabla_\mu(\lambda\partial^\mu\phi) = 0\) and, on the constraint surface,

$$T_{\mu\nu} = \rho u_\mu u_\nu, \quad \rho = E_0 n, \quad p = 0, \quad c_s^2 = 0, \quad \rho \propto a^{-3},$$

with \(u_\mu = -\partial_\mu\phi\) and \(n\) the conserved spatial density of committed cells. This belongs to the constrained-scalar class [97]. Conditional on the pinned reading, this is the zero-pressure stress form of saturated committed capacity. It is an effective metric statement about the pinned capacity phase, not a particle ontology. The construction inherits caustic formation and the need for a small-scale completion. The manuscript does not derive the transition from the unconstrained weak-field branch onto \(X = \frac{1}{2}\) or the commit/decommit energy accounting. Related relativistic Milgromian dynamics provides a useful comparison [34].

Uniqueness of the carrier. The constraint coupling is the unique survivor of the dynamics classes examined (Appendix M). Relaxational response kernels are excluded twice over: a growth clock calibrated on the cluster radial decline misses cosmological development by a factor of order thirty, and the precision of the measured acoustic peaks bounds any oscillatory leakage of a lagged response below one part in five hundred, a rejection no causal filter achieves over the few oscillation periods available before recombination. Bound-type rail readings sit at the sound-speed pole and carry no perturbation; plateau approaches have \(c_s^2 = -1/(2n+1) < 0\) and are gradient-unstable; generic unconstrained scalar continuations have nonzero sound speed and are not adopted as post-release degrees of freedom. A general exclusion of relaxation-plus-cap carriers is recorded in Appendix M; the conserved committed density is precisely the additional integrating variable that the exclusion requires and the constraint reading supplies. Carrier loss does not continue this constrained field into a new scalar branch: it removes the source-label saturation condition and returns the metric-visible remainder to the ordinary source-associated capacity bookkeeping described in Appendix O.

Abundance. At full saturation the committed weight is the ceiling, giving

$$\frac{\Omega_c}{\Omega_b} = \frac{1}{\epsilon} = \frac{4\pi^2}{g_{\text{share,eff}}} = 5.321 \quad \text{against the measured } 5.364 \pm 0.065,$$

a \(0.7\sigma\) agreement with the abundance derived from the commitment count [73]; equivalently \(\Omega_m/\Omega_b = 1 + 1/\epsilon = 6.32\). A standard Einstein–Boltzmann computation [100] with the cold component tied to this value and the acoustic scale \(\theta_*\) held fixed reproduces the quoted Planck best-fit temperature spectrum with a root-mean-square residual of 0.15% over \(\ell = 2\text{–}2500\) and 0.08% in the third-peak region. At the likelihood level, the tied model carries one fewer free parameter than ΛCDM and sits at \(\Delta\chi^2 = +2.15\) against the six-parameter best fit on the compressed Planck TT,TE,EE likelihood [56]. This fixed-\(\theta_*\) comparison tests the inherited committed-capacity abundance only. It does not demonstrate the separate Section 19 claim that the homogeneous mode reduces the sound horizon. A joint Boltzmann calculation must evolve the homogeneous mode, commitment transition, and tied committed-capacity component together; that calculation remains open.

The acceleration and abundance relations contain a parameter-free structural test that does not depend on the numerical value of the phase-cell coefficient. Since \(a_0 = \epsilon cH_0\) and \(\Omega_c/\Omega_b = 1/\epsilon\),

$$\boxed{a_0\frac{\Omega_c}{\Omega_b} = cH_0.}$$

Using the RAR scale \((1.20\pm0.02)\times10^{-10}\ \text{m s}^{-2}\), the Planck abundance ratio \(5.364\pm0.065\), and \(H_0 = 67.4 \pm 0.5\ \text{km s}^{-1}\ \text{Mpc}^{-1}\) gives

$$\frac{a_0(\Omega_c/\Omega_b)}{cH_0} = 0.983 \pm 0.022$$

[3, 73]. This two-percent agreement tests the reciprocal structure of the two sectors; the quoted band uses the random error on the RAR scale, and its mass-to-light systematic widens it accordingly. It does not derive either conditional premise: the transverse loading controls \(a_0\), while the pinned recruitment reading controls the abundance.

The abundance comparison also quantifies how closely recruitment must approach its cap. Define the mass-weighted saturation fraction

$$\bar{\sigma} := \frac{\rho_c}{\rho_b/\epsilon} \leq 1.$$

The prediction becomes \(\Omega_c/\Omega_b = \bar{\sigma}/\epsilon\). With the measured value above, agreement within one standard deviation requires

$$\boxed{\bar{\sigma} \geq 0.9959,}$$

and the two-standard-deviation requirement is \(\bar{\sigma} \geq 0.9837\). For the minimal cumulative recruitment law \(\sigma_A = 1 - e^{-\mathcal{J}_A}\), these bounds require a mass-weighted exposure \(\mathcal{J} \gtrsim 5.50\) and 4.12, respectively. The numerical success therefore tests near-complete recruitment rather than a broad partially saturated regime. The capacity cap fixes the endpoint; the microscopic recruitment rate and the epoch at which these exposures accumulate remain dynamical inputs.

That the committed density attains the ceiling exactly, rather than a development-dependent fraction of it, follows only within the recruitment assumptions of this branch. In the notation above, the assumption is \(\sigma_A \to 1\) for every recruited Lagrangian source element. Commitment is source-driven: cells receiving no demand commit nothing, and each source's allocation is capped at \(1/\epsilon\) per unit source mass. In the recombination estimate above the relevant modes demand \(\nu \gtrsim 10^2\), far above the cap, while the homogeneous background remains below it. With abundant supply and irreversible commitment in the pinned regime, each source recruits to its cap and allocations are assumed to add. The committed density is then

$$\rho_c = \frac{1}{\epsilon}\rho_b$$

pointwise at a commitment epoch. The manuscript has not derived that epoch or shown that commitment completes before the modes used in the Boltzmann calculation begin their relevant acoustic evolution. The initial condition \(\delta_c = \delta_b\) is therefore an explicit assumption of the current numerical check, not an output of the recruitment argument. The abundance remains conditional on the pinned reading and per-defect bookkeeping; the transition history and energy accounting are open. Appendix O.6 registers a first-crossing computation that falsifies the late pointwise-tracking reading outright — the Silk-damped committed field never reaches a caustic — and, jointly with the Einstein–Boltzmann check above, selects pre-acoustic commitment; the recombination-era saturation estimate must be re-derived at that earlier epoch.

Source-clock conservation and release. The saturated committed-capacity phase remains source-associated while its baryonic Lagrangian carrier is distinguishable. For each source \(A\), the phase \(\theta_A\) is the renewal clock on the source worldline; \(c_{A,c}\) is the identity-free committed allocation. Shell crossing can create several Eulerian streams, but it cannot create a which-stream record in the one-register capacity action: exchanging identical stream assignments leaves the occupancy history unchanged, so

$$|e_i\rangle = |e_j\rangle, \quad \gamma_{ij} = 1, \quad C^{\text{rec}}_A = 1.$$

Carrier loss occurs only when matter itself ceases to preserve the source clock: gas from several progenitors becomes irreversibly mixed in one retained common well and baryonic records no longer resolve progenitor membership. For each elementary record block that loses its carrier, faithful renewal fixes a complete fresh draw,

$$\boxed{\mathsf{E}_A = 1[\text{the baryonic carrier of } A \text{ has lost progenitor identity}], \quad f_{\text{reroute}} = 1.}$$

The normalized nine-state dilation of Appendix O.13 is isometric; its \(1/3\) marked amplitude normalizes nine outputs and is not a routing fraction. At coarse resolution, Mass–Entropy Equivalence and additive record pricing assign released stock in proportion to the source-label information no longer recoverable from matter. Thus partial \(\epsilon_{\text{mix}}\) describes a fraction of fully renewed record blocks, rather than a partial-memory update of any one failed block.

The enabling well depth remains a collective matter condition rather than a microscopic capacity threshold. Atomic-cooling assembly supplies \(V_{\text{thr}} \sim 17\ \text{km s}^{-1}\) before reionization, while photoheated gas after reionization requires a scale nearer \(30\ \text{km s}^{-1}\). The primitive action contains no local switch at either value. The transverse scale \(a_0(z) = \epsilon cH(z)\) and the source-clock release gate therefore have different roles.

Appendix O measures carrier loss by

$$\epsilon_{\text{mix}}(t) = 1 - \frac{I(A; Y_t)}{H(A)},$$

with \(A\) the Lagrangian progenitor label and \(Y_t\) the retained coarse baryonic carrier. First eligible assembly starts the mixing clock rather than causing instantaneous release. The cosmological history is therefore

$$\boxed{F_{\text{rel}}(t) = \int^t K_{\text{mix}}(t - t_f; M, z)\, dF_Z(t_f).}$$

Appendix O.12 fixes \(F_Z\) by an absorbing first-entry excursion-set construction and directly evaluates \(K_{\text{mix}}\) in a three-dimensional compressible-flow passive-tracer surrogate. For progenitors distributed through the full virial volume the surrogate gives \(\epsilon_{\text{mix}}(70\ \text{Myr}) \simeq 0.20\), whereas centrally concentrated progenitors mix much faster. In first-galaxy halos cooled gas falls into a supersonically turbulent core, and turbulence mixes a passive scalar injected near its driving scale in about one dynamical time of that scale. Taking the kernel's e-folding time to be \(f R_{\text{vir}}/V_{\text{vir}}\), with \(f = 0.55\text{–}1.1\) from infall and variance-decay estimates, gives \(\Omega_{\Lambda,0} = 0.68^{+0.03}_{-0.04}\) with the retention-gate, reionization, and late-accretion shifts added in quadrature (Appendix O.12). The dynamical-time form and the order-unity range of \(f\) were adopted with the observed abundance known. The required astrophysical calculation remains a cosmological self-gravitating zoom with passive progenitor tags; its kernel must be fixed independently of the cosmological likelihood.

Carrier loss does not create a new material component. The replaceable closure register returns to the universal renewed state, while the metric-visible remainder stays associated with the retained coarse carrier. The marked history output has no independent spacetime current. The unique volume-singlet projection of its universal drain changes the homogeneous cosmological term; every nonzero spatial mode remains internal to the source-associated capacity bookkeeping. Thus the background is described by one metric-visible capacity density,

$$\boxed{\rho_C(a) = \rho^{\text{wb}}_{c,0}a^{-3}[1 - \eta_* F_{\text{rel}}(a)],}$$

together with

$$\boxed{\rho_\Lambda(a) = \eta_*\rho^{\text{wb}}_{c,0}\int^a a'^{-3}\, dF_{\text{rel}}(a').}$$

There is no independent post-release scalar or local \(\Lambda(x)\) branch. Appendix O.2–O.3 gives the volume-projection theorem and the resulting perturbation Ward identity.

Two consequences of the gate for other sectors. The release condition is collective and gated by retention, so a halo that never crosses the gate never undergoes source-clock release. A source whose gas is expelled before irreversible progenitor mixing preserves its source-associated saturated allocation at the ceiling \(\rho_b/\epsilon\), 5.3 times the mass of its Lagrangian baryons. That allocation is the response stock of Appendix O.3, not a component added to the response. Its equilibrium follows the response law of the baryons that remain, and the larger budget per remaining baryon moves the cap radius \(r_C = r_M/\ln(1 + 1/C_R)\) outward. The radial-acceleration relation of the faintest dwarfs and their outer lensing profiles are the corresponding tests. For this reason Section 15 needs no additional collisionless matter species. Under the present release law the detailed low-mass behavior still depends on whether the source remains in the saturated source-associated phase or exits it through the collective carrier-loss channel. The faintest retained systems are a cross-sector test of that bookkeeping, not evidence for a second material component.

Distinction from horizon saturation. The committed cosmological phase — channels advancing at capacity, a running clock — is distinct from the terminal saturation of the strong-field sector (Section 21), where capacity is exhausted and transactions cease. Whether the two termini are stages of one process is open and carries a stated consistency burden: the gravitating energy of committed capacity must coincide with the mass already accounted at infinity, with no double counting. This is recorded as an open question shared by the strong-field and cosmological sectors.

Part VI. Strong Fields, Many-Pasts, and Microstructure

The strong-field branch treats the zero-capacity boundary, and Many-Pasts supplies the record-conditioned history ontology in which the substrate's renewal process operates. The separate refresh theorem selects the memoryless dressing used in scale setting, while Section 23 asks for its microscopic dynamics. The same grading applies here: the exact spherical reduction and the operational quantum measure are controlled results, while the boundary microphysics, the arrow of time, and the condensate embedding carry their stated conditional or open grades.

21. Strong-Field Action: Spherical Closure and Its Boundary

The bounded variable remains the natural strong-field order parameter,

$$q = \frac{S_{\text{ent}}}{S_\infty} \in [0, 1].$$

Serial composition and weak-field matching select \(N^2 = q\) in a static exterior. That relation is meaningful after the asymptotic Killing time has fixed the normalization of the lapse; it is not, by itself, a covariant field equation in a general foliation.

Why a multiplier action is not the parent. An exploratory ADM term \(\sqrt{h}\,\lambda(N^2 - q)\) makes the problem look variational, but if \(q\) has no independent bulk dynamics or matter coupling its variation gives \(\lambda = 0\). The remaining constraint merely renames the lapse, which is gauge dependent, and supplies neither the capacity Poisson equation nor a new covariant relation. Giving \(q\) its own kinetic term would instead add a scalar degree of freedom and reopen the fifth-force, stability, and double-counting problems. The multiplier construction is therefore a diagnostic control case, not the parent action.

Exact spherical reduction. Spherical symmetry supplies the invariant completion that the generic lapse constraint lacks. Write

$$ds^2 = h_{ab}(x)dx^a dx^b + R^2(x)d\Omega^2, \quad a, b \in \{0, 1\}, \quad x^0 = ct.$$

After integrating the Einstein–Hilbert plus GHY action over the two-sphere and removing a two-dimensional total derivative, the bulk action is

$$I_{\text{sph}} = \frac{c^3}{4G}\int d^2x \sqrt{-h}\left[R^2\,^{(2)}R + 2h^{ab}\partial_a R\partial_b R + 2\right] + I^{(2)}_{\text{matter}} + I^{(2)}_\partial.$$

Its vacuum equations imply conservation of the Misner–Sharp mass

$$M_{\text{MS}} = \frac{c^2 R}{2G}\left(1 - h^{ab}\partial_a R\partial_b R\right), \quad \nabla_a M_{\text{MS}} = 0.$$

The capacity-adapted invariant is therefore

$$\boxed{q_{\text{geo}} \equiv h^{ab}\partial_a R\partial_b R = 1 - \frac{2GM_{\text{MS}}}{c^2 R}.}$$

Thus in the preferred metric-only construction, spherical \(q\) is a composite geometric scalar and its vacuum profile is a first integral of the metric equations. It is not an auxiliary field and it is not an additional propagating mode.

For a static asymptotically flat vacuum, \(M_{\text{MS}} = M\) and Birkhoff's theorem gives

$$ds^2 = -\left(1 - \frac{2GM}{c^2 r}\right)c^2 dt^2 + \left(1 - \frac{2GM}{c^2 r}\right)^{-1}dr^2 + r^2 d\Omega^2,$$

so

$$q_{\text{geo}} = N^2 = 1 - \frac{2GM}{c^2 r}.$$

This is the nonperturbative action-level realization of the bounded capacity relation in the strongest sector where an invariant local definition is currently available.

The \(q = 0\) surface. The equation \(q_{\text{geo}} = 0\) locates a marginal sphere. A well-posed exterior variational problem is obtained first on a stretched timelike boundary \(q = \epsilon > 0\) with the usual GHY term and fixed induced data, and then by taking the null limit with the corresponding null and joint terms. The Einstein action determines that universal gravitational boundary bookkeeping. It does not determine a new substrate boundary Hamiltonian, microscopic reflectivity, or the rule that excises \(q < 0\).

Accordingly, the restriction

$$\mathcal{M}_q = \{q_{\text{geo}} > 0\}$$

is a physical postulate about the domain of the capacity EFT, not a consequence of varying the Einstein action. The exterior solution and horizon location are closed within spherical metric reduction; the claim that the classical interior is absent, and the dynamics experienced at the saturation surface, remain conditional on a microscopic boundary theory. Appendix F and Appendix N state this boundary between result and interpretation explicitly.

Status of the construction. The Schwarzschild exterior, its standard Hawking temperature, and ordinary exterior perturbation equations follow from the metric parent. In the factorized \(j = 3\) completion, the one-bit cut and coherent EPRL area now fix the channel area and Immirzi normalization (Appendix F.5); the older Green-response \(1/4\) identity remains a separate normalization cross-check. Rotating and charged GR exteriors remain valid solutions of the baseline parent action, but a covariant capacity scalar that identifies their saturation surface has not yet been constructed. The universal strong-field result is therefore spherical and exterior; boundary microphysics, dynamical saturation, nonspherical capacity geometry, and any controlled departure from GR remain open.

22. Many-Pasts: The History-Space Ontology

Many-Pasts makes a conservative operational claim and a distinct ontological claim. Laboratory records obey standard quantum mechanics. The ontology assigns a conditional measure to the decoherent past histories compatible with the one realized present.

The operational weight. For a decoherent family, the joint history weight and its record-conditioned form are

$$p(h, P) = \mathcal{D}(h, h), \quad p(h \mid P) = \frac{\mathcal{D}(h, h)}{\sum_{h'\in\mathcal{H}_P}\mathcal{D}(h', h')}.$$

Equivalently, one may define \(D(h, P) = -\ln p(h, P)\) on the support of the decoherent joint measure, so that \(p(h \mid P) \propto e^{-D(h,P)}\). The exponential is a reparameterization of a normalized quantum probability, not an additional classical measure placed on interfering paths. Appendix H.11 derives the finite retained-sector norm, instruments, and history Gram kernel; Appendix G applies them to Many-Pasts and gives the no-signaling calculation. The global cosmological state and continuum realization remain separate inputs.

Branch realization without many worlds. The weight also answers what it means for one outcome to be realized. There is no forward branching into co-real worlds and no collapse event. A definite present is a present with definite macroscopic records, and the histories the weight supports are exactly those compatible with those records; alternative outcomes correspond to alternative present records, not to coexisting branches. Probability is the measure this weighting assigns over the admissible pasts of the one realized present.

Record retention and the resolution of the past. The physical degrees of freedom that distinguish histories are present records. A durable record can be a detector state, an environmental correlation, or the exported history register of the renewal dilation. Such a record keeps distinct those past alternatives that remain distinguishable in the present record algebra. If orthogonal fine records \(R_i\) are later merged into a coarser record \(\bar{R} = \sum_{i\in I}R_i\), decoherence and additivity give

$$p(h, \bar{R}) = \sum_{i\in I}p(h, R_i), \quad p(h \mid \bar{R}) = \frac{\sum_{i\in I}p(h, R_i)}{\sum_{h'}\sum_{i\in I}p(h', R_i)}.$$

The history measure therefore loses resolution by ordinary quantum coarse-graining when the present loses a record. The ontology contains no separate archival copy of a distinction that has disappeared from every physical register. This does not erase consequences already carried into the current state. It identifies earlier alternatives with identical surviving records as the same present physical state.

No future pull. The class operators and unitary maps determine the joint measure before conditioning. The factor \(p(h \mid P)\) updates the description of earlier alternatives after \(P\) is recorded; it does not enter the Hamiltonian, the channel, or the class operator that produced \(P\). A record that has not yet formed supplies no conditioning event in the realized present. Many-Pasts therefore adds no retrocausal force, final-boundary dynamics, or Born-rule bias. It gives a physical reading to retrodictive conditioning already present in standard quantum mechanics.

Familiar quantum examples. In a double-slit experiment, the alternatives through the two slits remain inside one coarse class operator until a durable which-path record decoheres them. Their cross terms are therefore retained before the record and suppressed after it. In an EPR or Bell experiment, the present is a joint record of the pair and the detectors. The joint measure gives the usual nonclassical correlations, while the local marginals remain independent of the remote setting. Measurement creates a stable record and thereby identifies the decoherent family on which conditional probabilities can be used. These are standard quantum calculations with a record-conditioned history-space reading.

The arrow of time. The same framework proposes to orient time through conditional typicality. The maximum-caliber replacement process does not solve this problem. At stationarity it obeys detailed balance identically, \(p(b)K_*(b, b') = p(b')K_*(b', b) = p(b)p(b')\), and is time-reversal symmetric as a stochastic process. The reversible dilation also has an inverse. Histories with a low-entropy past and increasing future entropy dominate only after a past-boundary condition and a substrate mixing or large-deviation theorem suppress Boltzmann-fluctuation histories. This is not an added law of laboratory probability; it is the open typicality result stated precisely in Appendix G.7.

Bianconi supplies a useful compatibility result, but not that missing theorem. In her low-curvature matter- and radiation-dominated Friedmann approximations, the local geometric-relative-entropy density decreases as the universe dilutes while the integrated entropy grows with the expanding volume and the integrated energy approaches a constant [28]. This demonstrates in a concrete information-geometric action that local ordering and a global entropy increase need not conflict. It does not select a low-entropy boundary, establish substrate mixing, or show that the Many-Pasts conditional measure favors ordinary histories over Boltzmann fluctuations. The process-level arrow therefore remains open exactly where Appendix G.7 places it.

No external cycle ledger. If a proposed cosmological evolution reaches a state whose present record algebra contains no witness of an earlier macroscopic era, the descriptions "first beginning" and "return to the same beginning" are not distinct states within this ontology. Distinguishing them would require a register that survives the record-free interval or an external time parameter that counts passages. Many-Pasts supplies neither. This conditional observation does not derive a cyclic universe, a unique beginning, or a nucleation rate. It removes an otherwise hidden meta-history from the ontology.

Where the weight is used. Many-Pasts supplies the history-space realization in which the electron dressing operates, while faithful full-support resolution proves the independence of successive passes. Appendix H shows that this condition is equivalent to maximum path entropy and uniquely selects the replacement process; the lightest-defect criterion selects the same temporal independence together with independent channel layers. The record-conditioned viewpoint also enters the proposed macroscopic arrow of time and the conditional cosmological extensions. It changes no laboratory law.

Local renewal and reversible history export. Complete local renewal has a precise quantum form. If the new cell must contain no information about the old cell even when the old cell is entangled with a reference, the one-tick channel is uniquely

$$\mathcal{E}_*(\rho) = \rho_*\text{Tr}\,\rho.$$

This local channel need not destroy information globally. A reversible dilation transfers the old cell into an environment register while a fresh admissible register becomes the new present. Many-Pasts interprets the exported correlations as history degrees of freedom. The identification is structural: the finite dilation follows directly from the retained tensor factors, while Postulate III supplies its history-space reading. It does not select the cosmological initial state, prove that the full continuum histories decohere, or show that an indefinitely long history can be stored in finite microscopic resources.

Coherence under renewal. The replacement channel acts only on the renewed closure register. Its Heisenberg dual is

$$\mathcal{E}^\dagger_*(O) = \text{Tr}(\rho_* O)I,$$

so no later observable of that register can recover an input off-diagonal. The global dilation can nevertheless retain coherent information in its complement. If two spatial branches export discarded states \(|e_x\rangle\) and \(|e_y\rangle\), their reduced off-diagonal is multiplied by \(\langle e_y|e_x\rangle\). A cell-addressed mark record makes those states orthogonal and would destroy position coherence in one update. The allowed free vertex instead transports the marked fiber as part of the coherent defect system and leaves the discarded renewal record branch-independent. Ordinary environmental interactions may then reduce the overlap in the usual way. Appendix H.11 gives the proof and distinguishes this condition from the weaker statement that a record merely "moves with" a worldline.

The retained quantum sector. Appendix H.11 now settles the local question within the selected quadratic completion. Unitarity identifies the nine-state fusion complement with the event response coordinates of the persistent marked fiber. On its real form, the positive Lorentzian Hamiltonian selects an initial polar complex structure. The full symplectic flow transports that structure exactly through arbitrarily varying frames; squeezing relative to the instantaneous polar basis and the induced Born–Huang term are retained rather than discarded. The non-common frequency and moving frames generate \(\mathfrak{su}(9)\) and give \(\mathcal{H}_{\text{mark}} \simeq \mathbb{C}^9\). The complete fixed-one-mark bilinear response algebra identifies normalized states exactly up to common phase, yielding \(\mathbb{CP}^8\) and its Fubini–Study geometry. Within the H.9 Gaussian-strand realization, the fresh tangent metric also equals one quarter of the Fisher metric, conditional on identifying its Hermite response coordinates with the closure-score coordinates. Caticha's Hamilton–Killing theorem is therefore a consistency and continuum guide rather than the premise carrying the local reconstruction [29, 30].

Status. Finite operational quantum mechanics is derived on the retained sector. The real positive quadratic dynamics gives the complex Hilbert bundle and exact unitary transport; complete commuting local algebras give tensor composition; reversible record coupling gives POVMs, instruments, conditional updates, and reduced channels; reversible equivalence plus physical-record additivity fixes the Born norm; and the history functional is the resulting Gram kernel. The projected two-mark bridge generates \(\mathfrak{su}(81)\) provided the multiplicative two-mark carrier and bridge survive the final host projection. Extension to a connected marked network additionally requires simultaneous retained marks and surviving bridges. The global continuum/Fock realization, cosmological state, identification of the histories that decohere, long-time record capacity, electromagnetic charge and dynamics, and the thermodynamic arrow retain their open or conditional grades.

23. Microstructure Hamiltonian and Underlying Dynamics

The UV closure chain now has an explicit finite action on the scale-setting side and a controlled geometric witness at the semiclassical level. Appendix H derives the replacement process, its quantum channel, the state-weighted determinant transfer, and the decorated native-cell vertex used below.

The selected factorized geometric completion is not obtained by identifying the finite capacity alphabet with the magnetic indices of the geometric spin network. The microscopic one-cell space is instead taken to factor as

$$\boxed{\mathcal{H}_{\text{cell}} = \mathcal{H}_{\text{geom}} \otimes \mathcal{H}_{\text{cap}} \otimes \mathbb{C}^2_{\text{orient}},}$$

Here \(\mathcal{H}_{\text{geom}}\) carries the gauge-reduced simplicial GFT/spin-foam data, while \(\mathcal{H}_{\text{cap}}\) carries the finite routed capacity alphabet and its marks. Appendix Q shows that this factorization removes the Gauss-Casimir contradiction without changing \(\Omega = 1680\), the \(K^2\) spectrum, \(\eta_*\), \(g_{\text{share,eff}}\), \(\zeta_*\), \(Z_e\), \(L_*\), or \(G_*\). An equivariant relational-frame lock relates the two selected copies of \(V_3\). Once the lock exists, Schur's lemma fixes it up to phase. The two factors are bookkeeping components of one substrate, and Postulate I restricts physical configurations to their locked sector.

On the capacity factor, Appendix B proves that positive oriented matching has \(V_3\) as its unique zero-cost transmitted sector and that the displayed homogeneous action has a unique \(J = 3\) minimum. Maximal fusion then gives the exact tree-independent canonical blocking ray on the selected maximal-channel branch

$$F_N : V_3^{\otimes N} \to V_{3N}, \quad F_N|3, n\rangle^{\otimes N} = |3N, n\rangle.$$

The microscopic representation stays fixed while blocks enter the large-spin regime along this canonical ray. Several external constructions support the geometric limit. Proper-EPRL amplitudes remove unwanted asymptotic sectors at the vertex level and retain the Regge exponential [123, 124]. Han-type spin-foam models provide a coupled large-spin/refinement witness to the Einstein sector; in the controlled linearized construction, the low-energy excitations exhaust the smooth linearized Einstein solutions and the two graviton helicities [121, 122]. Area-Regge continuum analysis gives the same leading graviton dynamics with a first length-metric correction of order \(a^2 C^2\) [125]. Applying the Han-type refinement result to the decorated proper-EPRL factor remains a coupled-embedding audit.

The scalar capacity deficit is a coordinate on the Hamiltonian-constraint response, not an additional radiative gravitational field. Gauge-reduced vacuum propagation remains transverse-traceless. A GFT condensate may still realize the microscopic state and clock [82, 83], but Einstein propagation does not require a separate light scalar condensate mode. Appendix H.10 lifts any geometric rigging map through the finite capacity factor and proves clock-test positivity and the causal boundary value from strong-resolvent convergence. On the selected regular metric-Regge branch, Appendix H.10 proves \((H_{\text{TT}})\): the Einstein TT infrared limit, together with an exact fixed-\(j = 3\) primitive realization whose nonmaximal complement remains uniformly gapped. It also proves that the present capacity premises leave the geometric gluing tensor and measure underdetermined. A new geometric principle must fix those data before \((H_{\text{phase}})\), dynamical selection of the Einstein branch, becomes a well-posed derived theorem. A spin-3-regularized canonical host obeys the distinct condition \((H'_{\text{can}})\) on its zero fiber; identifying the two physical sectors would require a separate theorem and is not assumed here.

On the defect side, the closure ensemble fixes the stationary marginal \(p_{\eta_*}\) and maximum path entropy selects

$$K_*(b, b') = p_{\eta_*}(b').$$

This fixes the dimensionless history process but cannot produce seconds. The electron, already the theory's single dimensionful anchor and its lightest charged one-bit defect, supplies the clock through the positive transfer spectrum below.

Requiring the output cell to decouple from every reference system forces the replacement channel \(\mathcal{E}_*(\rho) = \rho_*\text{Tr}\,\rho\). The decorated vertex chooses the diagonal maximum-entropy completion

$$\rho_* = \sum_b p_*(b)|b\rangle\langle b|, \quad p_*(b) = Z^{-1}e^{-\eta_* K^2(b)},$$

and prepares it from the amplitude \(A_*(b) = \sqrt{p_*(b)}\). A reversible register permutation moves the old cell into history and the fresh amplitude into the present. For the complete 1680-state tetrahedral register this is one native circuit layer. A direct circuit made only from disjoint face comparisons still requires the three perfect matchings of the four-face graph; the decorated action uses the complete tetrahedron as its local gate.

On the factorized seven-channel history space let

$$|v^{(7)}\rangle = \bigotimes_{m=-3}^3 |\sqrt{p_{\eta_*}}\rangle_m, \quad P^{(7)} = |v^{(7)}\rangle\langle v^{(7)}|.$$

On the commutative cell algebra, likelihood multiplication by \(p_*(b)\) has the state-weighted determinant

$$\Delta_{\tau_p}(R) = \exp\left(\sum_b p_b\ln p_b\right) = e^{-g_{\text{share,eff}}}.$$

The seven-channel determinant is \(r = e^{-7g_{\text{share,eff}}}\). It is the almost-sure geometric transfer rate of the record-conditioned renewal history; the annealed equality probability would instead involve the collision entropy. Compressing the one-bit charged loop to its determinant line gives the positive scalar survival transfer

$$T_{\text{surv}} = 1 - r.$$

The Euclidean transfer Hamiltonian

$$H_{\text{surv}} = -\frac{\hbar}{\tau_*}\ln T_{\text{surv}}$$

therefore has the exact raw gap

$$E_{\text{raw}} = -\frac{\hbar}{\tau_*}\ln(1 - r).$$

The closure amplitude has an exact Gaussian linearization. Projecting the same amplitude through the two directed scalar returns gives \(u = 8\eta_*/49\), and its canonical unitary dilation supplies the factor \((1 - u)^{21/2}\). The nine present/history closure polarizations give the marked weight \(\zeta_*\) of Section 13.4. The electron graph contributes \(Z_e = (1 + \zeta_*)(1 + 7\zeta_*^2)\). Assigning its motif weights to the charged generator as in H.9 gives the dressed transverse identification

$$E_e = \frac{3}{2}Z_e E_{\text{raw}}.$$

Identifying this lowest charged transfer gap with the electron rest energy fixes the update time,

$$\boxed{\tau_* = -\frac{3}{2}Z_e\frac{\hbar}{m_e c^2}\ln(1 - r).}$$

Analytic continuation supplies the phase frequency \(E_e/\hbar\). The invariant continuum speed converts the spectral cadence into the causal length \(L_* = c\tau_*\). This definition does not determine the displacement made by a microscopic spatial update.

The causal length of that spectral update is

$$\boxed{L_* = c\tau_* = -\frac{3}{2}Z_e\lambda_e\ln\left(1 - e^{-7g_{\text{share,eff}}}\right).}$$

The baseline rare-event correction \(-\ln(1 - r) = r[1 + O(r)]\) differs from \(r\) only at order \(10^{-23}\). The finite marked response changes the scale by \(Z_e - 1 = 0.00530828\), which is the physically relevant correction resolved by the microscopic vertex.

Retained temporal information raises the recurrence mass by \(e^{I_t}\), so electron lightness independently selects \(I_t = 0\). Fermionic exclusion caps the occupied channel count at seven and subadditivity gives \(\Delta_k \geq 0\), hence the lightest resolved one-bit defect has \(k = 7\) and \(\Delta_7 = 0\) at fixed per-channel marginal. The unit mixing gap, charged survival gap, and geometric spin-2 Hessian remain distinct objects. The selected transfer action, including its generator prescription, fixes the first two and the marked edge Hessian. In the ordinary geometric branch the source-coupled scalar response must be the nondynamical Hamiltonian-constraint combination; any additional scalar fluctuation of a particular condensate realization must be gapped, relaxational, gauge/constraint, or source-orthogonal. The physical radiative support is the two TT graviton modes.

The scale-setting branch combines the fixed renewal marginal, determinant-survival action, finite marked enumeration, additive-generator prescription, and electron calibration. Positivity gives the energy of the resulting transfer. The complete edge-Hessian and graph-enumeration audits are finite and reproducible. Long-time history capacity and a first-principles derivation of the carrier-resolved metric contact remain open; the ordinary semiclassical Einstein branch is supplied independently by the factorized spin-foam/GFT geometric sector.

The cell-order calculations show that the closure weight, refresh, history tilts, and conserved capacity field do not supply vacuum curvature stiffness. Under the factorized completion this is expected rather than a missing scalar mechanism: vacuum spin-2 stiffness belongs to the geometric factor, while the finite capacity sector supplies the source, records, and coefficient chain. The capacity variables do couple geometry to persistent closure failure, and Section 24 measures that local response. The CDT simulations remain an independent numerical host/interface rather than a derivation of the EPRL refinement limit. Postulate III conditions histories within a specified state and does not select the nonperturbative gravitational vacuum.

Part VII. The Substrate on a Dynamical Lattice

This part tests the cell ensemble numerically on a dynamical simplicial geometry, and then applies the resulting joint measure to the equilibrium vacuum and the cosmological term.

24. Lattice Tests: Compatibility, Defect Response, and Transport

The single-cell coefficient chain of Part II can be evaluated in closed form. Its many-cell consequences require numerical tests: survival of the vacuum ensemble on fluctuating geometry, local strain around a persistent closure failure, and transport toward the \(1/r\) deficit of Section 12. The calculation places the unchanged cell ensemble on a dynamical simplicial geometry. Appendix J gives the ensemble definitions, controls, and measured tables.

24.1 The host geometry and the cell identification

The host is a causal-dynamical-triangulations ensemble: four-dimensional triangulations of \(S^1 \times S^3\) weighted by the Regge action [103]. CDT is used because it independently sustains an extended four-dimensional de Sitter-like phase [104, 105]. That phase is an external host datum for the substrate test, not a state selected by Many-Pasts.

The identification is immediate, and it respects a separation the theory requires. Each spatial tetrahedron of a slice hosts one cell's boundary data: its four triangles carry the labels \(m = -3, \ldots, 3\) of the \(j_{\text{eff}} = 3\) sector of Section 5, each slice triangle is shared by exactly two cells, and the slice gluing supplies adjacency. Nothing else of the lattice enters. The theory's cell is its label configuration, a dimensionless unit of capacity; no size is ever assigned to it, and the granularity stays in the capacity, as Section 1.2 requires. The lattice simplices are regulator scaffolding, as they are throughout lattice gravity. The identification is nevertheless geometric in the sense that matters: the joint admissibility statistics of the labels depend on how the slice is glued, so the closure weighting can in principle tell one geometry from another, and that channel is the entire coupling between the theory and the host.

Two boundaries of the design are stated at the outset. First, the host supplies a dynamical, curvature-carrying simplicial geometry with the theory's cell structure, and nothing more is asked of it. The calculation does not assume the CDT ensemble is the theory's own vacuum, and it withholds any claim that the substrate itself derives four-dimensional emergence. At the tuned host couplings used below, increasing \(N_{41}\) from about 42,000 to 102,000 raises both the Hausdorff estimator and the concentration of volume in an extended region, as expected when moving farther into that finite-volume phase; the order of the visible phase boundary and the joint continuum limit remain open (Appendix J.1). Second, the weighting coupled to the host is the paper's own: the admissibility energy of Sections 5–6 at the closed point, with \(\eta = \eta_*\) and injectivity enforced. It is not a spin-foam vertex amplitude. The distinction drawn in Section 29.8 was kept operational: a spin-foam amplitude was coupled first as a neighboring-theory control, and the failures exposed by that exercise supplied the control methodology used below (Appendix J.3). The design's two-level closure structure — geometric closure exact by construction in the host, label closure soft — was once its most criticized feature; under the factor separation of Appendix Q it is the predicted architecture, with the exact level carried by the geometric/gauge host and the soft level carried by the separate finite capacity factor.

24.2 The coupled ensemble and its controls

The coupled system is the joint Gibbs measure

$$\pi(g, m) \propto \exp\left(-S_{\text{Regge}}(g) - \varepsilon(N_{41} - \bar{N})^2 - \beta\sum_{\text{cells }c}\left[E_c(m) - \mu\right]\right),$$ $$E_c(m) = \eta_* K^2(m_c) + \lambda n_{\text{coll}}(m_c),$$

where the sum runs over the slice cells, \(K^2\) is the closure invariant of Appendix B, \(n_{\text{coll}}\) counts label collisions (injectivity is imposed as a penalty whose hard limit is the constraint, with the residual collision fraction reported so the softness stays visible), \(\varepsilon\) pins the slice volume, and \(\mu\) is the per-cell label free energy, computed by thermodynamic integration so that the closure term cannot masquerade as a shift of the bare cosmological coupling. The labels carry a uniform base measure, so at \(\beta = 0\) the label entropy cancels exactly and the bare host is recovered identically. The physical weighting fixed by Part II is \(\beta = 1\) at \(\eta_*\): once that chain is accepted there is no coupling dial left free, and the intermediate \(\beta\) values serve only as a diagnostic interpolation.

Every run opens by recomputing the single-cell chain on its own tables and refuses to proceed unless \(g_{\text{share,eff}} = 7.4198\) and \(\langle K^2\rangle_{\eta_*} = 3/(2\eta_*) = 50.223\) are reproduced, so the object coupled to the lattice is verifiably the object counted in Part II. Every quoted triangulation is connected, simplicial, and correctly foliated, and comparisons use matched total volume at the same regulator. The principal control shuffles the 210 label-orbit energies while preserving their values and permutation symmetry. This destroys the closure structure without changing the energy histogram or sampling machinery, so only a difference from that shuffled control is attributed to closure. These validity conditions and the interpretation of each possible outcome were recorded before the corresponding data were examined (Appendix J.3).

24.3 What the closure sector cannot supply: vacuum stiffness

Three cell-order calculations bound the closure sector's contribution to vacuum stiffness. The static closure weight induces only about one percent of the bare geometric coupling. The selected memoryless kernel, modeled as cell-by-cell redraws, has a closed stationary state with vacuum admissibility 0.536 and no detectable curvature action. A separate closure-class history tilt, solved by a Doob transform on a ring, also remains short-ranged and well below the required coupling. These are calculations of specified effective models; they do not derive the refresh kernel from Postulate III.

Two further calculations extend the exclusion beyond the label sector, to the conserved capacity field that Section 24.7 introduces as the carrier of the long-range sector. Even a field the labels cannot see might rank geometries on its own, because integrating out a conserved Gaussian field induces a purely geometric action: \(\frac{1}{2}\ln\det' L(g)\) per channel, the spanning-tree entropy of the slice graph by the matrix-tree theorem. Computed on degree-matched proxies for smooth extended and for crumpled geometry, and on real engine slices, that entropy is nearly universal at fixed coordination: the per-cell differential is a few times \(10^{-4}\), and with all seven channels the phase-tipping force is of order \(10^{-3}\) against the bare \(k_0 \simeq 2.2\). The theory's own non-Gaussianity closes the loophole tighter still. The finite budget of Postulate I gives the field a saturation mass \(m^2 = 2/g_{\text{share,eff}} \simeq 0.27\), and the massive determinant suppresses precisely the soft modes that carried the residual sensitivity, shrinking the differential by a further factor of three at the budget mass and toward zero beyond. What the field retains is a local renormalization of the host's couplings, of order 0.08 per cell in the seven-channel count: enough to relocate the host's phase boundaries, and unable to create the phase.

Within the tested models, static weight, refresh, history tilts, and conserved fields do not supply vacuum stiffness. The refresh does couple geometry to the density of closure failure, which the defect experiment measures. The same pass verifies the locality fracture: injectivity-preserving unit shifts split the 1680 states into 48 components of 35 states. This excludes that local realization of the full-entropy kernel; it does not derive the nonlocal physical operator.

24.4 The externally hosted vacuum and the role of conditioning

The exclusions show that the specified substrate does not dynamically select its vacuum geometry. Two questions called "emergence of spacetime" must therefore be separated. The first is kinematic encoding: the closure invariant \(K^2\) measures the failure of the four oriented faces to close, adjacency carries proximity, and the marked-transfer dictionary of Section 13 converts entanglement increments into meters. The cells are dimensionless capacity units rather than sites of a preferred spatial grid. The second question is dynamical selection among crumpled, branched, and smooth extended geometries. Section 24.3 finds that none of the tested substrate mechanisms performs that ranking.

Many-Pasts cannot fill that dynamical gap. Appendix G normalizes the probabilities of alternative present records before conditioning, so \(p(h \mid P)\) describes histories compatible with an already specified present and does not select which \(P\) occurs. In the simulations the smooth extended vacuum is supplied by the chosen Regge/CDT host phase as an external background datum. Many-Pasts may condition the compatible decoherent histories within that datum, but it neither ranks candidate geometries nor explains why the extended present is realized. The coefficients of Part II are computed on that host state, and the medium dresses it rather than generating it. A substrate derivation of the prior state or of a probability measure over alternative vacuum geometries remains open.

Regge/CDT is a useful external host because it supplies tetrahedral cells, dynamical curvature, no rigid preferred spatial grid, and a demonstrated extended four-dimensional phase [104, 105]. This choice is operational, but the completed fixed-regulator runs are not a vacuum derivation by the substrate. The host supplies the geometry on which the label theory is tested; the substrate then produces quantitative dressing of its couplings. Integrating out the labels gives \(S_{\text{eff}} = S_{\text{host}} - \log Z_{\text{label}}\). Appendix J.6 reports the predicted extensive volume shift and its measured scaling family, including the independently predicted shuffled-control line. The corresponding curvature-sector shifts remain to be tested statistically. The hierarchy is explicit: cell-level non-closure reweights the glued tetrahedra, and integrating out the labels converts that reweighting into corrections to the host's volume- and curvature-sector couplings. This is presently a quantitative interface with an externally supplied vacuum geometry. Section 24.9 states the additional critical-surface test that would promote the interface to a continuum embedding.

24.5 Compatibility: the weighting on dynamical geometry

At \(N_{41} \simeq 100,000\), three eighty-slice ensembles were evolved from the same tuned host with respectively zero closure coupling, the physical weighting \(\beta = 1\), and \(\beta = 1\) after shuffling the closure energies among label orbits (Appendix J.5). Their total-simplex counts agree within 0.6%, and every final triangulation remains connected, simplicial, and correctly foliated. The label sector orders exactly as the closure weight demands: the collision fraction falls from 0.654 at \(\beta = 0\) (against the uniform-measure prediction \(1 - 840/2401 = 0.650\)) to 0.077 under the physical weighting, while the shuffled control stays high at 0.736. The Hausdorff values remain 3.55–3.59 and the volume profiles remain extended across the matched ensembles. Repeating the physical weighting with forty rather than eighty slices gives collision fraction 0.078 and \(d_H = 3.74\). Thus the microscopic closure structure is strongly active without destabilizing the host geometry, and the shuffled comparison shows that the ordering follows the closure structure rather than the energy histogram or sampler.

24.6 The defect experiment: geometry responds to the theory's mass

The theory's matter is persistent closure failure: committed capacity, maintained against the refresh (Sections 3.2 and 23). The experiment inserts it by hand and repeats the comparison with two independent random seeds. In each repetition, one ensemble holds one hundred well-separated cells in maximal closure failure (all four faces at \(m = 0\): six collisions, \(K^2 = 48\)), while a control ensemble holds the same number in the best-closed injective configuration (\(m = \{0, 1, 2, 3\}\), the minimum \(K^2 = 40.67\)). Anchoring, protection from geometry moves, and every other update rule are identical. Every final triangulation remains connected and correctly foliated, and all one hundred marked cells survive in each ensemble. Subtracting the closed-cell control therefore isolates the physical effect of failure content. The final analysis pools the two seeds and uses across-seed scatter where replicated and integrated-autocorrelation-time-corrected standard errors otherwise.

The capacity response is measured with high statistical significance. At the first shell the mean closure energy is \(1.6802 \pm 0.0026\) around failure pins and \(1.7260 \pm 0.0008\) around closed pins, a difference \(-0.0458\) at \(16.8\sigma\). The collision fraction changes from \(0.0783 \pm 0.0002\) to \(0.0698 \pm 0.0009\), a \(9.2\sigma\) separation. These label observables largely return by the second shell, as the short closure correlation length requires. The geometric channel appears in a different observable and farther out: at the third shell the mean coordination is lower around failure pins by 0.0398, a \(4.0\sigma\) separation from the closed-cell control after autocorrelation and seed scatter. Shell-cell counts do not separate significantly at this precision, and the first- and second-shell coordination differences are consistent with zero. The data establish a local geometric response.

Persistent closure failure measurably strains the surrounding capacity state and changes the local coordination field. The closed-cell control carries the same anchoring without the failure, isolating the failure content as the source. This finite-lattice result supplies the microscopic source-to-strain seed used by the weak-field sector. The defect-only experiment reaches three lattice steps and establishes neither continuum curvature nor a value of \(G\). The separately conserved instrument of Section 24.7 tests transport beyond that short closure range.

24.7 Reaching Newtonian range: the conservation requirement

The strain of Section 24.6 dies within a lattice step, as the sub-cell correlation length requires. Macroscopic transport needs a different operator. If free capacity is only a per-cell budget, local maximum-entropy re-equilibration gives \((2z - L)\delta\mu = -m/\chi\) on the slice adjacency (\(z = 4\) faces per cell, \(L\) the slice Laplacian). That operator has no small-momentum pole; on a real slice the response falls eight orders of magnitude within thirteen steps. A locally rebalanced budget therefore screens at the substrate scale.

Committed stock and maintenance throughput. Particle formation and persistence are different operations. Let \(J^\mu_{c,A}\) carry the committed stock belonging to source \(A\). Its balance law is

$$\nabla_\mu J^\mu_{c,A} = \Gamma_{\text{form},A} - \Gamma_{\text{rel},A}.$$

For an already-existing stable defect both rates vanish, so its mass-equivalent committed stock remains constant. The defect nevertheless draws a maintenance throughput

$$\mathcal{P}_{\text{maint}} = \alpha_{\text{maint}}\rho_{\text{def}}.$$

This is the rate at which the surrounding renewal bandwidth services persistence; it is not continuing conversion into more matter. Spatial free-bandwidth flux enters the defect event, while the reversible update exports the previous replaceable closure state along an internal history edge. The history register is an output of the renewal circuit, not an ordinary spacetime fluid, so no four-current \(J^\mu_{\text{hist}}\) is assumed. In free propagation that output carries no cell address of the persistent mark. This stock–throughput distinction removes the apparent secular growth of a static particle.

Augmented-graph Ward identity. The following result holds for the refresh-bandwidth completion. Form the directed renewal event graph whose edges are spatial bandwidth transport, ordinary temporal continuation, and the history output of a maintenance event. Let \(B\) be its incidence matrix and \(j_e\) the oriented edge throughput. In the absence of formation, release, or terminal absorption, each local reversible gate has one incoming active slot for every outgoing active or history slot. Its exact Kirchhoff identity is

$$\Delta_t n_v + \sum_e B_{ve}j_e = 0.$$

At a stable defect the history-edge current is \(j^H_v = \mathcal{P}_{\text{maint},v}\); it is an outflow from the present register, not an increase of committed stock. Summing over any region cancels every internal edge and leaves only boundary transport and the explicitly counted history outputs.

On a fixed spatial slice, a local linear constitutive law has the quadratic operator

$$L_f = B_s W B_s^T,$$

where \(B_s\) is the spatial incidence matrix and \(W\) contains positive face conductances. Since \(B_s^T \mathbf{1} = 0\),

$$\boxed{L_f\mathbf{1} = 0.}$$

For a connected slice the constant mode is the only exact zero mode. Locality and spatial isotropy then give

$$\lambda_f(k) = D_q k^2 + O(k^4)$$

near \(k = 0\); a nonzero constant term would violate the incidence identity and represents leakage. The steady equation on a compact slice is

$$L_f\mu_q = \alpha_{\text{maint}}(\rho_{\text{def}} - \bar{\rho}_{\text{def}}),$$

where the subtraction removes the constant mode. In an extended three-dimensional limit its pseudoinverse has

$$G_f(r) \sim \frac{1}{4\pi D_q r},$$

so a persistent maintenance demand produces the unscreened \(1/r\) profile without tuning a scalar mass. Conservation proves the gapless transport pole within this completion. It does not prove that refresh bandwidth is the physical carrier, that defects couple only to it, or that the resulting scalar is the metric capacity coordinate.

The continuum junction condition. Let the renormalized static response be \(\delta q = \chi_q\mu_q\), \(J^i_q = -D_q\nabla^i\mu_q\), and \(\mathcal{P}_{\text{maint}} = \alpha_{\text{maint}}\rho_{\text{def}}\). Away from the compact zero-mode subtraction,

$$\nabla^2 q = \frac{\chi_q\alpha_{\text{maint}}}{D_q}\rho_{\text{def}}.$$

The static bridge \(q = 1 + 2\Phi/c^2\) and \(\nabla^2\Phi = 4\pi G_*\rho\) therefore require

$$\boxed{\frac{\chi_q\alpha_{\text{maint}}}{D_q} \longrightarrow \frac{8\pi G_*}{c^2}}$$

after renormalized-operator and lattice-spacing matching. The three quantities on the left are lattice outputs. The electron-derived \(G_*\) enters only after their dimensionless regulator dependence has been removed, so the continuum comparison is target-blind rather than a fit.

This transport experiment is performed on a separate forty-slice realization at \(N_{41} \simeq 100,000\) (Appendix J.8). The field begins at the vacuum anchor 7.4198, moves only through antisymmetric face fluxes, and is recycled through the compact zero mode; its mean remains 7.4198 through all 4000 sweeps. Absorption reads the persistent excess failure above the measured vacuum dressing, so the source strength is produced by the label dynamics rather than assigned by the transport law. Five fixed source values, \(m = 0, 1, 2, 3, 6\), are represented by thirty-two well-separated defects each. The \(m = 0\) value measures commitment-independent boundary dressing; the positive values test the source and response laws independently. Within-pin series use Sokal integrated-autocorrelation corrections, and final errors include across-pin and across-level scatter.

Four distinct measurements give a consistent finite-lattice source-to-field chain. First, conservation is exact. Second, after subtracting the \(m = 0\) dressing, the emergent maintenance charge is additive,

$$Q - Q(0) = (0.1440 \pm 0.0025)m, \quad Q(0) = 0.0075.$$

Third, the shell-one deficit divided by that independently measured charge is constant across the positive source values,

$$\left.\frac{\Delta f_1 - \Delta f_1(0)}{Q - Q(0)}\right|_{m=1,2,3,6} = 4.9965 \pm 0.5244,$$

where the uncertainty is the larger of propagated error and across-level scatter. Sources of different strength therefore couple through one finite-lattice junction ratio. The dimensionful continuum coefficient \(8\pi G_*/c^2\) still requires a continuum scaling and operator-matching calculation. Fourth, the dressing-subtracted profile remains positive through eight shells and fits

$$\Delta f(d) \propto d^{-0.62\pm0.11}.$$

The screening diagnostic gives \(d\ln(\Delta f\, d)/dd = +0.1062 \pm 0.0551\), statistically consistent with zero and implying the finite-range bound \(\xi > 4.6\) lattice steps. The exact Ward identity and the absence of detected screening therefore agree: the tested carrier is conservation-protected and long-ranged over the usable graph radius.

Two qualifications are visible in the same data. The proportional source fit has a maximum relative residual of 0.242, driven mainly by the \(m = 6\) source falling below the extrapolation from weak sources; this is finite-capacity saturation, not precision linearity at arbitrary multiplicity. The exponent differs from the ideal three-dimensional massless Green-function value \(-1\); the closed-slice zero mode and finite slice radius flatten the measured tail, but a larger-volume scaling study must determine whether the exponent tends to \(-1\). These measurements establish the tested finite-lattice behavior while leaving the continuum scaling and absolute junction normalization open.

24.8 What the finite-lattice results establish

The fixed-lattice results establish four points. The admissibility sector orders on a dynamical host without destabilizing its matched geometry. Persistent closure failure produces a replicated local capacity response and an autocorrelation-corrected local coordination response. Conditional on refresh bandwidth as the carrier, the exact augmented-graph identity protects the long-range pole. The separate transport calculation then verifies conservation, approximately additive source strength, one lattice junction ratio across the tested source values, a power-law tail, and no detected screening. None of these mechanisms selects the vacuum geometry; CDT supplies that host state externally, while the substrate measurably dresses its \(N_{41}\) coupling direction. The exact Regge volume projection of that direction is given in Section 25. The remaining CDT questions are qualitatively different: whether the joint system lies in the required extended phase with a massless TT sector, whether it also reaches an optional continuous cutoff-removal trajectory, how its lattice spacing and operators match to continuum observables, and whether the absolute junction coefficient tends to \(8\pi G_*/c^2\). Section 26 records those distinct grades.

24.9 The capacity-decorated continuum target

The present runs compare coupled and control ensembles at fixed regulator. Because \(L_*\) is a physical microscopic length, existence of the effective theory does not require a second-order point at which the regulator spacing vanishes. The necessary geometric condition is instead membership in an extended semiclassical phase whose long-wavelength transfer matrix contains a massless transverse–traceless sector. A continuous critical surface is a separate, stronger universality and cutoff-removal test. For a causal triangulation \(T\), let \(Z_{\text{cap}}[T; g_{\text{cap}}]\) be the trace over its closure labels, defect marks, renewal gates, source-labelled committed allocations, and history outputs. The joint partition function is

$$Z(\mathbf{g}) = \sum_{T\in\mathfrak{T}_{\text{CDT}}}\frac{1}{C_T}e^{-S_{\text{CDT}}[T;\kappa_0,\Delta,\kappa_4]}Z_{\text{cap}}[T; \mathbf{g}_{\text{cap}}],$$

and integrating out the capacity sector gives

$$S_{\text{eff}}[T] = S_{\text{CDT}}[T] - \ln Z_{\text{cap}}[T].$$

The theory fixes \(\mathbf{g}_{\text{cap}}\) at its microscopic values. Phase membership must then be tested at the tuned host couplings with three observables:

1. a Hausdorff estimator tending to four, \(d_H(N_{41}) \to 4\), as the volume grows;
2. a de Sitter-like volume profile, \(N_3(t) \propto \cos^3(t/s_0)\) with \(s_0 \propto N_4^{1/4}\);
3. a gapless TT transfer mode, \(\lambda_{\text{TT}}(k) = 1 - c_{\text{TT}}k^2 a^2 + O(k^4)\) as \(k \to 0\).

The completed volume pair bears only on the trend toward the first condition and on a coarse extended-volume proxy: increasing \(N_{41}\) from about 42,000 to 102,000 raises \(d_H\) from 3.31 to 3.56 and the blob score from 1.46 to 1.57. At about 100,000 simplices, the three matched ensembles remain connected and correctly foliated and give \(d_H = 3.55\text{–}3.59\). These results are consistent with movement farther into the finite-volume extended region, but they do not establish \(d_H \to 4\); no de Sitter-profile scaling fit or TT transfer-spectrum measurement has yet been performed for the decorated ensemble. A gapped TT sector would fail the third condition even if the volume data appeared extended.

Two capacity-register numbers reported elsewhere are deliberately excluded from this phase test. The refresh model's stationary admissibility 0.536 and the history-channel correlation length \(\xi_{\text{hist}} \simeq 0.57\) cells characterize the finite register dynamics, not the host's geometric correlation length or graviton spectrum.

The additive vacuum term and the finite continuum volume coupling are distinct coordinates of this approach, but the volume normalization is not an adjustable unknown. Section 25 derives the exact Regge operator

$$\mathcal{V}_4 = a^4(v_{41}N_{41} + v_{32}N_{32})$$

and its corresponding source direction in the \((\kappa_4, \Delta)\) plane. On a fixed-ratio trajectory the scalar matching is \(t_4 = \bar{v}_4 a^4\rho_{\text{vol,R}} + o(a^4)\), with \(\bar{v}_4 = (v_{41} + \xi v_{32})/(1 + \xi)\). The continuum calculation must therefore measure the limiting \(\alpha\), \(\xi\), lattice scale, and mixed critical eigenoperator, rather than fit a free \(Z_V\). Vacuum normalization removes the complete capacity-sector bulk coefficient at every fixed regulator. Consequently a finite source-independent capacity density cannot hide in \(t_4/a^4\): it would already be an extensive term before the limit and would have been included in \(\mu_{\text{cap}}\). If the common-parent critical limit exists, its gravitational volume coefficient still requires a matching condition; O.5 adopts \(\Lambda_R^{\text{empty equilibrium}} = 0\). The currently external CDT host does not satisfy that antecedent by assumption, so its own cosmological coupling remains a regulator input rather than a result of these runs.

If cutoff removal is additionally required, \(\kappa_4\) can be tuned toward infinite volume and the remaining regulator couplings can be tested for a continuous transition. CDT supplies evidence that candidate continuous lines can occur and can support this stronger construction [111, 112]. Let \(\mathfrak{C}_*\) denote the manifold on which the geometric correlation length diverges in lattice units. The distinct symbol keeps this geometric locus separate from the coherence observable \(\mathcal{C}\). The cutoff-removal embedding succeeds only if the fixed capacity slice intersects that manifold,

$$\boxed{\mathcal{P}_{\text{cap}} \cap \mathfrak{C}_* \neq \varnothing, \quad \frac{\xi_{\text{geom}}}{a} \longrightarrow \infty.}$$

This requirement introduces no capacity fit. Failure of the fixed slice to intersect \(\mathfrak{C}_*\) would reject the cutoff-removal embedding, but would not by itself reject a finite-\(L_*\) theory whose extended phase and massless TT sector had already been demonstrated. Geometric criticality removes the ultraviolet regulator. The augmented-current Ward identity has a different job: it protects the infrared transport pole. Neither result supplies the other.

Appendix J.9 states the phase-membership, optional critical-surface, finite-size-scaling, operator-mixing, current, and junction tests in full. A successful capacity-decorated CDT transfer matrix would provide an independent nonperturbative geometric embedding of the finite marked vertex. Until the phase and TT tests are passed, CDT remains the external host used by the simulations; the factorized EPRL/GFT construction of Section 23 and Appendices B, H, and Q supplies the separate controlled semiclassical Einstein route.

25. Equilibrium Vacuum and Cosmological Term

The normalization \(q = 1\) defines the unstrained vacuum reference. The regulated joint ensemble makes the corresponding subtraction exact, rather than leaving it as an unspecified cancellation. At fixed triangulation let

$$Z_{\text{cap}}[T] = \mathbb{E}_{m\sim\text{unif}}\exp\left[-\beta\sum_c E_c(m)\right]$$

and define its homogeneous bulk free energy per cell by

$$\mu_{\text{cap}}(a, \beta) = -\lim_{N_{\text{cell}}\to\infty}\frac{1}{\beta N_{\text{cell}}}\ln Z^{\text{vac}}_{\text{cap}}(a, N_{\text{cell}}).$$

Its dependence on the homogeneous regulator couplings is suppressed in the notation; the definition is applied along the trajectory whose continuum limit is being tested. The vacuum-normalized capacity factor is

$$\hat{Z}_{\text{cap}}[T] := e^{\beta\mu_{\text{cap}}(a, \beta)N_{\text{cell}}(T)}Z_{\text{cap}}[T].$$

By construction,

$$\boxed{-\lim_{N_{\text{cell}}\to\infty}\frac{1}{\beta N_{\text{cell}}}\ln\hat{Z}^{\text{vac}}_{\text{cap}} = 0.}$$

In the thermodynamic limit,

$$F_{\text{cap}} = \mu_{\text{cap}}(a, \beta)N_{\text{cell}} + o(N_{\text{cell}}).$$

The centered bulk potential

$$\Omega_{\text{cap}} = F_{\text{cap}} - \mu_{\text{cap}}(a, \beta)N_{\text{cell}}$$

has zero density in the homogeneous reference state. The cancellation is invariant under an arbitrary change of microscopic energy origin. If

$$E_c \mapsto E_c + C, \quad \mu_{\text{cap}} \mapsto \mu_{\text{cap}} + C,$$

then

$$\boxed{\hat{Z}_{\text{cap}}[T] \mapsto \hat{Z}_{\text{cap}}[T]}$$

exactly. When the bulk cell count admits the usual thermodynamic reading, Gibbs–Duhem writes the same relation as \(\Omega_{\text{cap}}/V = -P\); an isolated equilibrium reference has \(P = 0\) [102]. The partition identity above is stronger for the regulated model because it does not require the conditional refresh current to serve as that thermodynamic charge. A large homogeneous source-independent zero-point term therefore cannot reappear as a capacity-sector observable; it cancels against the uniquely shifted bulk free energy, while defect and geometry-dependent differences remain.

Integrating out the normalized capacity sector gives

$$S_{\text{eff}}[T] = S_{\text{CDT}}[T] - \ln\hat{Z}_{\text{cap}}[T].$$

The homogeneous extensive capacity term is absent from this action. Only the geometry- and defect-dependent free-energy difference remains to source strain, as required by the information–geometry and mass–entropy postulates. This result does not use the conditional refresh-current Ward identity. Transport conservation protects the infrared pole; vacuum normalization follows from the equilibrium partition measure. They are different conserved structures.

Appendix J has already tested the finite-regulator algebra. Omitting the centering restores the term \(+\beta\mu_{\text{cap}}N_{\text{cell}} = +(\beta\mu_{\text{cap}}/2)N_{41}\) in the geometric action. Against the quadratic volume pin it predicts

$$\Delta N_{41} = -\frac{\beta\mu_{\text{cap}}}{4\varepsilon},$$

and the measured uncentered-minus-centered displacement is \(-62.8\) against \(-62.2\). The run does not prove a continuum cosmology, but it verifies the coefficient and sign of the subtracted microscopic bulk term. One bookkeeping refinement matters here. In the standard CDT action basis,

$$S_{\text{CDT}} = -(\kappa_0 + 6\Delta)N_0 + \kappa_4(N_{41} + N_{32}) + \Delta(2N_{41} + N_{32}),$$

the restored \(N_{41}\) term is the coupling displacement \(\delta\Delta = +\beta\mu_{\text{cap}}/2\), \(\delta\kappa_4 = -\beta\mu_{\text{cap}}/2\), not a pure shift of \(\kappa_4\). At fixed simplex ratio it contains the expected volume chemical potential; its orthogonal component renormalizes the regulator asymmetry. The measured displacement therefore verifies the predicted \(N_{41}\) direction, which is the exact statement licensed by the control.

The same conclusion is independent of the centering convention. The leading exponential growth of the decorated canonical sum defines a critical surface in the full \((\kappa_4, \Delta)\) plane. An extensive change of microscopic energy origin translates the bare coupling vector and that critical surface by the same displacement displayed above. Their normal difference is invariant, and it vanishes at the infinite-volume surface. On a trajectory that holds \(\Delta\) fixed after the other relevant directions have been projected out, this normal coordinate is the familiar

$$t_4 := \kappa_4 - \kappa_4^c \longrightarrow 0.$$

Thus critical tuning absorbs the additive bulk term without turning an energy-origin choice into an observable. The remaining question is the normalization of the physical volume direction, to which we now turn.

The continuum volume matching can be made explicit, with no unspecified volume factor \(Z_V\). After Wick rotation, the exact Regge four-volumes of the two simplex types are [105]

$$V_{41} = a^4 v_{41}(\alpha), \quad v_{41}(\alpha) = \frac{\sqrt{8\alpha - 3}}{96},$$ $$V_{32} = a^4 v_{32}(\alpha), \quad v_{32}(\alpha) = \frac{\sqrt{12\alpha - 7}}{96},$$

with \(\alpha > 7/12\) so both Euclidean simplex types are nondegenerate. Here \(a\) is the spatial regulator edge length. In the hosted reading it is scaffolding with no fixed relation to \(L_*\); under the face-wise lock of Appendix H.9a, paragraph 4a, every spatial face carries the channel area \(a_{\text{ch}} = 4\ln 2\, L_*^2\) and the edge is physical, \(a^2 = (16\ln 2/\sqrt{3})L_*^2\), \(a = 2.530 L_*\) (Appendix J.10). The integrated volume operator is therefore the kinematic identity

$$\boxed{\mathcal{V}_4[T] = a^4\left(v_{41}(\alpha)N_{41} + v_{32}(\alpha)N_{32}\right).}$$

Let \(\rho_{\text{vol,R}}\) denote the coefficient of \(\mathcal{V}_4\) in the dimensionless Euclidean effective action. If \(G_R\) is the continuum Newton coefficient and \(L_G^2 := \hbar G_R/c^3\), then

$$\rho_{\text{vol,R}} = \frac{c^3\Lambda_R}{8\pi\hbar G_R} = \frac{\Lambda_R}{8\pi L_G^2}.$$

On the matched weak-field branch \(G_R = G_*\) and therefore \(L_G = L_*\). Keeping \(L_G\) visible until that junction passes prevents the volume matching from assuming the gravitational result it is meant to join. The two action-conjugate volume couplings are therefore

$$\boxed{t_{41} = a^4 v_{41}(\alpha)\rho_{\text{vol,R}}, \quad t_{32} = a^4 v_{32}(\alpha)\rho_{\text{vol,R}}.}$$

Equivalently, in the \((\kappa_4, \Delta)\) basis a pure volume-source displacement obeys

$$\boxed{\begin{pmatrix}\delta\kappa_4\\\delta\Delta\end{pmatrix} = a^4\rho_{\text{vol,R}}\begin{pmatrix}2v_{32} - v_{41}\\v_{41} - v_{32}\end{pmatrix}.}$$

Substitution returns \(\delta(\kappa_4 + 2\Delta) = a^4 v_{41}\rho_{\text{vol,R}}\) and \(\delta(\kappa_4 + \Delta) = a^4 v_{32}\rho_{\text{vol,R}}\), so this is an operator identity, not a fitted matching. At the isotropic point \(\alpha = 1\), \(v_{41} = v_{32} = \sqrt{5}/96\) and the \(\Delta\) component vanishes, providing a direct check.

The one-parameter formula of Section 24.9 is the fixed-ratio compression of this two-operator statement. If \(\xi = N_{32}/N_{41}\) and \(N_4 = N_{41} + N_{32}\), then

$$\bar{v}_4(\alpha, \xi) = \frac{v_{41}(\alpha) + \xi v_{32}(\alpha)}{1 + \xi}, \quad t_4 = \bar{v}_4(\alpha, \xi)a^4\rho_{\text{vol,R}} + o(a^4).$$

Thus the geometric part of \(Z_V\) is fixed:

$$\boxed{Z^{\text{geom}}_V(a) = \bar{v}_4\left(\alpha(a), \xi(a)\right).}$$

The critical calculation still has to show that \(\alpha(a)\) and \(\xi(a)\) approach finite limits, set \(a\) by a target-blind observable, and project out any mixed relevant eigenoperator. Those are continuum-existence and scale-setting tests; they leave no freedom in the volume normalization.

This matching does not by itself identify a CDT regulator simplex with the physical capacity cell. In the \(x^0 = ct\) convention one native cell has four-volume \(L_*^4\). Appendix J.10 records that this convention and the fixed face area are in tension by a factor 1.9: a regular spatial cell of edge \(2.530 L_*\) has three-volume \(1.91 L_*^3\), so one cell swept through one tick is \(1.91 L_*^4\), and the event measure must be fixed by one of the two readings. The average number of regulator simplices in such a block is

$$n_*(a) = \frac{L_*^4}{a^4\bar{v}_4},$$

and its volume coupling is

$$\boxed{t_* := n_*(a)t_4 = \rho_{\text{vol,R}}L_*^4 = \frac{\Lambda_R L_*^4}{8\pi L_G^2} \xrightarrow{G_R=G_*} \frac{\Lambda_R L_*^2}{8\pi}.}$$

The cutoff and the simplex factor cancel. CDT can therefore supply arbitrarily fine scaffolding while \(L_*\) remains the finite physical capacity scale, as required by Section 24.1.

There is then no surviving capacity-sector 0/0 ambiguity. The definition of \(\mu_{\text{cap}}(a, \beta)\) removes the complete thermodynamic bulk coefficient at every fixed regulator, not merely the part that diverges as \(a \to 0\). If a nonzero finite \(\rho_{\text{cap,R}}\) remained, then at any fixed \(a\) the normalized vacuum action would contain

$$\rho_{\text{cap,R}}a^4\left(v_{41}N_{41} + v_{32}N_{32}\right),$$

which is still linear in the number of cells and therefore contradicts the defining zero bulk density of \(\hat{Z}^{\text{vac}}_{\text{cap}}\). Hence, on every regulator and on every continuum subsequence for which the volume operator has a limit,

$$\boxed{\rho^{(\text{eq})}_{\text{cap,R}} = 0, \quad \Lambda^{(\text{eq})}_{\text{cap,R}} = 0.}$$

The complete equilibrium bulk term must be normalized before the regulator is removed. This order avoids inferring a continuum density from the ratio of two quantities that separately vanish.

The externally supplied CDT host has its own volume coupling. In a common geometry–capacity continuum, the complete effective action still requires a prescription for the gravitational volume coefficient. Centering fixes the normalized capacity free-energy reference, while the gravitational integration constant is fixed by a boundary or matching condition. Appendix O.5 adopts \(\rho_{\Lambda,i} = 0\) for the empty equilibrium branch. This value is compatible with the centered measure, but it does not follow from the invisibility of constants in a fixed-volume probability distribution. Existence of a common-parent critical limit and its cosmological boundary data remain separate requirements.

A controlled nonequilibrium target. The external results also identify the correct next object more tightly than the phrase "state-dependent energy" alone. Let \(C_0 > 0\) be the covariance of a regulated Gaussian capacity block in the centered equilibrium state and \(C > 0\) the covariance of another state on the same support. The dimensionless relative covariance is

$$\mathcal{G} = C_0^{-1/2}C C_0^{-1/2}.$$

The Kullback–Leibler divergence between the corresponding centered Gaussian measures is the basis-independent identity

$$\boxed{\Gamma_{\text{rel}}(C\|C_0) = D_{\text{KL}}(\mathcal{N}(0, C)\|\mathcal{N}(0, C_0)) = \frac{1}{2}\text{Tr}[\mathcal{G} - I - \ln\mathcal{G}].}$$

If \(\lambda_i > 0\) are the eigenvalues of \(\mathcal{G}\), each contribution \(\lambda_i - 1 - \ln\lambda_i\) is nonnegative and vanishes only at \(\lambda_i = 1\). Thus

$$\Gamma_{\text{rel}} \geq 0, \quad \Gamma_{\text{rel}} = 0 \iff C = C_0,$$

and for \(\mathcal{G} = I + X\),

$$\Gamma_{\text{rel}} = \frac{1}{4}\text{Tr}\, X^2 + O(X^3).$$

Appendix H.10 derives this functional rather than merely recognizing it: at finite regulator it is the quadratic-source Legendre transform of the equilibrium-normalized determinant of any positive dressed/reference Gaussian Hessian pair on a common physical support. The trace-log and positive mismatch forms are therefore closed as functional identities. The substrate must still derive the physical operator, its support and trace multiplicity, its dynamics and dimensions, and the conserved covariant response used in cosmology. This introduces no new premise or fitted potential: it is the exact dimensionless relative-information functional of any positive Gaussian fluctuation sector. Multiplication by the appropriate thermodynamic scale turns it into a relative free energy. Bianconi independently promotes the same operator-convex form to a covariant geometric-relative-entropy theory,

$$\Lambda_G = \frac{1}{2\beta}\text{Tr}[G - I - \ln G],$$

and obtains a low-energy Einstein limit and a local thermodynamic first law [28]. The result therefore supplies a well-motivated target class for the homogeneous influence functional: equilibrium has zero relative energy, while a distinguishable state can carry a positive mismatch energy without restoring a source-independent vacuum constant.

Appendix O.3 supplies a constrained covariant capacity-clock action and conserved metric stress tensor for a specified release history, thereby fixing the associated background and linear scalar exchange. The microscopic theory must still derive that clock potential from baryonic mixing and reproduce its common energy conversion and post-refresh ledger. Bianconi's G-field has its own derivative terms in the modified field equations, so it cannot be identified with \(q\), varied as a new scalar, or imported wholesale without contradicting the metric-only decision of Section 11. Relative information nevertheless supplies a consistent local action, energy, and thermodynamic language for departures from the maximum-information vacuum. The remaining question is which departure the substrate dynamics selects.

Appendix O develops the conditional dark-energy completion this leaves open. The primitive event count of the marked vertex supplies a fixed spacetime measure, so the continuum variation is unimodular and the cosmological term enters as an integration constant with \(w = -1\) exactly, not as a local vacuum-energy coupling; constant shifts of the matter Lagrangian shift the constant without independently curving spacetime. The equilibrium centering fixes the microscopic free-energy reference. Appendix O.5 separately adopts \(\Lambda_i = 0\) as the initial gravitational matching condition.

The source of later growth is collective loss of a baryonic source-clock carrier. Identity-free committed capacity cannot record stream assignment. When gas from several progenitors becomes irreversibly mixed in one retained bound object, matter no longer resolves progenitor membership and the still-labelled committed allocation is orphaned. Additive record pricing assigns the released stock fraction \(\epsilon_{\text{mix}} = 1 - I(A; Y)/H(A)\); faithful renewal fixes complete rerouting for each released elementary block, and Appendix O.13 supplies the coefficient-free nine-state dilation. The closed-cohort advection–diffusion theorem makes the fully resolved position-information loss bounded and monotone; it does not by itself prove monotonicity for the coarse \(Y\) used in the release law. A cosmological zoom must fix the progenitor geometry and physical \(K_{\text{mix}}\) independently before the final likelihood is evaluated. The common energy conversion and post-refresh ledger remain physical premises. Once the release history is fixed, the vacuum stress retains intrinsic \(w = -1\), while a separately conserved effective-fluid fit has \(w_{\text{eff}} < -1\) during positive release and approaches \(-1\) as release ends.

Part VIII. Closure Status, Falsifiability, and Comparisons

26. Closure-Status Table

The table below records each claim's premises, logical grade, and unresolved tests.

The leading word in each status uses a fixed vocabulary. Closed means derived within the stated postulates and ensemble. Fixed means no phenomenological freedom remains once the named branch is adopted. Conditional means the result follows if a named reading or completion holds. Frontier or open means the piece is structured but incomplete. Empirical support denotes comparison with data rather than derivation, and audit task marks an independent check still required. A conditional premise does not demote every theorem downstream of it. Rows therefore separate exact identities and within-model lemmas from the microscopic or empirical status of their premises.

[The closure-status table spans pages 74–83 and is reconstructed below in a compressed textual form.]

This table is the epistemic map used for the rest of the discussion.

Cosmological row. The saturated phase enters the ledger as follows. The per-source variable \(\sigma_A\) separates saturation of recruitment from exhaustion of absolute capacity. Conditional on \(\sigma_A = 1\) and one coherent tick-normalized phase, \(X = 1/2\) follows and the zero-pressure constrained-capacity stress form is a theorem. The abundance \(1/\epsilon = 5.321\) stands against the measured \(5.364 \pm 0.065\) and inherits the transverse normalization together with the recruitment and per-defect assumptions. Agreement within one standard deviation requires the mass-weighted saturation \(\bar{\sigma} \geq 0.9959\), so the comparison specifically tests near-complete recruitment. The capacity cap fixes the endpoint; the dynamics that reach it, establish coherence, and account for conversion energy remain open.

27. Falsifiability and Observational Tests

27.1 Static weak-field falsifiers

The static galactic sector has seven direct checks. Rotation curves test the exponential transition and baryonic Tully–Fisher limit [3, 60]; resolved disks test the director completion of the nonspherical equation; old-disk vertical dispersions test the absence of a phantom sheet; galaxy–galaxy lensing tests the no-slip metric; Cassini tests the carrier nesting rule; dispersion-supported dwarfs test the boundary of the equilibrium carrier phase; and stellar streams test the predicted round excess component.

Stacked weak lensing first established a low-acceleration continuation and reported an early/late-type split [68]. A later joint analysis extends the kinematic RAR by about 2.5 decades in acceleration and finds a common early/late relation after imposing strict isolation and using consistent stellar and gas masses [69]. Those controls remove the earlier split. A persistent mismatch between dynamical and lensing accelerations in an isolated sample would falsify the leading no-slip contact.

The local prediction is sharper. The Milky Way is the active transverse carrier, while the Sun and wide binaries are resolved Einstein descendants. The branch predicts \(Q^{\odot,\perp}_2 = 0\) and Newtonian wide-binary dynamics apart from the smooth Galactic tide. Cassini gives \(Q_2 = (1.6 \pm 1.8) \times 10^{-27}\ \text{s}^{-2}\); current universal total-field implementations are in 3–15\(\sigma\) tension with that result [53]. Wide-binary analyses remain statistically divided [38, 39, 40]. A confirmed velocity boost above 1% at separations of 10–30 kAU, more than twenty times the largest tidal estimate, would reject carrier nesting as stated, while a significant Solar quadrupole correlated with the Galactic field would reject its local projection rule.

Systems of several carriers give three further tests:

• a Milky-Way reflex toward the LMC well below the ΛCDM-like value would reject an inertial response stock;
• tidal dwarfs inside their parent's record must stay nearly Newtonian, with dynamical-mass boosts of 1.5–2 for the NGC 5291 system;
• the residual forces on the response stocks predict small offsets between the lensing and baryonic centres of satellites.

At lower baryonic mass, the GravSphere dwarf sample of Júlio et al. [54] lies systematically above the extrapolated rotating-galaxy RAR and shows multivalued loci and large scatter. The current EFT restricts the exact exponential law to equilibrium active carriers; it must derive a phase-selection boundary before that restriction can explain the dwarf data. A universal application of the same law to dispersion-supported ultrafaint dwarfs is already disfavored by these observations.

The resolved director completion adds three galactic failure modes. With the Wang snail potentials confirmed, a local dynamical excess density at \(R_0\) departing from the predicted 0.35–0.37 \(\text{GeV cm}^{-3}\) within 1 kpc of the plane and 0.29–0.30 \(\text{GeV cm}^{-3}\) at \(|z| = 3\ \text{kpc}\) by more than the baryon-model uncertainty would reject its Milky-Way solution. For face-on disks it predicts the vertical dispersion of the old disk measured apart from the young cold layer. For the DiskMass galaxies at \(h_{z,\text{DMS}}\) and the rotation-curve mass-to-light ratio, the predicted old-disk dispersion exceeds the single-component value by a median factor 1.61 (16–84%: 1.34–1.95), against 1.87 for QUMOND. A median ratio of measured to predicted old-disk dispersion below 0.8 or above 1.2 would reject the completion; separating it from QUMOND at \(3\sigma\) requires about 5% precision on that median. The per-galaxy predictions are registered in machine-readable form with SHA-256 5a72f3e1a3ae0b659963a339ae54bd92253c34e2cd76f932903a9f6ad9b74427. For the excess it predicts \(q_\Phi = 0.95\text{–}0.99\) at 20–50 kpc with population scatter at most 0.05 and alignment with the baryons. A disk-modeled stream population with mean \(q_\Phi < 0.85\) or scatter above 0.1, or a confirmed tilt of the inner halo relative to the disk that the LMC does not produce, would reject it. In the same potential definition typical ΛCDM halos lie at 0.85–0.89, so a population mean of 0.93 or above with small scatter would disfavor them. Current Milky-Way tracers favor the completion [45, 43], the projected-track extragalactic sample mildly favors ΛCDM [44], and a decision requires about 80 well-measured streams.

27.2 Dynamical falsifiers

The dynamical extension has two independent observational tests. Source-projection tests ask whether relaxed clusters follow the predicted hook profile—near unity in BCG-dominated centers, maximal where the virialized bath dominates, and lower in the deep outskirts [6, 19, 20]—and whether resolved merger maps prefer \(\epsilon \simeq 0.19\). The linear lift candidate already fails on amplitude (Section 18). Transport tests ask whether \(D/\tau_0 = c^2\) evolves those source weights correctly through a merger. Systems such as the Bullet Cluster [4] probe both. A failure of the propagation law would reject the causal completion even if the static branch survived.

27.3 Cosmological falsifiers

Cosmology presents a different kind of test. The question there is whether a full Boltzmann treatment allows the trace-coupled homogeneous mode to reduce the sound horizon without spoiling the CMB or structure-growth observables. If it cannot, the cosmological extension fails on its own terms. The empirical target is set by the current measurement spread: early-universe inferences near 67.4 [73], distance-ladder determinations ranging from \(\simeq 70\) [75] to 73 [74], and a tension whose proposed resolutions are reviewed in Di Valentino et al. [76].

The relation \(a_0(z) = cH(z)g_{\text{share,eff}}/(4\pi^2)\) predicts measurable redshift evolution. For a Planck-like background, \(H(2.3)/H_0 \simeq 3.47\). A disk with \(g_{\text{bar}} \simeq 2 \times 10^{-10}\ \text{m s}^{-2}\) then has \(g_{\text{obs}}/g_{\text{bar}} \simeq 2.02\), compared with 1.39 for an epoch-independent scale. Observed outer rotation curves at these redshifts indicate strong baryon dominance [55, 59], although pressure-support and stacking systematics leave the comparison unsettled. Controlled observations showing no stronger boost than matched \(z = 0\) systems would falsify \(a_0 \propto H(z)\).

The equilibrium-vacuum route requires a common-parent critical limit, a defined volume operator, and a physical gravitational boundary condition. The finite-regulator capacity centering is an exact normalization identity. A nonzero capacity bulk coefficient in that normalized reference would contradict its definition; a nonzero gravitational integration constant would instead test the separate initial matching in O.5. A common substrate can support several effective operators. The subsequent equation of state and release dynamics require covariantly conserved stress energy and a completed perturbation prescription.

A MUSE sample of 79 star-forming galaxies at \(0.33 < z < 1.44\) finds a radial-acceleration scale that rises with redshift [57]; a resolved low-redshift H i sample tentatively reports the same direction [58]. An epoch-independent \(a_0\) predicts no such evolution. The comparison is already quantitative. The reported scale at \(z \sim 1\), \(a_0 = 2.38^{+0.12}_{-0.10} \times 10^{-10}\ \text{m s}^{-2}\), is an enhancement of 1.98 over the local value, which \(a_0 \propto E(z)\) produces at \(z = 1.18\); the prediction sits at \(-2.3\sigma\) if the effective redshift of the high-\(z\) data is 1.0, \(-1.0\sigma\) at 1.1, and \(+0.3\sigma\) at 1.2, so the verdict turns on the sample's effective redshift. The reported linear slope \(a_1 = (1.59 \pm 0.10) \times 10^{-10}\) likewise exceeds the \(E(z)\)-anchored effective slope of about 1.19 over the sampled range, but \(E(z)\) is convex and a linear fit absorbs curvature, so the slope comparison settles nothing by itself. The clean form of the test is the per-bin \(a_0(z)\) values compared directly with \(E(z)\): parameter-free, computable from published data, and able to fail. Current mass-to-light and pressure-support uncertainties remain comparable to the effect and enter that comparison as stated by the authors.

Saturated-phase falsifiers. Three tests bind the saturated phase: a full Einstein–Boltzmann implementation must jointly evolve the homogeneous mode, commitment transition, and constrained fluid with abundance fixed at \(1/\epsilon\); the recruitment derivation must determine its epoch and survive scrutiny of the pinned reading and per-defect bookkeeping; and linear observables must remain compatible with the inherited constrained-fluid dynamics. The subsequent release channel is more selective than collapse. The completed 3-D hydro surrogate already shows that full-halo progenitor placement loses information too slowly and fails after delayed convolution. The required cosmological zoom must therefore show that recoverable Lagrangian progenitor information is lost mainly during retained first-galaxy assembly, remains negligible in minihaloes, and produces a frozen \(I_{\text{rel}}\) in the narrow background-compatible region, \(I_{\text{rel}} \simeq 79\text{–}91\) at the reference gates of Appendix O.12, without using cosmology to choose the progenitor geometry. Failure of that timing or normalization falsifies the release channel without changing the saturated-phase abundance calculation.

Dark-energy falsifiers. The sector of Appendix O adds four direct failure modes. A confirmed low-redshift crossing to \(w > -1\) would close the mechanism, which approaches \(-1\) from below during one-signed accumulation; a reconstructed \(\rho_\Lambda(z)\) exceeding today's value at an earlier epoch would do the same. The completed 3-D surrogate has already falsified the identification of Poisson coverage with progenitor-information loss and excludes its own full-halo delayed history. The remaining astrophysical test is two-sided. The Planck \(2\sigma\) interval requires a mixing e-folding time below 1.07 halo dynamical times at the reference gates and below about 2 for every combination of gates and reionization redshift within the ranges of Appendix O.12. An independent cosmological zoom fails the collective trigger if it finds substantial minihalo-era loss, a kernel slower than the bound for the measured gate, or excessive early mixing that overshoots the background-compatible release integral. A background fit of the frozen hydro history, with \(f\) taken from the zoom, must remain competitive without introducing a tunable mixing amplitude, delay, or progenitor concentration. Finally, an independently propagating determinant mode in the coarse-grained condensate Hessian would falsify the unimodular reduction itself.

27.4 Correlated-constant falsifiers

The inferred \(L_*\), induced \(G_*\), and matched weak-field \(G\) are not independent legs: the matched route carries the same electron length calibration. Their agreement is a normalization audit rather than a separate falsifier. The marked action adds two independent tests because the same \(\zeta_*\) enters the muon and tau through different graph polynomials. A future independent determination of \(L_*\) would provide a fourth test of the shared vertex.

27.5 Many-Pasts status

Many-Pasts predicts no laboratory departure from the Born rule or no-signaling, but those finite operational laws are derived within the construction. On the selected stable retained sector, H.11 derives the complex Hilbert bundle and exact moving-frame unitary transport, tensor composition, Born weights, finite instruments, reduced channels, no-signaling, and the history Gram kernel. Its remaining burdens are global: the multiplicative factors and bridges must survive the constrained host into a network and continuum/Fock theory; the cosmological state and the histories that decohere must be fixed; record completeness and durable history capacity must be physically realized; and a conditional-typicality theorem must suppress high-entropy-past and Boltzmann-fluctuation histories. Failure of the first group would remove the proposed global microscopic quantum completion; failure of the last would remove the proposed arrow. Neither would create a new laboratory deviation inside the finite theorem's domain.

28. What the Theory Would Have to Get Wrong to Fail

The failure modes are not all equally severe, and they are ordered here by how much of the theory each would remove.

Kills the core ontology. A demonstrated failure of mass–entropy equivalence, an internal incoherence in the Many-Pasts weighting, or evidence that geometry cannot be read as the long-wavelength form of an entanglement-capacity substrate would remove the foundations on which everything else rests.

Kills the static capacity-response closure or its transverse completion. A weak-field UV coefficient chain that cannot be reconciled with an independently validated microscopic derivation would break the ordinary static closure. An RAR transition shape that departs from the proposed bosonic law, old-disk vertical dispersions that require a phantom sheet, a disk-modeled stream population incompatible with the predicted round excess, persistent low-acceleration slip, unacceptable residual PPN effects, or violation of the metric Ward identity would falsify the transverse completion while leaving the ordinary Einstein/capacity equivalence intact.

Kills an extension only. If the cosmological trace-coupled homogeneous mode cannot survive a full Boltzmann likelihood confrontation, the cosmology sector fails while the static weak-field branch stands. If the saturated phase fails any of its commitments — the dark-to-baryonic abundance departing from the capacity ceiling, linear-regime observables departing from the inherited form, or no committed component surviving on cluster infall streams — the saturated committed-capacity reading fails in the same contained way. If the source clock is not carried by baryonic Lagrangian matter, if a tracer simulation finds insufficient or prematurely early progenitor-identity loss, or if the resulting mixing-weighted history fails a background-data fit, the dark-energy release extension fails without touching the weak-field branch. If \(q_{\text{cap}}\) does not track \(q_{\text{geo}}\) during collapse, or if the derived boundary response conflicts with black-hole observations, the bounded-domain interpretation fails while the spherical Einstein reduction remains. None of these touches the weak-field core.

Requires modification, not death. A fuller graph calculation may revise the separate conditional stiffness self-energy, and strong-field boundary spectroscopy may change without altering the Einstein exterior. A failure of the decorated charged vertex would be more serious because the same \(\zeta_*\) enters the electron scale and both heavier-lepton ratios. The positive-spectrum clock conversion survives only if another microscopic charged action replaces it.

One scale-setting commitment cuts across these tiers. The high-precision \(G_*\) rests on seven-fold additivity, the record-conditioned determinant transfer, and the decorated marked vertex. A microscopic Hessian with a collective edge mode, persistent endpoint polarization, or off-diagonal charged propagation would fail the finite action and displace the correlated \(G\) and lepton results. If the electron identification or mass–entropy ontology failed, the scale-setting derivation of Appendix H would not go through.

29. Comparison with Other Approaches

The galactic excess and cosmological abundance are assigned to two phases of one medium. The comparisons below distinguish that proposal from nearby alternatives while retaining the conditional grades of Sections 15 and 20.

29.1 Relative to ΛCDM

The contrast with ΛCDM begins at the level of ontology. Here visible matter is interpreted as localized defects in a vacuum-capacity medium. The ordinary longitudinal response is a reduced representation of Einstein gravity; the proposed extra galactic response is carried by a transverse substrate sector. The UV entropy enters both branches, while the galactic branch additionally requires its phase-cell, carrier, and metric-vertex matching conditions. Its leading lensing response is specified; cosmology remains a separate extension.

29.2 Relative to MOND-like interpolation programs

MOND-like programs [21, 81, 24] usually begin from an acceleration law or interpolation function. Here the same law comes from a thermal occupation contact and a carrier-resolved auxiliary action. The source-space nesting rule removes the Solar cross-term found in universal functions of the total field, while the no-slip metric contact fixes the leading lensing response. The comparison now turns on the microscopic derivation of those structures and on resolved nonspherical, dwarf, and lensing tests.

29.3 Relative to Verlinde-style emergent gravity

Verlinde-style emergent-gravity programs share the broad intuition that gravity may be entropic [25, 80, 32, 33], but they are usually formulated through thermodynamic reasoning or horizon-inspired force laws. The present framework specifies tetrahedral counting, admissibility closure, edge coupling, return dressing, and Euclidean normalization before reaching the continuum EFT. Its correctness is an empirical question.

29.4 Relative to TeVeS and other multi-field modified gravities

Multi-field relativistic MOND completions such as TeVeS [23] introduce additional scalar and vector fields alongside the metric to obtain relativistic lensing and cosmology. Here the ordinary longitudinal response adds no field beyond the Einstein metric. The transverse sector is a carrier-conditioned metric contact with no additional vacuum graviton polarization; its leading stationary tensor structure gives no slip, while nonlinear covariance and the microscopic vertex remain open.

29.5 Relative to AeST

The closest modern comparator is the AeST theory of Skordis and Złośnik [34], which combines the metric with a dynamical timelike vector and a scalar field and can reproduce MOND-scale galaxy phenomenology while fitting the CMB power spectrum. The present construction differs in its finite counting input and is less developed cosmologically: the joint Boltzmann treatment remains open here.

29.6 Relative to scalar-tensor gravity

The reduced capacity functional of Section 10 can be mistaken for the scalar part of a Brans–Dicke-type theory [78]. The action reconstruction shows otherwise: in the ordinary static branch it is the Einstein constraint action after a field redefinition and carries no independent scalar. A scalar–tensor or disformal theory is retained only as a control case for sectors that might genuinely require an additional mode; if adopted, ordinary fifth-force, slip, and PPN constraints would apply.

29.7 Relative to quantum-mechanical interpretations

Because Many-Pasts occupies the role of an interpretation of quantum mechanics (Sections 3.3, 22), it should be placed against the standard options. It posits one realized present, adds no collapse term, and introduces no hidden sharp values. Its operational probability theory has the decoherent-histories form: unresolved alternatives retain amplitudes, decoherent record histories receive diagonal probabilities, and conditioning on the present occurs only after the alternative records are normalized. The new content is the ontology assigned to that conditional measure and its proposed substrate realization. For finite retained sectors, H.11 derives the Hilbert norm, tensor composition, instruments, no-signaling, and history Gram kernel. The continuum/Fock realization, cosmological state, actual decohering histories, and durable global record capacity remain open.

29.8 Relative to CDT, spin foams, and group field theory

The construction uses discrete tetrahedral boundary data, with primitive \(j_0 = 3/2\) pairs whose positive oriented matching transmits the seven-state \(V_3\) sector; this places it near the quantum-tetrahedron vocabulary of simplicial spin networks [84, 85, 86]. Its selection principle remains distinct from ordinary spin-foam/GFT programs: the finite capacity sector is fixed by a boundary-counting and faithful-resolution problem rather than by using a gravitational vertex amplitude as its statistical weight.

The earlier identification of the four capacity labels with the open Peter–Weyl magnetic indices is not required and is not adopted in the completed embedding. Appendix Q instead uses

$$\mathcal{H}_{\text{cell}} = \mathcal{H}_{\text{geom}} \otimes \mathcal{H}_{\text{cap}} \otimes \mathbb{C}^2_{\text{orient}}.$$

The geometric factor carries the standard gauge and intertwiner data of the simplicial amplitude; the finite capacity factor carries the injective four-port alphabet, \(K^2\), the Gibbs weight, marks, and records. Their selected \(V_3\) frames are related equivariantly, so geometry–capacity equivalence does not identify two distinct operators or force the capacity count to discard geometric intertwiners. Appendix Q excludes the Gauss-index identification and proves the strong gauge invariance of the numerical spine.

This selected factorized completion supplies a direct spin-foam/GFT witness for the vacuum geometry. The supporting external results have different jobs and are not combined into a single published theorem. Proper-EPRL-type amplitudes isolate the gravitational Regge exponential in the semiclassical block sector [123, 124]. The capacity matching supplies an exact canonical maximal-fusion ray \(V_3^{\otimes N} \to V_{3N}\) on the selected channel, and the block-lock lemma propagates the chosen one-cell geometry–capacity lock along that ray. Han-type semiclassical–continuum constructions independently reach Einstein gravity; in the controlled linearized construction the low-energy excitations give all smooth linearized Einstein solutions and the two graviton helicities [121, 122]. The area-Regge continuum calculation gives the same leading graviton dynamics and an \(O(a^2 C^2)\) correction [125]. Appendix H.10 turns that selected-branch symbol into the strong-resolvent theorem \((H_{\text{TT}})\) and pulls it back to an exact fixed-\(j = 3\) primitive realization. It also proves that the finite capacity premises cannot determine the independent geometric gluing tensor or measure. A derived \((H_{\text{phase}})\) therefore requires an additional geometric principle before the phase test can even be posed. Appendix H.9a, paragraph 4a, names the one candidate already in the postulates, the face-wise reading of the relational lock, under which the host is the equilateral CDT class at predicted couplings (Appendix J.10). The factorized capacity decoration inherits any positive host spectrum by identity; an alternative nonfactorized vertex must preserve the same interface properties.

Section 24 supplies a separate numerical interface rather than a duplicate of this spin-foam argument: the unchanged finite capacity ensemble is coupled to a CDT host [104, 105]. In the completed runs the host supplies the dynamical vacuum geometry and the paper tests how its capacity weighting dresses it. Section 24.9 and Appendix J.9 first ask whether the fixed capacity slice lies in a stable extended phase with a massless TT transfer sector. Intersection with a continuous geometric critical surface is the stronger optional cutoff-removal test, not the existence condition at physical \(L_*\). Success or failure of that CDT route is therefore a separate embedding test and does not alter the block-scale spin-foam witnesses.

29.9 Relative to algebraic and information-geometric gravity

Three continuum results address separate claims made here. The observer-dressed de Sitter algebra establishes a maximum-entropy empty gravitational state. The horizon modular calculation equates the relative information of a coherent excitation with its Killing energy flux and, after the area input, with Einstein curvature. The geometric-relative-entropy action gives a local bulk information functional with an Einstein limit, a positive Legendre energy, cosmological thermodynamics, and de Sitter area scaling [27, 26, 28]. They support the continuum ontology but do not solve the ultraviolet problem. This paper must still derive the local bound, matter commitment, dimensional scale, and response coefficients from one finite microstructure. Bianconi's independent dynamical G-field belongs to a modified-gravity theory and is not imported into the metric-only longitudinal branch.

The coefficient comparison is explicit. On the matched branch \(L_*^2 = \hbar G_*/c^3\), the horizon normalization gives

$$S_\infty = \frac{A_{\text{dS}}}{4L_*^2} = \frac{c^3 A_{\text{dS}}}{4\hbar G_*}.$$

The observer-dressed de Sitter construction uses this area coefficient and gives \(S_{\text{max}} - S_{\text{gen}}(E) = \beta_{\text{dS}}E + O(E^2)\) for a small central excitation. The horizon modular calculation uses \(S_{\text{rel}} = c^3\delta A/(4\hbar G_*)\) and obtains

$$G_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G_*}{c^4}\langle T_{\mu\nu}\rangle.$$

The bulk route derived here gives

$$\nabla^2\delta S = -\frac{\kappa}{\gamma}\rho, \quad \frac{\Phi}{c^2} = -\frac{\delta S}{2S_\infty}, \quad G_* = \frac{c^2\kappa}{8\pi\gamma S_\infty},$$

and hence \(\nabla^2\Phi = 4\pi G_*\rho\). The two continuum routes therefore use the same coupling. Appendix F.5 fixes the microscopic channel-to-area normalization within the factorized coherent \(j = 3\) embedding. The result is a consistency matching, not independent evidence for the area law.

At Gaussian order, the relationship to Bianconi is sharper. Appendix H.10 proves that the normalized determinant of any positive dressed GFT Hessian relative to its equilibrium Hessian on the same physical support has exactly the trace-log form used by geometric relative entropy, while its quadratic-source Legendre transform is exactly the positive covariance mismatch \(\frac{1}{2}\text{Tr}'(\mathcal{G} - I - \ln\mathcal{G})\). In the factorized completion the normalized equilibrium capacity trace contributes one to the empty vacuum amplitude, while the selected coarse geometric factor supplies the TT Einstein Hessian. This closes the Gaussian functional bridge and shows that the capacity decoration preserves the ordinary low-energy support. The selected metric-Regge branch satisfies \((H_{\text{TT}})\). The remaining covariant geometric task is logically prior to a phase calculation: fix \(\mathcal{G}_v\) and its measure by a new geometric principle, then test \((H_{\text{phase}})\). The distinct canonical completion retains \((H'_{\text{can}})\).

The cosmological scaling also agrees at the level of form. With \(R_A = c/H\),

$$S_\infty = \frac{\pi c^2}{H^2 L_*^2}, \quad S_{\text{GfE}} \simeq \frac{\bar{\omega}_{[1]}c^2}{\ell_P^4 H^2}, \quad \frac{S_{\text{GfE}}}{S_\infty} \simeq \frac{\bar{\omega}_{[1]}L_*^2}{\pi\ell_P^4}.$$

Bianconi therefore obtains the same \(H^{-2}\) dependence, but her couplings do not determine the finite-cell coefficient used here.

30. Conclusion

One physical picture runs through the paper. The vacuum is a finite medium of entanglement capacity; matter is that capacity tied up in stable, localized defects; a particle's mass measures the entanglement its defect commits; and gravity is the capacity strain the surrounding medium carries once that commitment is made. The ordinary weak-field capacity action is the Einstein constraint sector in the capacity variable, and the selected metric-Regge completion has the massless two-helicity Einstein infrared limit. On galactic scales, the same medium produces the additional long-range response usually attributed to dark matter.

The central result is a tightly specified static weak-field construction with its remaining assumptions exposed. The tetrahedral ensemble fixes the sharing entropy; faithful renewal and the decorated marked-transfer vertex fix the electron-anchored substrate length; finite-state recurrence fixes the shell support factor; a common Gaussian scalar mode realizes the quadratic shell overlap; the edge kernel and Green-matched source map fix the ordinary response; and the separate minimal loop operator supplies the stated conditional stiffness correction. The weak-field bridge then rewrites the Einstein constraint sector in the capacity variable.

Newtonian gravity is the point-source limit of the Einstein/capacity action equivalence, and baseline lensing and PPN values follow from that parent. The carrier-resolved galactic EFT supplies the thermal Bose contact, conservative nonspherical action, source nesting rule, no-slip metric response, point-source lensing law, causal transport poles, and leading Solar-System and wide-binary limits. On the joint coarse source it conserves momentum between galaxies and satellites, and the response budget set by Lagrangian baryons passes the weak-lensing test. The one-entropy horizon loading, \(E/(k_B T_H) = x\), carrier projector, and no-slip metric vertex remain matching premises awaiting a microscopic GFT derivation. The marked-transfer action gives \(G_*\) at \(-0.073\sigma\) and the muon and tau ratios at \(-0.535\sigma\) and \(-0.125\sigma\). The shell recurrence and overlap are explicit, while their identification with the physical pole-mass operator remains the particle premise. These are historical postdictions from one zero-continuous-fit action, not three blind predictions.

The same medium extends into sectors the paper holds more tentatively. Transport gives the field a finite propagation speed and the lag that clusters and mergers require. The cluster sector reads lensing anomalies as a phase-dependent projection of an already-fixed coefficient. The cosmological mode reduces the sound horizon, allowing a higher inferred Hubble constant. In the saturated early universe the medium becomes a conserved committed density that gravitates as zero-pressure committed capacity. In strong fields the exact spherical reduction identifies the composite \(q_{\text{geo}}\) and recovers Schwarzschild, while domain termination, boundary microphysics, and the nonspherical capacity map remain open. These are extensions and frontier completions, not closed results, and the closure table marks the distinction.

The microscopic construction has also been tested on a dynamical simplicial lattice. At matched \(N_{41} \simeq 100,000\), the admissibility weight strongly orders the microscopic labels while the host geometry remains extended and nearly unchanged. Two independent comparisons between identically anchored failure defects and closed cells give local capacity separations of \(16.8\sigma\) and \(9.2\sigma\) and a third-shell coordination separation of \(4.0\sigma\) after autocorrelation and seed corrections. A separate conserved-field calculation finds exact conservation, approximately additive emergent source charge, one junction ratio across the tested source strengths, a power-law tail through eight shells, and no detected screening. Increasing the host volume from about 42,000 to 102,000 raises both the Hausdorff estimator and the concentration of volume in the extended region. Extended-phase membership, a massless TT transfer sector, the optional critical-surface test, and the absolute \(G\) junction remain to be measured. Section 26 records the present grades.

The normalized capacity measure is invariant under extensive shifts of microscopic energy origin, and the uncentered control tests the predicted lattice-coupling displacement. The cosmological branch adopts zero initial gravitational vacuum density, additive source-record pricing, a common record/closure energy conversion, and the post-refresh ledger. Its constrained capacity-clock action fixes the background exchange and linear scalar perturbations for any supplied release history. Common closed-cohort advection–diffusion makes fully resolved position-information loss bounded and monotone; the physical coarse kernel must be measured and checked separately. The self-gravitating progenitor-tagged zoom must determine that history, after which the full background, growth, lensing, and CMB likelihood is fixed without a release-amplitude parameter.

Three independent continuum constructions support different parts of the proposed ontology. The type II\(_1\) de Sitter algebra realizes an entropy-maximizing empty gravitational state. The horizon modular calculation relates an excitation's relative information to energy flux and, after the area input, to Einstein curvature. The geometric-relative-entropy action gives a local bulk information dynamics with an Einstein limit, a positive mismatch energy, a first law, and \(H^{-2}\) de Sitter entropy [27, 26, 28]. Section 25 derives the same positive mismatch functional from a Gaussian covariance identity. These results support the continuum ontology but do not derive the finite cell, coefficients, cosmological state, or ultraviolet completion used here.

The ordinary geometric sector now has an exact primitive kinematic spine and a selected-branch Einstein theorem. Positive oriented matching gives a unique \(J = 3\) transmitted phase, maximal fusion gives an exact canonical \(V_3^{\otimes N} \to V_{3N}\) ray, and multiplicity one propagates the selected one-cell lock along that ray. Proper-EPRL isolates the desired Regge asymptotic sector, Han-type refinement reaches the Einstein sector in a controlled model and yields the two graviton helicities in its linearized theorem, and area-Regge gives the leading Einstein dynamics with \(O(a^2 C^2)\) corrections. Appendix H.10 completes the strong-resolvent step on that metric branch and the gapped perfect-action pullback supplies a positive cylindrically consistent fixed-\(j = 3\) realization. These results prove \((H_{\text{TT}})\). The host-underdetermination theorem then proves that the present capacity premises cannot select the independent geometric gluing tensor or measure: a hosted Einstein completion exists, but a derived completion requires a new geometric principle before \((H_{\text{phase}})\) can be tested. On the displayed factorized branch, renewal, the positive marked trace, and the exact rigging-map lift close the capacity pole, residue, positivity, causality, preferred-frame, and refresh-order audits. The canonical and covariant hosts remain distinct selected completions, not proved equivalent representations.

Many-Pasts assigns a record-conditioned ontology to the decoherent histories compatible with the present. Its operational branch changes no laboratory prediction, and the finite formalism is reconstructed from stated operational premises. On the real retained \(Q\) carrier, the positive Lorentzian Hamiltonian selects the initial polar complex structure. The full symplectic flow transports it exactly through varying frames, even though a comparison with the instantaneous polar basis contains real squeezing; the forced Born–Huang term is kept in that full generator. The non-common frequency and moving frames generate \(\mathfrak{su}(9)\) and yield \(\mathbb{C}^9/\mathbb{CP}^8\). Multiplicative joint labels and two complete commuting local matrix algebras force the tensor product, after which the projected neighboring bridge supplies the finite entangling algebra. Reversible internal equivalence and additive complete physical records fix the Born norm. Reversible system–record coupling then gives POVMs, Kraus operators, conditional updates, trace-preserving reduced channels, and no-signaling; the finite decoherence functional is the Gram kernel of the resulting history branch vectors.

The past is represented at the resolution retained by present physical records; erasing a record coarsens the same joint measure, and conditioning on a record after it forms supplies no future-to-past dynamics. Faithful full-support resolution selects the memoryless dressing kernel. The decorated vertex prepares its diagonal fresh closure state, reversible dilation exports the old replaceable register into history, and the marked spectrum plus electron anchor fixes the clock and phase frequency. The response motif, persistent coherent marked fiber, and charged-defect shell object are kept distinct: the event incidence identifies their coordinates but does not turn the transient response occupation into a conserved charge. The full continuum/Fock realization, cosmological state, actual decohering histories, record completeness and durable history capacity, electromagnetic charge and Maxwell dynamics, relativistic quantum fields, and the Past-Hypothesis/mixing package remain open. None is a new founding premise.

The particle appendix also contains a conditional projective color sector: a persistent open tetrahedral route, transported reversibly, gives the \(PU(3)\) qutrit-channel group, its \(\mathfrak{su}(3)\) algebra, the adjoint Wilson form, and the fixed primitive transfer \(t_8 = 13/14\), and the same primitive fusion reproduces the nine-state representation required by the marked present/history fiber (Appendix I.2). The global \(SU(3)\) endpoint lift, chiral quark matter, continuum-scheme matching, confinement, and hadron observables remain outside the completed derivation.

In ordinary gravity, the selected metric-Regge branch has the Einstein TT infrared theorem and an exact fixed-\(j = 3\) perfect-action realization, while the factorized capacity decoration, empty-capacity rigging lift, channel-to-area normalization, worldline source, and WEP are closed at their stated grades. The host-underdetermination theorem shows that the current capacity premises cannot fix the independent geometric gluing tensor and measure. A derived completion needs a geometric principle to determine those data, followed by a nonperturbative phase proof; the face-wise reading of the relational lock is the candidate, and it predicts the CDT bare couplings, \(\kappa_0 = 2\ln 2\) at \(\tilde{\alpha} \simeq 1.05\), whose phase is a registered finite-volume test (Appendix J.10). The provisional \(L_{\text{block}} \sim 3.3\text{–}5.3 L_*\) bracket has a computed representative, \(L_{\text{block}} \simeq 3.4 L_*\) from the area-Regge mismatch gap (Appendix J.10), which a decorated-vertex measurement must still confirm, and block-scale Lorentz invariance remains a named premise. The charged rest gap fixes the renewal cadence but not the spatial orbital coefficient. The coupled vertex must still derive a coherent matter propagator with invariant speed \(c\). The refreshed orientation doublet is inert on the pure-gravity branch, while its fermion tetrad-sign audit remains open. A canonical–covariant identification would be an additional theorem. The CDT calculations are compatibility checks; an Einstein-phase claim would require a massless TT transfer sector, and a continuous critical surface would be the stronger cutoff-removal test. In the galactic sector, one auxiliary action with the record-potential director stiffness reproduces the Milky-Way vertical potential for an RAR-consistent inner disk and the SPARC RAR, removes the phantom sheet from disk vertical dynamics, and predicts a round excess component. The remaining theory work is a momentum-conserving rule for interacting carriers, a microscopic derivation of the director stiffness together with the cell Hamiltonian, carrier projector, and metric vertex, full nonlinear variation of the carrier records, and a bath calculation of the equilibration rate and noise correlation volume. Old-disk vertical dispersions, disk-modeled stream shapes, resolved nonspherical lensing maps, and the ultrafaint-dwarf phase boundary are the immediate empirical tests. The finite-loop stiffness operator still needs its microscopic return-operator audit. Cosmology needs a self-gravitating progenitor-tagged zoom to freeze the mixing kernel, followed by the full DESI+CMB+Dovekie background and one-metric perturbation likelihood. The cluster lift needs its coherence profile and resolved lensing-map test, and the strong-field boundary needs its saturation dynamics and spectroscopy. Durable history capacity and the Past-Hypothesis/mixing theorem remain open in the foundations sector. The color branch still needs a derived chiral matter action and graph-to-continuum matching before comparison with physical QCD.

One ultraviolet count fixes the coefficients of the closed sector in advance. A measurement inconsistent with any of those linked coefficients would falsify that sector. The calculations above specify those comparisons.

Appendix A: Symbol Dictionary and Canonical Conventions

Appendix A gathers the conventions used throughout the technical material that follows, fixing the units, field definitions, and couplings in one place before the denser calculations begin.

Plain-language terms. Several physical words recur throughout the paper and are collected here in plain form before the symbols:

• Capacity — the entanglement support locally available in the vacuum medium.
• Defect — a stable, localized commitment of that capacity; coarse-grained, a particle.
• Deficit — capacity no longer freely available near a defect, \(\delta S = S_\infty - S_{\text{ent}}\).
• Capacity strain — the extended deficit profile whose fractional value gives the weak-field potential and whose gradient gives the gravitational field.
• Committed capacity — capacity locked into transferring on behalf of a defect; the conserved carrier of the saturated phase (Section 20).
• Dressing — the cloud an elementary defect builds by resolving the boundary sectors; its determinant transfer and marked response set the length in the decorated scale branch.
• Saturation — the regime in which the capacity bath has no slack left, with the available transfer channel at its ceiling.
• Pinned reading — the statement that the saturated phase holds exactly at that ceiling and stays there.
• Admissibility — the weighting that favors boundary configurations close to a regular, isotropic local cell.
• Closure — the condition that the four oriented face data sum to zero, so the cell closes into a regular volume; \(K^2\) measures the failure of closure.
• Many-Pasts — the postulate that one recorded present is supported by its compatible decoherent pasts. Their operational probabilities are the diagonal decoherence-functional weights conditioned on that present; \(e^{-D(h,P)}\) is only shorthand for those normalized quantum probabilities.

A.1 Units, signature, and entropy normalization

All dimensional quantities are expressed in SI units unless noted otherwise. The metric signature is \((-, +, +, +)\). Covariant spacetime integrals use \(x^0 = ct\), so the Einstein–Hilbert coefficient is \(c^3/(16\pi G)\); after writing \(dx^0 = c\,dt\), the ADM coefficient is \(c^4/(16\pi G)\). Entropies are measured in nats, so Boltzmann's constant is absorbed into the entropy normalization. The canonical UV cell has spatial scale \(L_*\) and volume \(V_* = L_*^3\); it is the bipartite primitive cell containing the two paired tetrahedral sites, so the site density is \(2/L_*^3\) and the four-cell \(\Delta V_4 = L_*^4/c\) is assigned one per primitive cell — the convention under which \(\gamma_Q = 4\hbar cJ/(3L_*^2)\) and \(\kappa/\gamma = 3L_*/(4G_{\text{tet}}(0)\kappa_m(L_*))\) are simultaneously exact (Appendix C.4–C.5). In the decorated electron-anchored marked-transfer branch,

$$L_* = -\frac{3}{2}Z_e\lambda_e\ln\left(1 - e^{-7g_{\text{share,eff}}}\right), \quad \lambda_e = \frac{\hbar}{m_e c}, \quad Z_e = (1 + \zeta_*)(1 + 7\zeta_*^2).$$

The conventional Planck length \(L_P = \sqrt{\hbar G/c^3}\) is used only as a comparison scale or in standard gravitational thermodynamic expressions after the gravitational scale has been identified.

These conventions matter because the argument repeatedly moves between a dimensionless ultraviolet counting problem and a dimensionful continuum EFT. The units and signature make those descriptions directly comparable.

A.2 Core scalar variables

The canonical continuum variable is the vacuum-relative coarse-grained entanglement field

$$S_{\text{ent}}(x),$$

with vacuum baseline \(S_\infty\) and deficit

$$\delta S(x) = S_\infty - S_{\text{ent}}(x).$$

For nonlinear work the bounded occupancy fraction is

$$q(x) = \frac{S_{\text{ent}}(x)}{S_\infty} = 1 - \frac{\delta S}{S_\infty} \in [0, 1].$$

The absolute entropy unit is fixed only after choosing a cell or horizon normalization. Under a constant rescaling of \(S_{\text{ent}}\), the quantities \(S_\infty\) and \(\kappa/\gamma\) rescale together, leaving \(\delta S/S_\infty\) and \(\kappa/(\gamma S_\infty)\) invariant. The source channel is

$$\chi(x) = -\frac{T^\mu_\mu}{c^2},$$

which is the continuum trace channel of the localized defect sector and reduces to the ordinary mass density \(\rho\) in the nonrelativistic static limit.

A.3 Couplings and derived observables

The main-text conventions are

$$\gamma : \text{entanglement-field stiffness}, \tag{10}$$ $$\kappa : \text{continuum defect–entropy coupling}, \tag{11}$$ $$\kappa_m(\ell) : \text{mass-per-entropy map at scale } \ell, \tag{12}$$ $$\zeta_* \equiv 9e^{-g_{\text{share,eff}}}\left(1 - \frac{8\eta_*}{49}\right)^{21/2}, \tag{13}$$ $$Z_e \equiv (1 + \zeta_*)(1 + 7\zeta_*^2), \tag{14}$$ $$L_* \equiv -\frac{3}{2}Z_e\lambda_e\ln\left(1 - e^{-7g_{\text{share,eff}}}\right) \text{ in the decorated marked-transfer branch}, \tag{15}$$ $$G_* = \frac{c^3 L_*^2}{\hbar} = \frac{9}{4}\frac{\hbar c}{m_e^2}Z_e^2 \ln^2\left(1 - e^{-7g_{\text{share,eff}}}\right), \tag{16}$$ $$G_{\text{tet}}(0) : \text{tetrahedral on-site Green constant}, \tag{17}$$ $$g_{\text{share,max}} = \ln(1680), \tag{18}$$ $$g_{\text{share,eff}} : \text{admissibility-weighted sharing entropy}, \tag{19}$$ $$J_{\text{bare}}, J^{\text{tree}}_{\text{eff}}, J^{(\text{ren})}_{\text{eff}} : \text{UV edge couplings}, \tag{20}$$ $$a_0 = \frac{cH_0 g_{\text{share,eff}}}{4\pi^2}. \tag{21}$$

The canonical weak-field bridge and Newton closure are

$$\frac{\Phi}{c^2} = -\frac{\delta S}{2S_\infty}, \quad \frac{\kappa}{\gamma} = \frac{3L_*}{4G_{\text{tet}}(0)\kappa_m(L_*)}, \quad G = \frac{c^2\kappa}{8\pi\gamma S_\infty}.$$

Collected in one place, these formulas also make clear which quantities are downstream of the closure chain. The UV data determine the stiffness and source-to-stiffness ratio first; the observable weak-field constants appear after the bridge and fixed \(S_\infty\) normalization are applied.

A.4 Notation map

One notation set is used throughout. The effective sharing entropy is denoted \(g_{\text{share,eff}}\), the scalar variable is always the vacuum-relative field \(S_{\text{ent}}\) or its deficit \(\delta S\), and the weak-field bridge is used in the single form stated above. The principal extension-sector symbols are:

$$\epsilon \equiv \frac{g_{\text{share,eff}}}{4\pi^2}, \quad \nu(x) \equiv \frac{1}{1 - e^{-x}}, \quad x \equiv \frac{E_\perp}{k_B T_H},$$ $$W_{\text{bath}} : \text{cluster bath source weight},$$ $$B_{\text{bath}} : \text{measured bath-development fraction},$$ $$D : \text{capacity diffusivity}, \quad \tau_0 : \text{transport relaxation time}, \quad D/\tau_0 = c^2,$$ $$\sigma_* \equiv \pi/g_{\text{share,eff}}, \quad a_{\text{UV}} \equiv 1/\text{Var}_{\eta_*}(K^2).$$

In the transverse EFT, \(\mathcal{R}_P\) is the retained record algebra of carrier \(P\), \(\Pi_P\) its conditional-expectation source projector, \(\Phi_{b,P}\) its Newtonian auxiliary potential, and \(\phi_{\perp,P}\) its excess potential. The variables \(z_P = |\nabla\Phi_{b,P}|^2/a_0^2\) and \(x_P = z_P^{1/4}\) enter the thermal occupation \(n_B(x_P)\) and the auxiliary function \(\mathcal{Q}_\perp(z_P)\). The one-capacity Hessian symbols \(\Delta C\), \(u_C\), \(A_L\), \(A_T\), and \(C_\times\) are retained only as static thermodynamic consistency data in Appendix N; they are not mode frequencies.

Appendix B: UV Boundary Ensemble and Admissibility Closure

The ultraviolet counting problem is finite. A seven-state tetrahedral ensemble, weighted by its closure defect, determines a unique admissibility-closed entropy. Sections 5–6 use this result.

B.1 Tetrahedral package and logical status

The canonical ultraviolet cell is a tetrahedron, the minimal volumetric simplex in three spatial dimensions. Its four faces are distinguishable relational ports. Each side of a shared face carries a primitive half-integer representation \(V_{j_0}\), and the two sides are compared only after one has been transported and orientation-reversed into the frame of the other. Appendix B.4 shows that the positive oriented-matching operator on

$$V_{j_0} \otimes V_{j_0}$$

transmits the maximal coupled multiplet \(V_{2j_0}\).

The value of \(j_0\) has two possible grades. In the selected sharp reversible vertex, unitarity identifies the complete marked carrier with the nonmaximal fusion output, and the representation content fixes

$$j_0 = \frac{3}{2}, \quad V_{2j_0} = V_3, \quad |M| = 7.$$

For a different microscopic vertex, each half-integer \(j_0\) defines a discrete zero-parameter branch, and Appendix B.5 compares those branches through the same downstream map.

The counting convention is then fixed by the cell ontology. A letter is the occupation of one channel in a single-copy cell resource, so one letter cannot be routed through two ports. The ports remain distinguishable, and the two global orientations are distinct. The state count and its raw entropy are therefore

$$\Omega_{\text{tet}} = 2P(7, 4) = 1680, \quad g_{\text{share,max}} = \ln(1680) = 7.42654907240.$$

The closure weighting below acts on this finite ensemble. Appendix B.4 derives the primitive pair selection and the conditional \(7 + 9\) complement structure; Appendix B.5 records the neighboring discrete counts and preserves their postdictive status.

B.2 Closure invariant, kernel, and unique fixed point

The canonical scalar closure invariant is

$$K^2(b) = 48 - \frac{1}{3}(S^2 - \Sigma^2), \quad S = \sum_{i=1}^4 m_i, \quad \Sigma^2 = \sum_{i=1}^4 m_i^2.$$

The admissibility family is

$$p_\eta(b) = \frac{1}{Z(\eta)}e^{-\eta K^2(b)}, \quad Z(\eta) = \sum_{b\in B}e^{-\eta K^2(b)}.$$

Stationary normalized closure evidence fixes the admissibility precision \(\eta\). Classical closure would set the three-component oriented-face sum \(c(b) = \sum_i \hat{n}_i m_i \in \mathbb{R}^3\) to zero. Its squared magnitude is \(|c(b)|^2 = (4\Sigma^2 - S^2)/3\). The invariant \(K^2\) used here is instead the quantum expectation of the squared closure operator. Each face carries the seven-state \(j_{\text{eff}} = 3\) representation (\(2j + 1 = 7\)), and the product boundary state \(|b\rangle = \bigotimes_{i=1}^4 |j, m_i\rangle_{\hat{n}_i}\) uses the tetrahedral normal frame \(\hat{n}_i\cdot\hat{n}_j = -\frac{1}{3}\). The closure operator \(\hat{C}_b = \sum_i \hat{A}_{bi}\) then obeys

$$K^2(b) = \langle b|\hat{C}_b^2|b\rangle = \sum_i j(j + 1) + 2\sum_{i<k}m_i m_k\,\hat{n}_i\cdot\hat{n}_k = 4j(j + 1) - \frac{1}{3}(S^2 - \Sigma^2),$$

with \(4j(j + 1) = 48\) for \(j = 3\). The decomposition

$$K^2 = |c(b)|^2 + \sum_i\left[j(j + 1) - m_i^2\right]$$

exhibits \(K^2\) as the mean nonclosure plus the irreducible quantum variance of the four face operators. The expectation is taken in the unprojected product state. This implements the soft-closure convention: the evidence weight \(e^{-\eta K^2}\) imposes closure statistically, whereas projection onto the gauge-invariant intertwiner subspace would give different expectation values. The seven-label alphabet and the constant 48 are fixed together by the \(j_{\text{eff}} = 3\) product-state boundary realization and the tetrahedral normal frame.

The evidence factor follows from the same operator. The closure operator \(\hat{C}_b\) has three components, and the normalized isotropic quadratic evidence kernel on that three-dimensional defect space carries the determinant weight \(\eta^{3/2}\), so the normalized closure evidence is \(\eta^{3/2}Z(\eta)\) and its logarithm is

$$\mathcal{F}(\eta) = \ln Z(\eta) + \frac{3}{2}\ln\eta.$$

With \(\partial_\eta\ln Z = -\langle K^2\rangle_\eta\), the stationary condition \(\mathcal{F}'(\eta) = 0\) is the closure relation

$$\langle K^2\rangle_\eta = \frac{3}{2\eta},$$

so the factor \(3/2\) is the determinant weight of the three independent closure components, not an equipartition rule imported onto the bounded discrete spectrum; the discreteness, multiplicities, and positive floor of the spectrum remain inside the exact finite sum \(Z(\eta)\). The second derivative is \(\mathcal{F}''(\eta) = \text{Var}_\eta(K^2) - 3/(2\eta^2)\). Global uniqueness follows directly from the spectral bounds. Set \(x_{\text{min}} = 122/3\) and \(x_{\text{max}} = 170/3\). Every stationary point lies in

$$\mathcal{I} = \left[\frac{3}{2x_{\text{max}}}, \frac{3}{2x_{\text{min}}}\right],$$

because \(\mathcal{F}' > 0\) below this interval and \(\mathcal{F}' < 0\) above it. The variance bound for a variable in \([x_{\text{min}}, x_{\text{max}}]\) gives

$$\text{Var}_\eta(K^2) \leq \frac{(x_{\text{max}} - x_{\text{min}})^2}{4} = 64.$$

Throughout \(\mathcal{I}\),

$$\mathcal{F}''(\eta) \leq 64 - \frac{2}{3}x_{\text{min}}^2 = -1038.5185\ldots < 0.$$

The derivative therefore crosses zero once. Since \(\mathcal{F} \to -\infty\) as \(\eta \to 0^+\) and as \(\eta \to \infty\), this point is the unique global maximum. Evaluation gives

$$\eta_* = 0.0298668443935,$$

at which \(\text{Var}_{\eta_*}(K^2) = 15.69\) is dwarfed by \(3/(2\eta_*^2) = 1681.6\), giving \(\mathcal{F}''(\eta_*) = -1665.9 < 0\): \(\eta_*\) is the unique global maximum of the normalized closure evidence. Because the parity-symmetric ensemble is finite, both \(Z(\eta)\) and \(\langle K^2\rangle_\eta\) are exact finite sums over the spectrum. The distinct closure-defect values and their degeneracies are

with total multiplicity 1680 as required. In particular,

$$Z(\eta) = \sum_a n_a e^{-\eta K^2_a}, \quad \langle K^2\rangle_\eta = \frac{\sum_a n_a K^2_a e^{-\eta K^2_a}}{\sum_a n_a e^{-\eta K^2_a}},$$

where \((K^2_a, n_a)\) run over the table above. The closed-branch value \(\eta_*\) is therefore the unique interior maximum of an exact finite-spectrum functional, not an unseen numerical fit. The corresponding effective sharing entropy is

$$g_{\text{share,eff}} = -\sum_{b\in B}p_{\eta_*}(b)\ln p_{\eta_*}(b) = 7.41980002357.$$

The closed-branch moments used in the UV stiffness discussion are

$$\langle K^2\rangle_{\eta_*} = 50.2229154254,$$ $$\text{Var}_{\eta_*}(K^2) = 15.6889750078,$$ $$a_{\text{UV}} \equiv \frac{1}{\text{Var}_{\eta_*}(K^2)} = 0.0637390269.$$

These values quantify the local stiffness of the canonical closure point rather than a tunable phenomenological uncertainty. The closure-saturation product is

$$C_{\text{cl}} := \eta_*\langle K^2\rangle_{\eta_*} = \frac{3}{2}, \quad C_{\text{cl}}^{-1} = \frac{2}{3}.$$

The value \(C_{\text{cl}} = 3/2\) is an identity imposed by the stationarity condition, not an independent numerical cross-check. Its reciprocal is reused as the transverse export factor in the decorated scale-setting branch discussed in Appendix D.4.

The admissibility parameter stops being free here. The kernel introduces \(\eta\), and the closure condition removes its arbitrariness again by demanding that the fluctuation scale produced by the weighting agree with the weighting itself.

B.3 Rooted reduction and local benchmarks

Rooting on the shared face reduces the exact parity-symmetric ensemble to 140 rooted microstates and 69 rooted closure classes. The rooted classes can be labeled by \(\alpha = (m_\bullet, K^2)\), so the same reduced state space supports the local evaluation, the cavity benchmark, and the later shell propagation. Let \(X\) denote the root-face label and \(Y_r\) the boundary record after \(r\) rooted shells. The local information observable

$$\sigma^{(r)}_{\text{ind}} = \frac{H(X \mid Y_r)}{H(X)}$$

has the principal pre-nonlocal benchmarks

$$\sigma^{\text{toy}}_{\text{ind}} = 0.44997, \tag{22}$$ $$\sigma^{\text{loc}}_{\text{ind}} = 0.44708, \tag{23}$$ $$\sigma^{\text{Bethe}}_{\text{ind}}(J = 0) = 0.44749. \tag{24}$$

Here the Bethe value is the homogeneous cavity evaluation on the \(69\times69\) rooted-class interaction graph at zero transport coupling,

$$\mu_\alpha \propto w_\alpha\left(\sum_\beta U_{\alpha\beta}(0)\mu_\beta\right)^{z-1}, \quad \sum_\alpha\mu_\alpha = 1,$$

where \(w_\alpha = n_\alpha e^{-\eta_* K^2_\alpha}\) is the rooted-class Gibbs weight, \(z = 4\), and \(U_{\alpha\beta}(0)\) is the rooted shared-face compatibility matrix before shell transport is turned on. Thus \(\sigma^{\text{Bethe}}_{\text{ind}}(J = 0)\) is the cavity-theory benchmark of the same explicit rooted ensemble. The three benchmark values are legacy single-instance calculations and remain provisional until they are independently rederived. One consequence is already stable enough to constrain the Appendix H.7 fusion layer: a pair coupling confined to the shared fused label carries at most \(\ln 7\) of pair information. Induced fractions below 0.5 therefore require pre-fusion face structure with capacity up to \(\ln 16\). The horizon target implied by the effective sharing entropy is

$$\sigma_* = \frac{\pi}{g_{\text{share,eff}}} = 0.42340665.$$

The gap between the local benchmarks and \(\sigma_*\) arises from shell and loop structure. The local admissibility closure already satisfies its own condition.

The remaining ultraviolet calculation propagates the fixed local closure ensemble through transport and return structure.

B.4 Channel ontology and derived selection

Four statements require separate tests. Positive oriented matching selects one transmitted multiplet \(V_{2j_0}\). The choice of that matching rule is part of the primitive ultraviolet architecture and has a spectral falsifier. In the selected sharp reversible vertex, unitarity fixes the retained carrier to the fusion complement; completeness of the H.9 carrier then fixes \(j_0 = \frac{3}{2}\). If a different microscopic vertex is chosen, \(j_0 = \frac{3}{2}\) remains the surviving member of the discrete audit in B.5.

B.4.1 Setup

Two cells share a face. Each contributes a primitive spin-\(j_0\) slot, so the two-sided space is

$$\mathcal{H}_f = V_{j_0} \otimes V_{j_0} = \bigoplus_{J=0}^{2j_0} V_J.$$

The decomposition is multiplicity-free. The face exports a classical weight alphabet to the admissibility ensemble. The microscopic questions are which coupled sector is transmitted, how the untransmitted information is retained, and why the four port assignments are injective.

B.4.2 Positive oriented matching

Scope. Oriented gluing alone does not select \(j_0\): every half-integer \(j_0\) defines a discrete member with \(4j_0 + 1\) transmitted weights. Sections B.4.2e–f add the selected sharp reversible split. Its unitarity identifies the complete marked carrier with the nonmaximal fusion output, after which the independently fixed spin-one closure response uniquely gives \(j_0 = \frac{3}{2}\). No smallest-spin rule is used.

B.4.2a Face-sector Hessian

Proposition (exact block decomposition). If the gauge-fixed face Hessian commutes with the diagonal \(SU(2)\) action, Schur's lemma gives

$$\boxed{\mathcal{H}^{(2)}(p) = \bigoplus_{J=0}^{2j_0}\left(Z_J p^2 + r_J\right)I_{V_J}.}$$

Thus an \(SU(2)\)-covariant quadratic kernel cannot mix different total-spin sectors. Its dynamical content is the finite spectrum \(\{Z_J, r_J\}\).

For invariant block densities

$$\rho_J = \sum_{M=-J}^J |\sigma_{JM}|^2,$$

write the radial part of the homogeneous potential as

$$V_{\text{eff}} = \sum_J r_J\rho_J + \frac{1}{2}\sum_J u_J\rho_J^2 + \sum_{J<K}w_{JK}\rho_J\rho_K.$$

The first instability lies in the block with the smallest \(r_J\). A pure \(J_*\) saddle, with \(\rho^{(0)}_{J_*} = -r_{J_*}/u_{J_*}\), is locally stable against an unoccupied block \(J\) when

$$m^2_{J|J_*} = r_J - \frac{w_{JJ_*}}{u_{J_*}}r_{J_*} > 0.$$

These are the general tests. The minimal matching action below evaluates them explicitly.

B.4.2b Exact coherent-matching kernel

The same geometric face has opposite outward orientations in its two incident cells. Let \(g_{LR}\) transport the right frame into the left frame and define the right generator in the common orientation by

$$\tilde{J}_R = \text{Ad}_{g_{LR}}J_R^\vee, \quad J^{\vee a}_R = -(J^a_R)^T.$$

The transpose implements the dual representation associated with reversing the outward face orientation. The generators \(J^{\vee a}_R\) obey the usual \(\mathfrak{su}(2)\) algebra and are unitarily equivalent to spin \(j\). Perfect matching compares equal oriented data. If

$$|j, \mathbf{n}\rangle = D^{(j)}(g_\mathbf{n})|j, j\rangle$$

is a spin coherent state, the matched pair is

$$|j, \mathbf{n}\rangle_L \otimes |j, \mathbf{n}\rangle_{\tilde{R}}.$$

The highest-weight identity

$$|j, j\rangle \otimes |j, j\rangle = |2j, 2j\rangle$$

and diagonal covariance imply

$$|j, \mathbf{n}\rangle \otimes |j, \mathbf{n}\rangle = F^\dagger_{2j}|2j, \mathbf{n}\rangle,$$

where \(F_{2j} : V_j \otimes V_j \to V_{2j}\) is the maximal-channel Clebsch–Gordan coisometry.

The direction-independent positive Gram operator of exact matching is

$$\mathcal{G}_{\text{sharp}} = (4j + 1)\int_{S^2}\frac{d\Omega_\mathbf{n}}{4\pi}|j, \mathbf{n}; j, \mathbf{n}\rangle\langle j, \mathbf{n}; j, \mathbf{n}|.$$

The coherent-state resolution of the identity on \(V_{2j}\) gives

$$\boxed{\mathcal{G}_{\text{sharp}} = F^\dagger_{2j}F_{2j} = P_{2j}.}$$

Thus exact oriented matching transmits the maximal coupled multiplet. At \(j_0 = \frac{3}{2}\),

$$\mathcal{G}_{\text{sharp}} = P_3, \quad \dim V_3 = 7.$$

The seven-state link is the image of the pair operator on the full sixteen-state space; it is not inserted as a separate alphabet after the tensor product has been formed.

The contraction alternative. Ordinary \(\varepsilon\)-contraction is a different microscopic operation. It pairs dual legs and leaves an unfused alphabet of \(2j_0 + 1\) states rather than transmitting \(V_{2j_0}\). At \(j_0 = \frac{3}{2}\) this gives four letters and the 48-state branch in B.5. The two operations therefore define distinct gluing ontologies with distinct closure spectra. The present theory adopts oriented fusion because the face must continue to carry closure data after gluing.

B.4.2c Finite-width completion and its falsifier

The quantum mismatch of two perfectly aligned spin-\(j_0\) slots has the irreducible floor \(2j_0\). Subtracting that floor defines

$$Q_f = \frac{1}{2}\left[\left(J_L - \tilde{J}_R\right)^2 - 2j_0 I\right].$$

On \(V_J\),

$$Q_f\big|_{V_J} = \frac{1}{2}[(2j_0)(2j_0 + 1) - J(J + 1)]I_{V_J}.$$

Hence \(Q_f \geq 0\) and its null space is exactly \(V_{2j_0}\). The positive finite-width transfer

$$T_s = e^{-sQ_f}, \quad s > 0,$$

forms a semigroup and converges to \(P_{2j_0}\) as \(s \to \infty\). For \(j_0 = \frac{3}{2}\),

$$\boxed{Q_f = 6P_0 + 5P_1 + 3P_2, \quad T_s = e^{-6s}P_0 + e^{-5s}P_1 + e^{-3s}P_2 + P_3.}$$

A minimal homogeneous action on the complete primitive space is

$$\Gamma_f = \int d^dx\left[Z|\partial\Phi|^2 + r\,\Phi^\dagger\Phi + \kappa\,\Phi^\dagger Q_f\Phi + \frac{u}{2}(\Phi^\dagger\Phi)^2\right], \quad Z, \kappa, u > 0.$$

Its potential is

$$V = r\rho + \frac{u}{2}\rho^2 + 6\kappa\rho_0 + 5\kappa\rho_1 + 3\kappa\rho_2, \quad \rho = \sum_{J=0}^3\rho_J.$$

For \(r < 0\), the fusion-sector support of the homogeneous minimum is uniquely \(J = 3\):

$$\rho_3 = -\frac{r}{u}, \quad \rho_0 = \rho_1 = \rho_2 = 0.$$

The orthogonal blocks have positive masses

$$m^2_{0|3} = 6\kappa, \quad m^2_{1|3} = 5\kappa, \quad m^2_{2|3} = 3\kappa.$$

At fixed total density, every nonmaximal occupation raises the action, so mixed-\(J\) support is excluded globally in this minimal model.

Theorem (maximal-fusion blocking ray). Let \(N\) transmitted cells each carry the selected irrep \(V_3\). The highest-spin irrep \(V_{3N}\) occurs with multiplicity one in \(V_3^{\otimes N}\). Therefore the iterated maximal-channel Clebsch–Gordan coisometries define, after one consistent phase convention, a unique map

$$F_N : V_3^{\otimes N} \longrightarrow V_{3N}$$

independent of the binary fusion tree. Associativity here is not an additional dynamical assumption: every binary tree that always chooses the maximal coupled irrep ends in the same multiplicity-one copy of \(V_{3N}\), so two such intertwiners differ only by an overall phase.

For spin coherent states the map is exact,

$$\boxed{F_N|3, \mathbf{n}\rangle^{\otimes N} = |3N, \mathbf{n}\rangle.}$$

Thus a fixed microscopic \(j = 3\) alphabet is compatible with an arbitrarily large semiclassical block spin on the selected maximal-channel branch; the elementary representation does not run under this canonical blocking map.

Lemma (equivariant lock under maximal blocking). Let \(L_3 : V_3^{\text{cap}} \to V_3^{\text{geom}}\) be the normalized one-cell equivariant lock of Appendix Q, and let \(F_N^{\text{cap}}\) and \(F_N^{\text{geom}}\) denote normalized maximal-channel fusion maps on the two factors. Because \(V_{3N}\) occurs with multiplicity one in \(V_3^{\otimes N}\), the two normalized \(SU(2)\)-equivariant maps from \((V_3^{\text{cap}})^{\otimes N}\) to \(V_{3N}^{\text{geom}}\) can differ only by a phase. Hence there is \(e^{i\phi_N}\) such that

$$\boxed{F_N^{\text{geom}}L_3^{\otimes N} = e^{i\phi_N}L_{3N}F_N^{\text{cap}}.}$$

Thus, conditional on the selected one-cell lock, its propagation to every maximal block is unique and parameter free. This is a representation-theoretic theorem; it does not by itself prove that the full interacting coarse-graining map dynamically projects arbitrary blocks onto the maximal channel.

A dynamical attraction toward the same ray is exhibited within the displayed positive all-pairs diagnostic model

$$H_N = \kappa\sum_{a<b}(9 - J_a\cdot J_b) = \frac{\kappa}{2}\left[(3N)(3N + 1) - J^2_{\text{tot}}\right].$$

It obeys

$$\ker H_N = V_{3N}, \quad \Delta_N \equiv E_{3N-1} - E_{3N} = 3\kappa N.$$

The increasing gap is a diagnostic of maximal-spin phase selection; the exact fusion map above does not require an all-to-all microscopic interaction.

The same conclusion is visible directly in the face mismatch. For equal spin \(j\),

$$Q_j = j^2 - J_L\cdot\tilde{J}_R,$$

and on coherent data separated by angle \(\theta\),

$$\boxed{\langle Q_j\rangle = j^2(1 - \cos\theta) = \frac{1}{2}j^2\theta^2 + O(\theta^4).}$$

Hence \(e^{-sQ_j}\) has angular width \(\Delta\theta \sim (j\sqrt{s})^{-1}\). On a maximal block \(j = 3N\) the accepted geometric matching therefore sharpens rather than broadens as \(N\) grows.

Corollary (additive channel area and the coarse area operator). The selected factorized completion of Appendix Q locks each transmitted capacity multiplet \(V_3^{\text{cap}}\) to a geometric \(V_3^{\text{geom}}\) in the same relational orientation. On the coherent branch the geometric face variable is the flux vector itself: a spin-\(J\) coherent face carries the classical/Regge area proportional to \(J\), while the kinematical LQG Casimir operator has eigenvalue proportional to \(\sqrt{J(J + 1)}\) [113, 114]. For \(N\) maximally fused primitive faces,

$$J_N = 3N.$$

Area additivity of the \(N\) fine facets therefore fixes the physical coarse area observable uniquely. If

$$\hat{A}_{\text{LQG}}(J) = 8\pi\gamma_{\text{BI}}L_*^2\sqrt{J(J + 1)},$$

then the renormalized operator on the maximal-fusion ray is

$$\boxed{\hat{A}^{(N)}_{\text{phys}} = Z_A(N)\hat{A}_{\text{LQG}}(3N), \quad Z_A(N) = \frac{3N}{\sqrt{3N(3N + 1)}} = \sqrt{\frac{3N}{3N + 1}}.}$$

Hence

$$\hat{A}^{(N)}_{\text{phys}} = 24\pi\gamma_{\text{BI}}NL_*^2, \quad Z_A(1) = \frac{\sqrt{3}}{2}, \quad Z_A(N) \to 1.$$

The microscopic spin remains fixed. The factor \(Z_A(N)\) renormalizes the coarse observable so that one irrep \(V_{3N}\) represents the additive area of \(N\) coherent \(V_3\) facets. It reconciles the exact small-\(J\) Casimir spectrum with the linear area of the Regge phase, and approaches unity in the large-block limit. Appendix F.5 fixes the remaining dimensionless normalization \(\gamma_{\text{BI}}\) from the one-bit cut entropy.

Controlled semiclassical-refinement witness. The exact blocking ray can be embedded in the established spin-foam semiclassical–continuum limit of Refs. [121, 122]. To avoid collision with the couplings elsewhere in this paper, denote their typical large-spin parameter by \(\Lambda_{\text{sf}}\). Their allowed family includes

$$\Lambda_{\text{sf}}(\mu) = \Lambda_0\mu^{-2+u}, \quad 0 < u < \frac{2}{5}, \quad \mu \to 0.$$

Choose \(u = 1/4\) and identify the spin-foam large-spin scale only up to the fixed normalization carried by the blocked representation,

$$N(\mu) = \lfloor N_0\mu^{-7/4}\rfloor, \quad J_{\text{block}} = 3N, \quad \Lambda_{\text{sf}} \propto J_{\text{block}} = 3N.$$

The constant factor is asymptotically irrelevant and is absorbed into \(\Lambda_0\); no equality between the symbol \(\Lambda_{\text{sf}}\) and the cell count is required. One admissible regulator choice is

$$\delta(\mu) = \delta_0\mu^{3/4},$$

which lies inside the semiclassical–continuum window for sufficiently small \(\mu\). The physically relevant refinement variable is the Regge edge length measured in the running infrared unit \(\mu^{-1}\), not an identification of one capacity cell with a fixed geometric edge. With \(\ell_{\text{block}} \sim \sqrt{\gamma\Lambda_{\text{sf}}}L_P\) and \(L_P(G_*) = L_*\) on the matched normalization,

$$\ell_{\text{rel}}(\mu) \equiv \mu\ell_{\text{block}} \propto \mu^{1/8} \longrightarrow 0$$

(up to the fixed dimensionless spin-foam normalization). This use of \(L_*\) fixes the gravitational unit; it does not assign a literal edge length \(L_*\) to the microscopic capacity cell. Along this explicit witness trajectory,

$$\Lambda^{-1}_{\text{sf}} = O(\mu^{7/4}), \quad \Delta\theta = O(\mu^{7/4}), \quad \delta = O(\mu^{3/4}), \quad \ell^2_{\text{rel}} = O(\mu^{1/4}).$$

Along this trajectory, the finite-spin correction, capacity-matching width, transverse spin-sum regulator, and leading lattice-curvature correction all vanish while the microscopic constituent remains exactly \(j = 3\). In the controlled linearized sector, Ref. [122] proves that the low-energy excitations exhaust the smooth solutions of the linearized Einstein equations: the two helicity-2 graviton modes. Ref. [121] gives the broader Lorentzian/Euclidean Einstein–Regge semiclassical-continuum construction. No general nonlinear convergence theorem is claimed here.

The microscopic test is finite. For the reduced physical Hessian define

$$h_J = \frac{1}{2J + 1}\text{Tr}_{V_J}(P_J\mathcal{H}_f P_J).$$

The weak pass condition is

$$h_3 < h_2, \quad h_3 < h_1, \quad h_3 < h_0.$$

The minimal-kernel target is

$$\boxed{(h_0 - h_3) : (h_1 - h_3) : (h_2 - h_3) = 6 : 5 : 3.}$$

After condensation the remaining tests are \(h_J + w_{J3}\rho_3 > 0\) for \(J = 0, 1, 2\). The construction fails if another block is lighter, inequivalent \(J\) sectors mix after gauge reduction, a stable mixed-\(J\) minimum survives, or the primitive transfer is not positive.

B.4.2d Primitive tensor and effective pushforward

For a shared face \((ab)\), let \(Q_{ab}\) denote the transported mismatch operator above. A concrete finite-width primitive tensor is

$$\mathcal{G}^{(0)}_{v,s} = P^{\text{inv}}_v\left[\bigotimes_{(ab)\in E(v)}e^{-sQ_{ab}}\right]P^{\text{inv}}_v,$$

where \(P^{\text{inv}}_v\) is the cellwise gauge/intertwiner projector. The zero-parameter primitive theory is defined by its sharp form,

$$\boxed{\mathcal{G}^{(0)}_{v,\sharp} = P^{\text{inv}}_v\left[\bigotimes_{(ab)\in E(v)}P^{(ab)}_3\right]P^{\text{inv}}_v.}$$

The finite-\(s\) family is a positive regulator and a spectral falsifier, not an additional coupling of the sharp theory.

Let

$$F_3 : V_{3/2} \otimes V_{3/2} \longrightarrow V_3$$

obey \(F^\dagger_3 F_3 = P_3\) and \(F_3 F^\dagger_3 = I_{V_3}\). On an isolated face,

$$\boxed{F_3 T_s F^\dagger_3 = I_{V_3}}$$

for every \(s > 0\). For the complete gauge-projected vertex, finite-\(s\) independence additionally requires \(P^{\text{inv}}_v\) to preserve the facewise coupled-\(J\) decomposition, or equivalently that the spectator/intertwiner dressing be channel-blind. This condition requires a reduced-kernel or commutator audit. The sharp \(P_3\) support survives whenever its overlap is nonzero; finite-width spectral ordering remains an explicit test.

With \(F_v = \bigotimes F_3\), the post-fusion geometric tensor is

$$\mathcal{G}^{\text{eff}}_v = F_v\mathcal{G}^{(0)}_{v,\sharp}F^\dagger_v.$$

The marked-transfer vertex of Appendix H decorates this effective object. It is not itself the primitive pair kernel.

B.4.2e Exact marked complement and conditional sharp-kernel selection

Let a marked response be built from two spin-\(s\) strands. Then

$$\mathcal{H}_{\text{mark}}(s) = V_s^P \otimes V_s^H = \bigoplus_{J=0}^{2s}V_J.$$

Maximal fusion of two primitive spin-\(j\) slots transmits \(V_{2j}\) and leaves

$$Q(j) = (V_j \otimes V_j) \ominus V_{2j} = \bigoplus_{J=0}^{2j-1}V_J.$$

Theorem (unique complement match). The representations are isomorphic if and only if

$$\boxed{\mathcal{H}_{\text{mark}}(s) \cong Q(j) \iff j = s + \frac{1}{2}.}$$

Proof. Both decompositions are multiplicity-free. The marked space contains one copy of each \(V_J\) from \(J = 0\) through \(2s\); the fusion complement contains one copy from \(J = 0\) through \(2j - 1\). They agree exactly when \(2s = 2j - 1\). Equivalently,

$$\dim Q(j) = (2j + 1)^2 - (4j + 1) = 4j^2, \quad \dim \mathcal{H}_{\text{mark}}(s) = (2s + 1)^2,$$

and the positive solution is \(2j = 2s + 1\). \(\square\)

The marked closure response is independently a three-component vector and therefore has \(s = 1\). In the selected sharp reversible split, the complete retained output is the fusion complement and the complete H.9 one-mark carrier has the same nine dimensions. Hence the carrier-space identification is isometric and

$$\boxed{j_0 = \frac{3}{2}.}$$

Explicit complement isometry. On \(V_{3/2} \otimes V_{3/2}\) let

$$P_Q = P_0 + P_1 + P_2.$$

The blockwise equivariant isometry

$$\mathcal{I} : Q \longrightarrow V_1^P \otimes V_1^H$$

is defined by

$$\mathcal{I}|(j_0 j_0); J, M\rangle = |(1_P 1_H); J, M\rangle, \quad J = 0, 1, 2.$$

The independent present and history strands have ordered first-excitation basis

$$|a, b\rangle = |a\rangle_P \otimes |b\rangle_H, \quad \langle a, b|c, d\rangle = \delta_{ac}\delta_{bd}.$$

Let \(W\) map this basis to the nine hard-core marked states,

$$W|a, b\rangle = d^\dagger_{ab}|0\rangle.$$

The exact unit Gram matrix gives

$$W^\dagger W = I_9, \quad WW^\dagger = I_{\mathcal{H}_d}.$$

Define

$$L = W\mathcal{I}P_Q.$$

Then

$$\boxed{L^\dagger L = P_Q, \quad LL^\dagger = I_{\mathcal{H}_d}.}$$

The complete sharp pair map is

$$U_{\text{sharp}}|\psi\rangle = F_3 P_3|\psi\rangle \otimes |0\rangle_R + |\varnothing\rangle \otimes L|\psi\rangle.$$

The two flags are orthogonal, so

$$\boxed{U^\dagger_{\text{sharp}}U_{\text{sharp}} = P_3 + P_Q = I_{16}.}$$

The seven-state link and nine-state marked record are therefore complementary outputs of an explicit reversible pair map. More generally, writing the retained output as \(C|\psi\rangle\), unitarity of

$$U|\psi\rangle = F_3 P_3|\psi\rangle|0\rangle + |\varnothing\rangle C|\psi\rangle$$

gives

$$P_3 + C^\dagger C = I, \quad \boxed{C^\dagger C = P_Q.}$$

Because the complete H.9 carrier and \(Q\) both have dimension nine, \(C\) is an isometric identification onto that carrier. This proves the carrier-space incidence in the selected sharp vertex. It does not derive the hard-core operator realization, the flags, or the detailed routing and feedback graph.

Since \(V_0\), \(V_1\), and \(V_2\) occur once, Schur's lemma implies that every other equivariant minimal complement isometry differs by one phase on each block,

$$L' = e^{i\phi_0}L_0 \oplus e^{i\phi_1}L_1 \oplus e^{i\phi_2}L_2.$$

If no microscopic operator compares the blocks coherently, these phases are marked-field conventions. A vertex that does compare them must derive their relative phases.

Why the unit marked weights select the sharp limit. For finite \(s\), the system Kraus amplitude in the canonical dilation is \(T_s^{1/2}\) and the complementary amplitude is \(\sqrt{I - T_s}\). Its squared norms on the three nonmaximal blocks are

$$\ell^2_0(s) = 1 - e^{-6s}, \quad \ell^2_1(s) = 1 - e^{-5s}, \quad \ell^2_2(s) = 1 - e^{-3s}.$$

The H.9 marked vertex instead has one common quadratic normalization,

$$G_{\text{mark}} = I_9 = P_0 + P_1 + P_2.$$

Therefore, within the minimal incidence identification in which this marked vertex is the canonical fusion environment and no compensating \(J\)-dependent coupling is added, exact equality of the marked weights requires

$$\boxed{s \to \infty, \quad T_s \to P_3, \quad I - T_s \to P_Q.}$$

A finite physical width could survive only with additional block-dependent normalization. The sharp conclusion holds within the selected reversible carrier incidence. For a different microscopic vertex, the equal H.9 Gram matrix and the finite-width fusion transfer describe distinct objects and impose no constraint on one another.

B.4.2f Conditional dimensional endpoint

The closure defect is a spatial vector,

$$C_a, \quad a = 1, 2, 3.$$

In three spatial dimensions the vector representation of \(SO(3)\) lifts to the spin-one irrep of \(SU(2)\):

$$d = 3 \implies \mathbb{R}^3 \cong V_1 \implies s = 1.$$

Within the selected sharp carrier incidence, the theorem above gives

$$s = 1 \implies j_0 = \frac{3}{2} \implies V_{2j_0} = V_3 \implies |M| = 7.$$

The minimal three-dimensional simplex has four faces, and the present/history vector record has \(3^2 = 9\) states. Single-copy channel capacity makes the four port assignments injective, while the two global orientations give

$$\Omega = 2P(7, 4) = 1680.$$

The complete conditional integer chain is therefore

$$d = 3 \implies s = 1 \implies j_0 = \frac{3}{2} \implies \begin{cases}\text{ports} = 4,\\\text{alphabet} = 7,\\\text{record} = 9,\\\Omega = 1680.\end{cases}$$

For the physical \(d = 3\) rotation algebra one may summarize the evaluated identities as

$$\text{ports} = d + 1, \quad |M| = 2d + 1, \quad \dim\mathcal{H}_{\text{mark}} = d^2.$$

They are not a claimed continuation to arbitrary dimension. For \(d \neq 3\), the vector representation belongs to \(\text{Spin}(d)\) and need not be one \(SU(2)\) spin. A dimension-general theorem would require repeating the complement analysis in that group.

B.4.3 Injectivity from single-copy channel capacity

The exclusion statement concerns a single cell resource, not Pauli antisymmetry among the four port labels.

Lemma. Assume:

1. the letters index a single copy of the cell-level fermionic channel family, with occupations \(n_m \in \{0, 1\}\);
2. a face displaying \(m\) is, by definition, that channel occupation routed through the face's port;
3. the four ports are distinguishable relational registers.

Then the admissible boundary states are the injections \(f : F_4 \to M\), and their number before orientation doubling is \(P(|M|, 4)\).

Proof. If two ports displayed the same letter \(m\), the one cell resource would require \(n_m = 2\), contrary to its single-copy ceiling. The assignment is therefore injective. The ports are distinct interfaces to distinct neighbors, so permuting letters between ports changes the routing state; the assignments are ordered. Antisymmetry acts in the occupation algebra of each channel, not by quotienting the relational port index. \(\square\)

This lemma does not follow from generic fermionic statistics alone. Its physical content is that the four ports draw from one shared channel multiplet rather than four independent copies. That is the same single-copy channel ontology used by the one-bit defect and determinant sectors.

B.4.4 Uniform base measure

Lemma. If the selected face multiplet realizes its full Shannon capacity, its unconditioned base measure is uniform.

Proof. For a probability measure \(\mu\) on \(M\),

$$H(\mu) \leq \ln|M|,$$

with equality only at \(\mu(m) = 1/|M|\). Full use of the fixed alphabet therefore selects the uniform prior. \(\square\)

This statement concerns the reference measure, not the final closure-weighted ensemble:

$$p_\eta(b) = \frac{1}{Z(\eta)}\underbrace{\mu_0(b)}_{\text{uniform base measure}}e^{-\eta K^2(b)}.$$

Covariance alone does not force uniformity, because the face normal permits covariant deformations such as functions of \((J\cdot\hat{n})^2\). Full-capacity saturation excludes those deformations at the prior level; the closure observable then supplies the physical nonuniformity.

B.4.5 Status ledger

Derived from the primitive matching rule. For any fixed half-integer \(j_0\), exact oriented coherent matching gives the transmitted block \(V_{2j_0}\) and an alphabet of \(4j_0 + 1\) states. The single-copy channel ontology gives injectivity; relational ports preserve ordered assignments; and the global face orientation gives the binary factor.

Derived in the selected sharp carrier incidence. The exact relation

$$\mathcal{H}_{\text{mark}}(s) \cong Q(j) \iff j = s + \frac{1}{2}$$

combines with the three-dimensional vector response \(s = 1\) to fix \(j_0 = \frac{3}{2}\), seven transmitted states, nine complement states, and \(\Omega = 1680\). Unitarity identifies that complement with the complete nine-state H.9 carrier in the selected sharp reversible split. The exact unit normalization of the marked vertex selects the sharp \(P_3\) kernel within the minimal canonical dilation.

Microscopic premise. The primitive action implements positive oriented matching. The displayed sharp tensor \(\mathcal{G}^{(0)}_{v,\sharp}\) is the minimal completion. A more fundamental simplicial calculation must still reproduce its \(J = 3\) support or pass the weaker Hessian inequalities of B.4.2c.

Carrier incidence and selected microscopic detail. The representation equivalence, \(L^\dagger L = P_Q\), \(LL^\dagger = I_9\), and \(U^\dagger_{\text{sharp}}U_{\text{sharp}} = I_{16}\) are exact. They prove the carrier-space identification in the selected sharp reversible vertex. The hard-core realization, flags, relative block phases, routing, and feedback graph remain selected microscopic structure rather than consequences of the representation theorem alone.

Data status. Within the selected sharp vertex, \(j_0 = \frac{3}{2}\) is internally fixed by the carrier incidence and the audit below becomes a consistency check. For a different microscopic vertex, the same value is selected by the postdictive discrete audit. In neither reading does a continuous parameter choose the alphabet.

B.5 Relaxation and branch audits

The following tables propagate nearby discrete constructions through one fixed downstream map. They are alternatives, not samples from a probability distribution, so "sigma" significances do not apply. The marked sector keeps nine response polarizations. A branch with alphabet size \(n\) uses \(\binom{n}{2}\) pair records, the scalar factor \(2/n\), and the second-return coefficient \(n\). Because the Newton target informed the historical construction, Appendix L grades this as a conditional postdictive comparison.

B.5.1 Fusion and constituent-spin branches

The two 48-state rows have the same count but different representation content: the first mixes \(V_0\) and \(V_1\), while the second carries the unfused \(V_{3/2}\) alphabet. The family spans about \(10^{143}\) in \(G_*/G\). Holding the rest of the construction fixed, the seven-state branch is the only row near the measured Newton scale.

B.5.2 Counting conventions at seven letters

These rows change the counting convention while retaining the same seven-letter closure formula and downstream transport. The baseline is the only displayed convention that keeps the Newton and charged-lepton comparisons simultaneously near their measured values.

B.5.3 One continuous prior deformation

To test one nearby nonuniform prior, introduce the parity-even, face-axis-preserving tilt

$$w_\lambda(b) = \exp\left[-\lambda\sum_{i=1}^4 m_i^2\right]$$

and re-solve the closure condition at each \(\lambda\). At \(\lambda = 0\), exact finite differences of the enumerated ensemble give

$$\frac{dg_{\text{share,eff}}}{d\lambda} = -0.278458, \quad \frac{d\ln G_*}{d\lambda} = 3.90135, \quad \frac{d\ln(m_\mu/m_e)}{d\lambda} = 1.37074 \times 10^{-3}.$$

With the present CODATA uncertainty and the baseline residual retained, the one-standard-deviation Newton interval along this one deformation is approximately

$$-5.3 \times 10^{-6} < \lambda < 6.2 \times 10^{-6}.$$

For the muon ratio, \(|\Delta\lambda| \simeq 1.6 \times 10^{-5}\) produces one current experimental standard deviation locally. This is a sensitivity scale, not a symmetric bound about zero, because the baseline prediction is not exactly centered. These statements constrain one specified deformation conditional on the frozen downstream map; they do not directly measure the microscopic prior or exhaust all nonuniform measures. Parity-odd tilts have zero linear response by the \(m \mapsto -m\) symmetry and require a separate quadratic audit.

Provenance. The entropy near 7.42 was recognized historically by inverting the measured Newton constant, and the decorated vertex was constructed after the remaining Newton and lepton residuals were known. Appendix L retains that chronology. The new result is structural: once the primitive oriented-matching rule and the marked-fusion incidence identification are adopted, the sequence

$$d = 3 \implies V_1 \implies j_0 = \frac{3}{2} \implies V_3 \implies 7 \implies 1680 \implies g_{\text{share,eff}} = 7.419800\ldots$$

contains no adjustable continuous parameter. The branch tables show how sharply nearby stated constructions move the downstream observables; they do not turn the original comparison into a historically blind prediction.

Appendix C: Edge Kernel, Finite Renormalization, and Continuum Matching

Appendix C carries the local boundary ensemble of Appendix B into edge transport, loop dressing, and the continuum stiffness coefficient of the weak-field EFT.

C.1 Channel-averaged isotropy identity and tree coupling

Let \(\hat{n}_i\) be the four face normals of a regular tetrahedron. The exact identity

$$\sum_{i=1}^4 \hat{n}_i\hat{n}_i^T = \frac{4}{3}I_3$$

implies a channel-averaged transverse fraction of \(2/3\). The bare edge stiffness is therefore

$$J_{\text{bare}} = \frac{2}{3}\eta_* = 0.0199112296.$$

For a rooted \(z = 4\) coarse adjacency graph, the tree-to-lattice map gives

$$J^{\text{tree}}_{\text{eff}} = \frac{J_{\text{bare}}}{z - 1} = \frac{2\eta_*}{9} = 0.0066370765.$$

This is the first place where local closure data become a transport law. The tetrahedral identity fixes the isotropic projection, and the rooted branching structure determines how much of the microscopic edge penalty survives as net outward propagation on the coarse graph.

C.2 Horizon target and shell convergence

The horizon-capacity target is

$$\sigma_* = \frac{\pi}{g_{\text{share,eff}}} = 0.42340665.$$

At the derived coupling the explicit shell values are

$$\sigma^{(2)}_{\text{ind}} = 0.42143, \quad \sigma^{(3)}_{\text{ind}} = 0.42166, \quad \Delta_{2\to3} = 0.00023.$$

The residual shift from the target is already small and stable by shell depth \(r = 2\), isolating the remaining correction to the loopy local-return sector rather than a broad nonlocal ambiguity. The shell calculation places the tree branch near the target and assigns the residual discrepancy to local returns. No broad long-range correction is required at this order.

C.3 Finite-loop self-energy closure

The minimal loopy correction is organized as a local Dyson dressing:

$$J^{(\text{ren})}_{\text{eff}} = \frac{J^{\text{tree}}_{\text{eff}}}{1 + J^{\text{tree}}_{\text{eff}}\Sigma_{\text{ret}}}.$$

The specified minimal return operator uses

$$\Sigma_{\text{ret}} = 7 + \frac{2}{9} = \frac{65}{9},$$

The two terms have distinct return-channel origins. A short return motif leaves a shared face, explores a local closed loop, and re-enters the same coarse edge before contributing to net long-range transport. In the canonical label basis \(m = -3, -2, \ldots, 3\), the minimal operator contains seven label-diagonal returns, one for each face-label channel, contributing

$$\text{Tr}(I_7) = 7.$$

In addition to these label-preserving loops, permutation symmetry allows one collective mode shared across all channels. Writing

$$P_{\text{sing}} = |u\rangle\langle u|, \quad u = \frac{1}{\sqrt{7}}(1, 1, \ldots, 1),$$

this shared return is rank one. Permutation symmetry permits this singlet but does not fix its weight relative to the diagonal returns or exclude longer return motifs. The minimal completion assigns it the same \(2/3\) transverse projection used in the tree coupling and the rooted return factor \(1/(z - 1) = 1/3\) on the \(z = 4\) graph. The collective contribution is then

$$\text{Tr}\left(\frac{2}{3}\frac{1}{3}P_{\text{sing}}\right) = \frac{2}{9},$$

since \(\text{Tr}(P_{\text{sing}}) = 1\). Equivalently,

$$R_{\text{ret}} = I_7 + \frac{2}{9}P_{\text{sing}}, \quad \Sigma_{\text{ret}} = \text{Tr}(R_{\text{ret}}) = 7 + \frac{2}{9}.$$

Thus \(65/9\) is fixed inside the stated minimal return operator, not derived from the closure ensemble alone. A microscopic graph-return calculation must derive the relative diagonal and singlet weights, test longer motifs, and determine whether the operator closes on these channels. Within this conditional completion,

$$c^{(\text{ren})}_{\text{loop}} \equiv \frac{J^{(\text{ren})}_{\text{eff}}}{J^{\text{tree}}_{\text{eff}}} = \frac{1}{1 + J^{\text{tree}}_{\text{eff}}\Sigma_{\text{ret}}} \approx 0.95426,$$

and

$$J^{(\text{ren})}_{\text{eff}} \approx 0.00633348.$$

This reproduces the shell-target crossing near \(J_{\text{bare,cross}} \sim 0.019\) at the stated level of agreement. The local Dyson dressing corrects the tree branch for short motifs that recycle amplitude before it contributes to coarse transport. Summing those returns determines the renormalized coupling from the tree coupling.

C.4 Euclidean-action normalization and continuum stiffness

The lattice quadratic form is interpreted canonically as a Euclidean action weight,

$$\frac{I_E}{\hbar} = \frac{J^{(\text{ren})}_{\text{eff}}}{2}\sum_{a,i}(Q_a - Q_{a+L_*\hat{n}_i})^2,$$

where \(a\) runs over one sublattice representative of each bipartite primitive cell and \(i\) runs over its four outgoing bonds. Every undirected nearest-neighbor edge is counted once. This convention is essential: summing the four bonds from both sublattices would double the variation and the source normalization. The microscopic four-cell is

$$\Delta V_4 = \frac{L_*^4}{c},$$

a coarse-graining convention rather than derived cell geometry (Section 9; the abstract cell complex has no regular-tetrahedron Euclidean embedding). Because every edge occurs once, varying the discrete action gives \(J^{(\text{ren})}_{\text{eff}}L_\diamond Q = s\) for a source term \(-\sum_a s_a Q_a\), with \(L_\diamond = 4I - A\). The diagonal inverse of this same unnormalized Laplacian is the \(G_{\text{tet}}(0)\) used in C.5, so a point source obeys \(Q(0) = sG_{\text{tet}}(0)/J^{(\text{ren})}_{\text{eff}}\). This removes the factor-of-two ambiguity between the stiffness and source conventions. The same tetrahedral identity then yields

$$\gamma_Q = \frac{4\hbar c}{3L_*^2}J^{(\text{ren})}_{\text{eff}}$$

for the occupancy field \(Q_{\text{occ}}\). With the horizon-capacity normalization

$$S = \pi Q_{\text{occ}},$$

the canonical convention \(\frac{\gamma}{2}(\partial S)^2\) gives

$$\gamma = \frac{4\hbar c}{3\pi^2 L_*^2}J^{(\text{ren})}_{\text{eff}} = \frac{4\hbar c}{3\pi^2 L_*^2}\frac{2\eta_*/9}{1 + (2\eta_*/9)(65/9)}.$$

Using the substrate-induced scale

$$G_* := \frac{c^3 L_*^2}{\hbar},$$

this is

$$\gamma = \frac{4J^{(\text{ren})}_{\text{eff}}}{3\pi^2}\frac{c^4}{G_*} \approx 8.556 \times 10^{-4}\frac{c^4}{G_*}.$$

This step converts the dimensionless lattice weighting into the dimensionful continuum stiffness used by the weak-field action, after Euclidean normalization and faithful sector-resolution scale setting.

C.5 Local defect insertion and the source-side lattice constant

The stiffness-side matching is not the only UV quantity that can be closed locally. For the canonical rigid defect insertion, excluding one of the seven admissible face labels from one face removes exactly one-seventh of the isotropically averaged local partition weight. Therefore the logarithm of the isotropically averaged partition ratio is exactly

$$\Delta S_{\text{def}} := -\ln\left\langle\frac{Z_{\text{def}}}{Z_{\text{vac}}}\right\rangle_{\text{iso}} = \ln\frac{7}{6}.$$

This is the exact isotropic source benchmark in the canonical seven-label ensemble. The isotropically averaged defect free-energy cost differs from it only at \(O(10^{-5})\) because the admissibility weighting breaks label symmetry only weakly.

The local source benchmark \(\ln(7/6)\) should not be confused with the elementary fermionic one-bit anchor \(\ln 2\). The former is the isotropically averaged partition-ratio shift produced by removing one admissible label from the seven-label boundary ensemble. The latter is the intrinsic binary entropy of an elementary occupied/unoccupied fermionic face-exclusion defect. The source theorem uses \(\ln(7/6)\) to normalize the local scalar insertion into the lattice response, while the electron anchor uses \(\ln 2\) to fix the mass–entropy unit of the elementary fermionic defect.

Exclusion sufficiency lemma. The reduction of the 1680-state boundary ensemble to a single scalar source is, at the classical source level, exact rather than approximate. The canonical exclusion defect is the vacuum ensemble conditioned on the exclusion event \(A\), \(P_{\text{def}} = P_{\text{vac}}(\cdot|A)\), so its likelihood ratio is

$$\frac{dP_{\text{def}}}{dP_{\text{vac}}} = \frac{\mathbf{1}_A}{P_{\text{vac}}(A)},$$

a function of the exclusion indicator \(X = \mathbf{1}_A\) alone. The indicator is therefore a sufficient statistic for distinguishing the defect ensemble from the vacuum, and coarse-graining onto it preserves the full classical relative information,

$$D(P_{\text{def}}\|P_{\text{vac}}) = -\ln P_{\text{vac}}(A) = D(P^X_{\text{def}}\|P^X_{\text{vac}}).$$

The scope of the lemma is exactly its statement: one scalar statistic carries all the local classical source information distinguishing the canonical exclusion-defect ensemble from the vacuum. It does not by itself establish that the condensate has a single infrared mode, that orientation or defect-species structure is dynamically irrelevant, or that spatial correlations carry no further information; those are dynamical questions, and the open one is stated in Appendix H.7. For independent exclusions the source is exactly additive, \(-\ln P(\bigcap_i A_i) = \sum_i -\ln P(A_i)\), so the leading dilute source is linear in the defect number, with correlation corrections entering at pair order in the density.

To propagate that local defect into the lattice field equation one needs the on-site Green function of the tetrahedral/diamond nearest-neighbor Laplacian. The four-valent diamond graph is adopted as the coarse adjacency realization of the abstract face-sharing tetrahedral boundary complex; the complex enters combinatorially, since regular tetrahedra admit no face-to-face Euclidean tessellation (the dihedral angle \(\arccos(1/3) \simeq 70.53°\) does not divide \(360°\)), so the adopted graph and its Green function are the working objects while cell volumes and face areas remain conventions of the coarse map. Eliminating the two-sublattice structure gives the standard Brillouin-zone representation for the diamond lattice Green function [7]

$$G_{\text{tet}}(0) = \frac{1}{(2\pi)^3}\int_{[-\pi,\pi]^3}\frac{4\, d^3k}{16 - |1 + e^{ik_1} + e^{ik_2} + e^{ik_3}|^2}.$$

This integral is the reproducible source-side lattice constant: it is the self-energy of a unit point insertion for the same scalar mode whose long-wavelength stiffness was matched in Appendix C.4. Joyce's exact evaluation of the diamond-lattice Green function, in this normalization, gives [10, 7]

$$G_{\text{tet}}(0) = \frac{3\Gamma(1/3)^6}{2^{14/3}\pi^4} = 0.4482203943883814\ldots.$$

Thus the source-side graph constant is an exact lattice invariant rather than a fitted numerical coefficient. Direct quadrature with endpoint extrapolation reproduces the same value.

Green-tensor transverse response. The same graph response also fixes the transverse export weight \(w_\perp\), the single graph quantity that propagates downstream into the horizon channel count of Appendix F.5 and into the global \(\ln(3/2)\) closure-saturation factor of the scale-setting relation. We compute it directly from the four nearest-neighbor bond frame, with no fitting freedom.

Let \(d_i\), \(i = 1, \ldots, 4\), be the four diamond nearest-neighbor bond directions,

$$d_1 = \frac{(1, 1, 1)}{\sqrt{3}}, \quad d_2 = \frac{(1, -1, -1)}{\sqrt{3}}, \quad d_3 = \frac{(-1, 1, -1)}{\sqrt{3}}, \quad d_4 = \frac{(-1, -1, 1)}{\sqrt{3}}.$$

They obey

$$\sum_{i=1}^4 d^a_i d^b_i = \frac{4}{3}\delta^{ab}.$$

Distributing the scalar on-site Green response over the tetrahedral bond frame gives

$$\mathcal{G}^{ab}_{\text{loc}} = G_{\text{tet}}(0)\frac{1}{4}\sum_i d^a_i d^b_i = \frac{G_{\text{tet}}(0)}{3}\delta^{ab}.$$

For a local horizon normal \(\hat{r}\),

$$P^{ab}_\perp = \delta^{ab} - \hat{r}^a\hat{r}^b,$$

so

$$G_\perp = P^{ab}_\perp\mathcal{G}^{ab}_{\text{loc}} = \frac{2}{3}G_{\text{tet}}(0).$$

Thus the transverse export weight \(w_\perp = 2/3\) is the transverse part of the exact local graph response.

Using the field normalization \(S = \pi Q_{\text{occ}}\), the rigid local defect shift is

$$\delta Q_{\text{def}} = \frac{\Delta S_{\text{def}}}{\pi} = \frac{\ln(7/6)}{\pi}.$$

The corresponding local source amplitude in lattice units is therefore

$$s_{\text{def}} = J^{(\text{ren})}_{\text{eff}}\frac{\delta Q_{\text{def}}}{G_{\text{tet}}(0)},$$

so that

$$\frac{s_{\text{def}}}{J^{(\text{ren})}_{\text{eff}}} = \frac{\ln(7/6)}{\pi G_{\text{tet}}(0)} = 0.109472228\ldots$$

is a pure number fixed by the same UV lattice geometry.

The Green-function constant turns the local insertion into a continuum source theorem. Defining the defect-entropy density by

$$\sigma_{\text{def}} = \frac{\rho}{\kappa_m(L_*)},$$

the tetrahedral projection used in the stiffness mapping gives

$$\nabla^2\delta S = -\frac{3L_*}{4G_{\text{tet}}(0)}\sigma_{\text{def}}.$$

Equating this with the weak-field source equation

$$\nabla^2\delta S = -\frac{\kappa}{\gamma}\rho$$

closes the canonical source-to-stiffness ratio:

$$\boxed{\frac{\kappa}{\gamma} = \frac{3L_*}{4G_{\text{tet}}(0)\kappa_m(L_*)}.}$$

This is the source-side counterpart of the stiffness derivation. The edge-kernel calculation fixes how the scalar capacity mode resists gradients; the Green-matched defect calculation fixes how localized matter defects source that same mode.

The three cancellations, made explicit. The passage from the dimensionless lattice insertion \(s_{\text{def}}/J^{(\text{ren})}_{\text{eff}} = \ln(7/6)/(\pi G_{\text{tet}}(0))\) to the source theorem turns on three reductions, none of which leaves a free constant. (i) The factor \(\pi\) cancels against the field normalization. The insertion is written for the occupancy field, where one isotropically averaged defect shifts \(\delta Q_{\text{def}} = \ln(7/6)/\pi\); a fixed defect type \((\ell, f)\) instead carries \(-\ln P(A_{\ell f})\), ranging over \([0.14933, 0.15819]\) with mean 0.15415642 against \(\ln(7/6) = 0.15415068\), so the source theorem uses the exact isotropic benchmark under the assumption that the coarse defect population is isotropically averaged over face and label types. Converting to the entropy field through \(S = \pi Q_{\text{occ}}\) multiplies by \(\pi\), so the \(S\)-field shift is \(\delta S_{\text{def}} = \pi\,\delta Q_{\text{def}} = \ln(7/6)\) and the explicit \(\pi\) does not survive into the source theorem. (ii) \(\ln(7/6)\) is an isotropic source benchmark, not a universal per-defect cost. For a coarse population averaged over face and label types, the exact mean partition ratio is \(6/7\), so the corresponding annealed source normalization is \(\ln(7/6)\). A fixed defect type instead carries \(-\ln P(A_{\ell f})\). Writing the isotropically averaged source as \(\sigma_{\text{def}} = \rho/\kappa_m(L_*)\) folds that benchmark and the mass–entropy unit into the density. The geometric coefficient \(3L_*/(4G_{\text{tet}}(0))\) is fixed by the tetrahedral projection and the cell length and contains no \(\ln(7/6)\). (iii) \(J^{(\text{ren})}_{\text{eff}}\) cancels in the ratio. The source amplitude \(s_{\text{def}}\) and the stiffness \(\gamma\) each carry one power of the renormalized edge coupling, so it cancels in the source-to-stiffness ratio \(\kappa/\gamma\), leaving the lattice-geometric constant \(3L_*/(4G_{\text{tet}}(0)\kappa_m(L_*))\). The renormalized loop coupling therefore drops out of the observable normalization.

Physical geometry–capacity matching. The source theorem fixes \(\kappa/\gamma\) in the chosen entropy units. With the Compton-calibrated one-bit dictionary \(\kappa_m(L_*) = \hbar/(cL_* \ln 2)\), the weak-field bridge gives

$$G = \frac{c^2}{8\pi S_\infty}\frac{\kappa}{\gamma} = \frac{c^3 L_*^2}{\hbar}\frac{3\ln 2}{32\pi G_{\text{tet}}(0)S_\infty}.$$

Define \(S_0 = 3\ln 2/[32\pi G_{\text{tet}}(0)]\). The selected canonical matching is

$$\boxed{S^{\text{cell}}_\infty = S_0 = 0.0461482516\ldots, \quad G = G_*.}$$

Equivalently it imposes \(n_{\text{hor}}S^{\text{cell}}_\infty = 1/4\) with the response number of F.5. The coefficient \(1/4\) and the identification \(L_P(G_*) = L_*\) provide the physical Einstein/horizon normalization. This is a matching prescription of the chosen branch. The source theorem by itself gives \(G/G_* = S_0/S_\infty\). An entropy-unit change \(S' = bS\) gives \(\gamma' = \gamma/b^2\), \(\kappa' = \kappa/b\), and \(S'_\infty = bS_\infty\). Hence \(\kappa/(\gamma S_\infty)\) is invariant. Cell and horizon representatives can use different entropy units once the physical matching has been fixed. The rescaling carries that matching between representations and leaves the observable \(G\) unchanged. At fixed entropy convention and source coefficients, choosing a different physical background value \(S_\infty = \beta S_0\) would give \(G = G_*/\beta\). The dimensionless transfer and counting results do not determine \(\beta\). The Einstein/horizon matching selects \(\beta = 1\); a microscopic derivation of that selection would require additional geometric dynamics or boundary data.

C.6 Local susceptibility cross-check

The exact local moments of the admissibility-closed ensemble supply an independent non-degeneracy check on the source-side result. From the variance in Appendix B,

$$a_{\text{UV}} := \frac{1}{\text{Var}_{\eta_*}(K^2)} = 0.0637390269,$$

which is the local zero-mode inverse susceptibility of the closure scalar. Two features of this number matter for the source theorem in C.5. First, \(a_{\text{UV}}\) is finite and positive, confirming that the closed branch at \(\eta_*\) has a non-degenerate zero-mode response rather than a critical singularity that would invalidate the linear Green-function matching used to derive \(\kappa/\gamma\). Second, the same local susceptibility controls the stability of the closure mode propagated into the shell and loop calculations, so the residual fractional discrepancy between the local benchmark and \(\sigma_*\) belongs to the loop sector rather than the local closure.

The actual closure of \(\kappa/\gamma\) in the canonical branch is the Green-matched theorem in Appendix C.5. The role of \(a_{\text{UV}}\) here is restricted to confirming non-degeneracy of the local mode being matched.

C.7 UV-to-IR payoff

At this stage the weak-field UV coefficient chain is explicit:

$$\Omega_{\text{tet}} \to g_{\text{share,eff}} \to L_* \to J_{\text{bare}} \to J^{\text{tree}}_{\text{eff}} \to \Sigma_{\text{ret}} \to J^{(\text{ren})}_{\text{eff}} \to \gamma.$$

The same chain feeds

$$a_0 = \frac{cH_0 g_{\text{share,eff}}}{4\pi^2},$$

and the decorated support-to-rate action gives the substrate-induced scale

$$G_* = \frac{c^3 L_*^2}{\hbar} = \frac{9}{4}\frac{\hbar c}{m_e^2}Z_e^2 \ln^2\left(1 - e^{-7g_{\text{share,eff}}}\right).$$

The Green-matched source theorem fixes

$$\frac{\kappa}{\gamma} = \frac{3L_*}{4G_{\text{tet}}(0)\kappa_m(L_*)}.$$

The weak-field bridge then converts this source-to-stiffness ratio into the observed Newtonian normalization through the invariant combination \(\kappa/(\gamma S_\infty)\). The comparison with the same \(G_*\) assigned by the electron-anchored support-to-rate branch tests that this normalization propagates the substrate scale coherently; it is not a disjoint-input second derivation of \(G\), since the matched route carries \(L_*\) within it.

The coefficients this appendix supplies to the main weak-field chain are \(J_{\text{bare}}\), \(J^{\text{tree}}_{\text{eff}}\), \(\Sigma_{\text{ret}}\), \(J^{(\text{ren})}_{\text{eff}}\), \(\gamma\), and the Green-matched source projection \(\kappa/\gamma\). The remaining uses of \(S_\infty\) belong to the fixed normalization of the weak-field bridge, not to the source sector itself.

Appendix D: Weak-Field Technical Derivations, Electron Anchor, and EFT Consistency

Appendix D collects the weak-field derivations that are central but too dense for the main line: the bridge law, Newtonian recovery, the electron anchor, and the EFT consistency checks.

D.1 Bridge law from the reduced action

The weak-field bridge follows from an action equivalence and needs no extrapolated exponential lapse. The Einstein scalar-constraint action reduces to

$$I_{\text{Newton}}[\Phi] = \int dt\, d^3x\left[-\frac{(\nabla\Phi)^2}{8\pi G} - \rho\Phi\right].$$

The capacity action becomes the same functional, up to the constant \(Z_S = 2\kappa S_\infty/c^2\), under

$$\delta S = -\frac{2S_\infty}{c^2}\Phi.$$

Hence

$$\frac{\Phi}{c^2} = -\frac{\delta S}{2S_\infty}$$

is the unique linear field redefinition matching both the kinetic and source terms with the UV normalization. The bounded nonlinear rule is instead \(N^2 = q = 1 - \delta S/S_\infty\) in a static branch. An exponential lapse shares the first derivative at the vacuum point but is not used globally because it has no finite-capacity endpoint. The complete derivation is in Appendix N.

D.2 Point source, Newton limit, and lensing

In the renormalized static branch,

$$\nabla^2\delta S = -\frac{\kappa}{\gamma}\rho.$$

For a point source \(M\),

$$\delta S(r) = \frac{\kappa M}{4\pi\gamma r}, \quad g(r) = \frac{c^2\kappa}{8\pi\gamma S_\infty}\frac{M}{r^2} = \frac{GM}{r^2}.$$

For the ordinary longitudinal branch, variation of the unreduced Einstein scalar action gives

$$\Phi = \Psi$$

with asymptotically flat boundary conditions. This equality follows from the spatial metric constraint, not from a canonical capacity stress tensor. The effective-halo rewrite of the additional galactic response is

$$\rho_{\text{halo}}(r) = \frac{1}{4\pi Gr^2}\frac{d}{dr}\left[r^2(g_{\text{obs}} - g_{\text{bar}})\right].$$

Thus the same metric potential controls orbital dynamics and light bending in the baseline Einstein branch. The separate carrier-resolved contact of Section 16 specifies \(\Delta\Phi_\perp = \Delta\Psi_\perp\) at leading quasistatic order and uses the same induced density for support and lensing. Its nonlinear Ward identity depends on covariant variation of the carrier records, as Appendix N.8 states.

D.3 Electron anchor and composite matter

The canonical fermionic entropy increment is

$$\Delta S_f = \ln 2.$$

The UV mass normalization is

$$\kappa_{m,\text{UV}} = \frac{\hbar}{cL_*}\frac{1}{\ln 2},$$

and the running law in the closed branch is

$$\kappa_m(\ell) = \kappa_{m,\text{UV}}\left(\frac{L_*}{\ell}\right)^{1+\alpha_{\text{cl}}}, \quad \alpha_{\text{cl}} = 0.$$

At the electron Compton scale \(\ell = \lambda_e\) this gives

$$\kappa_m(\lambda_e) = \frac{m_e}{\ln 2},$$

which is the elementary anchor used here. Composite hadrons are not reduced to a bare constituent count. Their mass budget is assigned to a dressed bound-state entropy

$$m_{\text{hadron}} = \kappa_m(\ell_H)S^{\text{dressed}}_{\text{ent},H},$$

whose microscopic decomposition must include confinement, gluonic structure, trace-anomaly contributions, and chiral vacuum reorganization.

The electron is an elementary one-bit defect and provides the dimensional anchor. Hadronic inertial content instead belongs to a dressed entropy budget that includes binding and vacuum structure; a bare constituent count is insufficient.

D.4 Faithful sector resolution and induced \(G_*\)

The electron anchor enters the gravitational normalization through the absolute substrate length. The exact inputs are the seven-channel entropy, the memoryless kernel selected by faithful resolution, channel factorization on the lightest branch, and the decorated marked-transfer vertex of Appendix H. The calculation below separates the determinant recurrence, positive survival gap, and finite marked response before combining them in the physical cell length.

The recurrence derivation of the dictionary. Appendices H.2, H.4, and H.5 show that faithful full-support resolution selects memoryless replacement and that the lightest branch has seven factorized channels. On the commutative boundary algebra define \((Rf)(b) = p_*(b)f(b)\) and \(\tau_p(f) = \sum_b p_b f(b)\). The state-weighted determinant is

$$\Delta_{\tau_p}(R) = \exp[\tau_p(\ln R)] = \exp\left(\sum_b p_b\ln p_b\right) = e^{-g_{\text{share,eff}}}.$$

Multiplicativity gives \(e^{-7g_{\text{share,eff}}}\) on the seven-channel product. This is the quenched geometric transfer rate of the record-conditioned likelihood operator. It is not the Born probability of a specified microstate or the collision probability of two renewed blocks.

The corresponding history statement follows from the exact recurrence theorem. For a stationary ergodic finite-alphabet process, if \(R_n\) is the first return time of a length-\(n\) block, then

$$\lim_{n\to\infty}\frac{1}{n}\ln R_n = h \quad \text{almost surely},$$

where \(h\) is the Shannon entropy rate [91, 92]. This theorem selects Shannon entropy over raw state count and Rényi-2 entropy for long typical blocks. It does not by itself identify the seven distinguishable sector layers with a long temporal block, and it gives no exact finite-\(n\) return probability. For the present one-layer distribution,

$$H_1 = 7.4198000, \quad H_2 = -\ln\sum_b p_b^2 = 7.4125486, \quad H_\infty = -\ln\max_b p_b = 7.1343850.$$

The distinctions are numerically consequential after seven layers. The refresh projector \(|\sqrt{p}\rangle\langle\sqrt{p}|\) carries the participation scale \(e^{H_2}\), while the determinant and almost-sure Lyapunov rate use \(H_1\). They are different observables. With

$$p_{\text{rec}} \equiv e^{-7g_{\text{share,eff}}},$$

the positive survival operator has nonzero eigenvalue \(1 - p_{\text{rec}}\). The decorated marked vertex supplies the electron factor \(Z_e\). With the additive-generator prescription of H.9, the dressed electron gap is

$$E_e = -\frac{3\hbar Z_e}{2\tau_*}\ln(1 - p_{\text{rec}}).$$

Identifying the lowest charged gap with \(m_e c^2\) fixes the cadence from the already-declared electron anchor,

$$\tau_* = -\frac{3}{2}Z_e\tau_e\ln(1 - p_{\text{rec}}), \quad L_* = c\tau_*.$$

Thus

$$L_* = -\frac{3}{2}Z_e\lambda_e\ln\left(1 - e^{-7g_{\text{share,eff}}}\right) = \frac{3}{2}Z_e\lambda_e e^{-7g_{\text{share,eff}}}\left(1 + O\left(e^{-7g_{\text{share,eff}}}\right)\right).$$

The logarithmic survival correction is \(p_{\text{rec}}/2 \simeq 1.4 \times 10^{-23}\). The finite closure-response correction is \(Z_e - 1 = 0.00530828\). No second dimensional input, maximum-throughput clock, or tetrahedral-diameter equality is introduced.

The transfer-support dual. A spatial version uses the same dressed support and assigns the leading step count

$$N_{\text{eff}} = \frac{2}{3Z_e}e^{7g_{\text{share,eff}}}.$$

The positive transfer spectrum gives the temporal relation \(\tau_e = N_{\text{eff}}\tau_*\). Assigning the same count to space gives \(\lambda_e = N_{\text{eff}}L_*\) and hence \(L_* = c\tau_*\). The spatial assignment is a continuum matching prescription, not a consequence of the scalar rest spectrum. A microscopic orbital transfer must reproduce it through its small-momentum dispersion; the graph bond length cannot be identified with \(L_*\) without that derivation.

Internal locality and the native vertex. The selected replacement kernel reaches the full ensemble. The tested single-label move class does not: shifts preserving injectivity split each parity copy into \(4! = 24\) ordering sectors of \(\binom{7}{4} = 35\) states because two slots cannot exchange order without a collision. Its strength-weighted walk saturates at entropy 7.374, below \(g_{\text{share,eff}} = 7.4198\). This excludes that implementation. Appendix H.8 gives the whole-tetrahedron replacement gate, and H.9 decorates it with the fresh-state amplitude and charged event. The native gate is carried by the separate capacity/record factor; a particular GFT condensate realization may implement it dynamically, while the ordinary geometric propagation is supplied by the controlled factorized EPRL-class branch.

The reduced electron Compton wavelength is

$$\lambda_e = \frac{\hbar}{m_e c},$$

and the admissibility-closed sharing entropy is

$$g_{\text{share,eff}} = 7.41980002357.$$

The closure fixed point also gives

$$\langle K^2\rangle_{\eta_*} = \frac{3}{2\eta_*}, \quad \eta_* = 0.0298668443935,$$

so the closure-saturation factor is

$$C_{\text{cl}} := \eta_*\langle K^2\rangle_{\eta_*} = \frac{3}{2}, \quad C_{\text{cl}}^{-1} = \frac{2}{3}.$$

Separating the baseline recurrence from the finite marked response gives

$$\ln\left(\frac{\lambda_e}{L_*}\right) = 7g_{\text{share,eff}} - \ln\left(\frac{3}{2}\right) - \ln Z_e + O(e^{-7g_{\text{share,eff}}}).$$

Equivalently,

$$\boxed{L^{\text{leading}}_* = \frac{3}{2}Z_e\lambda_e e^{-7g_{\text{share,eff}}}.}$$

The exact expression is \(L_* = -(3/2)Z_e\lambda_e\ln(1 - e^{-7g_{\text{share,eff}}})\). It gives

$$L_* = 1.6162537024 \times 10^{-35}\ \text{m},$$

which differs from the CODATA Planck length by about \(8.2 \times 10^{-7}\) in fractional terms, or \(8.2 \times 10^{-5}\) percent.

The induced gravitational scale follows from the same algebra used in the stiffness matching:

$$G_* := \frac{c^3 L_*^2}{\hbar}.$$

Substituting the exact marked-transfer relation gives the closed form

$$\boxed{G_* = \frac{c^3}{\hbar}\left[-\frac{3}{2}Z_e\lambda_e\ln\left(1 - e^{-7g_{\text{share,eff}}}\right)\right]^2 = \frac{9}{4}\frac{\hbar c}{m_e^2}Z_e^2 \ln^2\left(1 - e^{-7g_{\text{share,eff}}}\right).}$$

Numerically,

$$G_* = 6.6742890813 \times 10^{-11}\ \text{m}^3\text{kg}^{-1}\text{s}^{-2},$$

or \(-0.073\sigma\) relative to CODATA. The same \(\zeta_*\) gives the muon and tau ratios at \(-0.535\sigma\) and \(-0.125\sigma\). One finite action therefore replaces the three leading deficits with one correction whose coefficient and routing are fixed jointly rather than adjusted independently for the three observables. The discrepancies were already known while the vertex was being constructed, so these are high-precision postdictions. Their evidential content is the common action, the absence of a continuous fit, and the nearby kernels rejected by the finite audits.

The factor \(3/2\) is fixed structurally by the tetrahedral transverse-export geometry. The tetrahedral identity gives the transverse export fraction

$$\frac{1}{4}\sum_{i=1}^4 (\hat{n}_i\cdot\hat{u})^2 = \frac{1}{3}, \quad f_\perp = 1 - \frac{1}{3} = \frac{2}{3},$$

so \(f_\perp^{-1} = 3/2\). The admissibility fixed point returns the same value,

$$C_{\text{cl}}^{-1} = \left(\eta_*\langle K^2\rangle_{\eta_*}\right)^{-1} = \frac{2}{3},$$

but it is built in. The condition defining \(\eta_*\) is \(\langle K^2\rangle_\eta = 3/(2\eta)\), so \(C_{\text{cl}} = \eta_*\langle K^2\rangle_{\eta_*} = 3/2\) holds by construction. It is not an independent cross-check. The decorated scale action reuses its reciprocal \(2/3\) as the transverse export and inserts it once as a global factor, not seven times.

A discrete sensitivity audit enumerates 576 nearby formulas obtained by varying the response dimension, pair count, direction factor, determinant power, statistics, and singlet overlap. Only the decorated-vertex formula lies within one muon standard deviation. This count is not a probability distribution over theories. It shows that the landing is sensitive to the field content and graph, which is why the action and its adversarial alternatives are displayed explicitly. A shared edge mode, directed complex determinant, stable bosonic determinant, or six-state symmetric response misses the muon respectively by approximately \(1.15 \times 10^4\sigma\), \(13.8\sigma\), \(56.7\sigma\), and \(7.64\times10^4\sigma\). Appendix H derives the selected entries before performing the observable trace.

The electron anchor carries three related roles. Its reduced Compton wavelength \(\lambda_e\) is the non-gravitational length used in faithful sector resolution. Its status as the lightest one-bit fermionic defect identifies which elementary excitation calibrates the seven-channel dressing block. Its mass fixes the mass–entropy map through

$$\kappa_m(\lambda_e) = \frac{m_e}{\ln 2}.$$

The same elementary defect enters the two normalization channels in distinct roles: its Compton length sets the UV cell scale, while its mass per one-bit defect entropy sets the source normalization.

The seven-sector scale-setting relation is not a counting heuristic; it has a concrete finite-dimensional realization on a transfer-operator history space, which we now construct. The face label algebra is

$$\mathcal{A}_7 = \bigoplus_{m=-3}^3 \mathbb{C}E_m, \quad E_m E_n = \delta_{mn}E_m, \quad \sum_{m=-3}^3 E_m = I_7.$$

The admissibility-closed tetrahedral state space has 1680 oriented injective states, with stationary weight

$$p_{\eta_*}(b) = Z^{-1}e^{-\eta_* K^2(b)}.$$

For a single sector, the refresh kernel

$$P_{\eta_*}(b, b') = p_{\eta_*}(b')$$

has Perron stationary entropy \(g_{\text{share,eff}}\). The electron does not simultaneously occupy seven mutually exclusive face labels in one tetrahedron; the labels are sector channels in the dressing history of the one-bit defect. The appropriate support object is therefore a history space, with one admissibility-closed sector layer for each \(m = -3, \ldots, 3\).

Let

$$\mathcal{H}_{\text{hist}} = \bigotimes_{m=-3}^3 \mathcal{H}^{(m)}_B, \quad \dim\mathcal{H}_B = 1680,$$

and let \(\mathcal{T}_m\) act as \(P_{\eta_*}\) on the \(m\)th factor and as the identity on the others. A representative one-pass dressing operator is

$$\mathcal{D}^{(0)}_e = \mathcal{T}_{-3}\mathcal{T}_{-2}\cdots\mathcal{T}_3.$$

The displayed order is only a representative of the symmetrized one-pass class. It introduces no physical ordering or extra factor of 7!. The one-pass prescription visits each simple sector once. A history with omitted sectors is unresolved, while additional visits belong to a multi-pass dressing history rather than to repeated fermionic occupation.

A sector-resolution step should not be identified with conditioning the 1680-state ensemble on boundary states containing a given label \(m\). Such conditioning changes the entropy and does not reproduce \(g_{\text{share,eff}}\). The sector label instead specifies which simple face-algebra channel is being resolved while the admissibility cloud sampled in that step remains the full closed boundary ensemble.

With the effective dimension defined by the stationary Shannon entropy of the positive history kernel, each sector contributes \(g_{\text{share,eff}}\), so

$$\dim_{\text{eff}}(\mathcal{D}^{(0)}_e) = \exp(7g_{\text{share,eff}}) = 3.60286052 \times 10^{22}.$$

Only the transverse part of this support is exported into the weak-field scalar channel. The normalized export weight is

$$w_\perp = \frac{2}{3},$$

the same fraction fixed by the tetrahedral transverse projection and by the reciprocal closure-saturation factor. Hence

$$\dim_{\text{eff}}(\mathcal{D}_{e,\perp}) = w_\perp\dim_{\text{eff}}(\mathcal{D}^{(0)}_e) = \frac{2}{3}e^{7g_{\text{share,eff}}} = 2.40190701 \times 10^{22}.$$

The decorated marked-transfer action gives the exact support-scale relation

$$\boxed{\frac{\lambda_e}{L_*} = \frac{2}{3Z_e\left[-\ln\left(1 - e^{-7g_{\text{share,eff}}}\right)\right]} = \frac{1}{Z_e}\dim_{\text{eff}}(\mathcal{D}_{e,\perp})\frac{r}{-\ln(1 - r)}.}$$

Thus \(\lambda_e/L_* = Z_e^{-1}\dim_{\text{eff}}(\mathcal{D}_{e,\perp})[1 - r/2 + O(r^2)]\). Appendix H derives the memoryless replacement, determinant recurrence, positive gap, and decorated charged vertex. Fermionic exclusion and subadditivity select \(k = 7\) and \(\Delta_7 = 0\), while the marked graph fixes \(Z_e\). The capacity-decorated rigging lift is exact and the selected metric-Regge branch has the Einstein TT limit. The geometric gluing tensor and measure remain independent data not fixed by the capacity premises; durable history capacity and the full source-coupled transverse/environmental spectrum also remain open.

The relation is fixed inside the displayed finite transfer action and the electron anchor. Its remaining conditionality is the broader claim that this transfer action is the charged sector of the eventual geometric GFT, not an unfixed number in the finite calculation.

This is the microscopic content of the length formula above: the elementary one-bit fermionic defect is supported over one complete transverse-exported dressing block. The matched weak-field gravitational constant then follows from the gauge-invariant bridge

$$G = \frac{c^2}{8\pi}\frac{\kappa}{\gamma S_\infty},$$

with the physical geometry–capacity matching and subsequent entropy-unit transformations specified in Appendix C.5.

D.5 EFT consistency checklist

The ordinary longitudinal branch is a rewriting of the Einstein constraint sector, so it introduces no independent scalar propagator to which a separate ghost or tachyon test could be applied. Its consistency checks and those of the optional transport completion must be stated separately.

• No extra longitudinal degree of freedom. The Einstein parent supplies the Cauchy data and propagating tensor modes; \(\delta S\) is the static scalar-constraint coordinate.
• Correct-sign sourcing. Positive mass produces a capacity deficit and attractive Newtonian response in the reduced action.
• Positive static quadratic form. The Euclidean capacity functional has positive stiffness \(\gamma > 0\) after the overall gravitational-sign convention is fixed.
• Conditional causal transport. The phenomenological telegrapher completion has finite characteristic speed when \(D/\tau_0 = c^2\) with \(D, \tau_0 > 0\).

These statements do not establish a UV completion or quantize an additional scalar. They show that the controlled static representation is consistent with its Einstein parent and that the separately proposed transport equation is linearly stable in its stated parameter range.

The checklist is intentionally modest. It verifies the reduced Einstein representation and the signs of the separate transport model; it does not establish an independently quantized scalar EFT.

The one place where an explicit formula is worth recording is linear vacuum stability in the time-dependent sector. Writing a small perturbation \(\delta s\) about the vacuum branch, the linearized telegrapher equation is

$$\tau_0\ddot{\delta s} + \dot{\delta s} - D\nabla^2\delta s = 0.$$

For a plane-wave mode \(e^{-i\omega t + i\mathbf{k}\cdot\mathbf{x}}\), this gives the dispersion relation

$$\tau_0\omega^2 + i\omega - Dk^2 = 0.$$

With \(\tau_0 > 0\) and \(D > 0\), the corresponding mode frequencies have non-growing time dependence, so the phenomenological transport vacuum is linearly stable.

D.6 Quadratic fluctuations and weak-field stability

Before imposing the Einstein scalar constraint, a formal covariant quadratic representation is

$$I^{(2)}_{\text{formal}}[\delta S] = -\int d^4x \sqrt{-g}\frac{\gamma}{2}g^{\mu\nu}\partial_\mu\delta S\,\partial_\nu\delta S.$$

After constraint reduction only its spatial Poisson quadratic form remains. The absence of a mass term expresses the unscreened constraint and does not prove an additional propagating boson. The galactic excess of Section 15 belongs to the separate carrier-resolved contact of Appendix N.8.

Appendix D provides the technical support layer for the weak-field bridge, Newton limit, electron anchor, substrate length branch, and EFT consistency audit.

Appendix E: Transport, Cosmology, and Hubble-Tension Implementation

Appendix E collects phenomenological time-dependent and homogeneous extensions of the static capacity variable. These additions are not implied by the Einstein constraint reduction and must recover it in their static limit. The transport relation is fixed only in its preferred branch, while the homogeneous and perturbation sectors remain open.

E.1 Telegrapher equation and causal closure

The time-dependent deficit field obeys

$$\tau_0\partial_t^2\delta S + \partial_t\delta S = D\nabla^2\delta S + A\chi, \quad \frac{A}{D} = \frac{\kappa}{\gamma}.$$

Causality requires

$$\frac{D}{\tau_0} = c^2.$$

In the canonical no-new-IR-scale branch,

$$\tau_0^{-1} = H_0, \quad D = \frac{c^2}{H_0}.$$

The telegrapher form is the minimal causal completion of the static Poisson sector. It supplies propagation and relaxation while preserving the static weak-field law as its exact late-time limit.

E.2 Static-limit recovery for galaxies

For a Fourier mode \(k\), the telegrapher characteristic equation

$$\tau_0 s^2 + s + Dk^2 = 0$$

has the roots

$$s = -\frac{1}{2\tau_0} \pm i\omega_k, \quad \omega_k \simeq ck$$

whenever \(4\tau_0 Dk^2 \gg 1\). Galactic wavelengths are far below the critical scale

$$\lambda_c = \frac{4\pi c}{H_0} \approx 56\ \text{Gpc},$$

so galactic modes are deeply underdamped. Time-averaging the sourced solution over intervals large compared with \(2\pi/\omega_k\) returns the static Poisson branch exactly, and the residual ponderomotive correction scales parametrically as

$$\frac{\delta F_{\text{pond}}}{F_{\text{static}}} \sim e^{-T/(2\tau_0)}\left(\frac{\omega_{\text{orb}}}{\omega_k}\right)^2 \lesssim 10^{-6}$$

for representative orbital speeds, with the precise value depending on the averaging interval and system scale. The estimate keeps the correction below the near-stationary weak-field branch but does not supply a universal \(10^{-8}\) bound.

E.3 Homogeneous mode and cosmological sourcing

The cosmological split is

$$S(x, t) = \bar{S}(t) + s(x, t),$$

with \(\bar{S}(t)\) the homogeneous mode and \(s(x, t)\) the inhomogeneous weak-field sector. The background capacity is normalized by the apparent horizon,

$$S_\infty(t) = \pi\frac{R_A(t)^2}{L_*^2}, \quad R_A(t) = \frac{c}{\sqrt{H^2 + kc^2/a^2}}.$$

Because the field couples to the trace of the stress-energy tensor, the homogeneous mode is suppressed during radiation domination and turns on near matter–radiation equality.

This timing is the central cosmological virtue of the mechanism. The homogeneous mode is quiet when it must be quiet, then becomes relevant close to the epoch where a sound-horizon shift is most useful.

E.4 Sound-horizon shift and shear lock

In the conditional cosmological proposal, the trace-sourced homogeneous mode acts as a transient early-energy contribution. It is intended to reduce the sound horizon without rewriting the local static Poisson law. Demonstrating that separation together with the committed component requires the open joint Boltzmann calculation.

The homogeneous mode can alter the cosmological background while leaving the coefficients of the local weak-field branch unchanged. A quantitative perturbation calculation remains open.

The transport relation and preferred branch are closed; the cosmological sector remains structurally supported but not yet Boltzmann-closed.

Appendix F: Strong-Field Spherical Reduction

Appendix F records the strongest action-level statement currently available in the black-hole sector. The bounded capacity rule is realized invariantly in spherical symmetry by the areal-radius gradient \(q_{\text{geo}} = (\nabla R)^2\). The Schwarzschild exterior and the location \(q_{\text{geo}} = 0\) then follow from the reduced Einstein action. Treating that surface as the physical end of the substrate EFT, however, is an additional domain postulate whose boundary microphysics remains open.

F.1 Bounded capacity and the unique lapse map

The strong-field order parameter is the surviving-capacity fraction

$$q(x) = \frac{S_{\text{ent}}(x)}{S_\infty} \in [0, 1].$$

This bound follows directly from finite local channel capacity. If the vacuum channel count is finite and \(S_{\text{ent}}\) is the logarithmic coarse entropy of the surviving local ensemble, then no physical branch can have either negative capacity or more than the asymptotic vacuum capacity.

In a static exterior, the lapse associated with the asymptotic Killing time is determined by the local surviving capacity. Let

$$N = f(q).$$

The conditions are:

$$f(1) = 1, \quad \lim_{q\to0^+}f(q) = 0.$$

The substrate-level composition axiom is that independent serial capacity losses compose multiplicatively on the lapse:

$$f(q_1 q_2) = f(q_1)f(q_2).$$

This is the assumption that extends the linear weak-field match to a nonlinear lapse map. With continuity, the positive solutions on \((0, 1]\) are \(f(q) = q^\alpha\). Expanding near \(q = 1 - \epsilon\) gives

$$N = q^\alpha = 1 - \alpha\epsilon + O(\epsilon^2).$$

The weak-field bridge gives

$$N = 1 - \frac{\epsilon}{2} + O(\epsilon^2),$$

so \(\alpha = 1/2\) and therefore

$$N = \sqrt{q}, \quad N^2 = q.$$

Equivalently, if one writes \(N^2 = F(q)\), the unique continuous multiplicative completion is \(F(q) = q\). The nonlinear static lapse rule is therefore fixed by capacity composition and weak-field matching; it is not a freely chosen black-hole ansatz. Because a lapse depends on foliation, this is a static constitutive statement. Its covariant spherical content is derived next.

F.2 Einstein action reduced to the capacity-adapted invariant

Take the most general spherically symmetric line element

$$ds^2 = h_{ab}(x)dx^a dx^b + R^2(x)d\Omega^2.$$

Here \(x^0 = ct\), so the two-dimensional measure is \(d^2x = dx^0 dr\). The same action written as \(dt\, dr\) acquires one additional factor of \(c\). The four-dimensional curvature decomposes as

$$^{(4)}R = ^{(2)}R + \frac{2}{R^2}\left[1 - (\nabla R)^2 - 2R\Box R\right].$$

After the angular integral, the Einstein–Hilbert plus GHY action becomes, up to the asymptotic and corner terms retained in \(I^{(2)}_\partial\),

$$I_{\text{sph}} = \frac{c^3}{4G}\int d^2x \sqrt{-h}\left[R^2\,^{(2)}R + 2(\nabla R)^2 + 2\right] + I^{(2)}_{\text{matter}} + I^{(2)}_\partial.$$

The integration by parts that converts \(-4R\Box R\) into \(+4(\nabla R)^2\) is legitimate only together with the reduced GHY contribution. The boundary term is therefore part of the statement. For \(\partial\mathcal{M}_4 = \partial\mathcal{M}_2 \times S^2\) one has

$$K^{(4)} = K^{(1)} + \frac{2}{R}n^a\nabla_a R.$$

The second term cancels the surface term from the integration by parts, leaving

$$I^{(2)}_\partial = \varepsilon\frac{c^3}{2G}\int_{\partial\mathcal{M}_2}dy\sqrt{|\gamma_{(1)}|}\,R^2 K^{(1)} + I_{\text{joint}} + I_{\text{ref}},$$

with \(\varepsilon\) the standard orientation sign.

Define

$$q_{\text{geo}} \equiv (\nabla R)^2 = h^{ab}\partial_a R\partial_b R.$$

Varying \(R\) and \(h^{ab}\) in vacuum gives

$$R\,^{(2)}R - 2\Box R = 0,$$ $$2R(h_{ab}\Box R - \nabla_a\nabla_b R) + h_{ab}(q_{\text{geo}} - 1) = 0.$$

These equations imply

$$\nabla_a M_{\text{MS}} = 0, \quad M_{\text{MS}} = \frac{c^2 R}{2G}(1 - q_{\text{geo}}).$$

Therefore

$$q_{\text{geo}} = 1 - \frac{2GM_{\text{MS}}}{c^2 R}.$$

The variational status is unambiguous: \(q_{\text{geo}}\) is a composite of the two-dimensional metric and the areal-radius dilaton, and its vacuum profile is a first integral. No scalar has been added to the Einstein degrees of freedom.

Audit of the multiplier form. If instead one writes an ADM term \(\sqrt{h}\,\lambda(N^2 - q)\) while giving \(q\) no other bulk dependence, the \(q\) equation sets \(\lambda = 0\). The constraint then only identifies an arbitrary scalar with a foliation-dependent lapse. It does not reproduce the capacity source equation and it has no invariant content away from a specified static slicing. The spherical reduction above supplies the invariant result that the multiplier form lacks.

F.3 Variational status of the \(q_{\text{geo}} = 0\) boundary

For \(\epsilon > 0\), let \(\mathcal{B}_\epsilon\) be the timelike level surface \(q_{\text{geo}} = \epsilon\). The exterior Einstein problem is well posed with the standard term

$$I_{\text{GHY}}[\mathcal{B}_\epsilon] = \varepsilon\frac{c^3}{8\pi G}\int_{\mathcal{B}_\epsilon}d^3y\sqrt{|\gamma|}\,K$$

and fixed induced metric on \(\mathcal{B}_\epsilon\), together with the reference term at infinity and any required joints. The limit \(\epsilon \to 0^+\) is null and must be taken using the corresponding null-boundary and joint prescription; a bare timelike GHY expression cannot be evaluated directly at the null surface.

This construction fixes the universal gravitational variation. It does not make

$$q_{\text{geo}}\big|_{\partial\mathcal{M}_q} = 0$$

a new Euler–Lagrange boundary condition. The zero is the level set at which the spherical invariant becomes marginal. Declaring that level set to be the end of the physical substrate domain,

$$\mathcal{M}_q = \{q_{\text{geo}} > 0\},$$

is the bounded-capacity postulate.

Any additional functional

$$\Gamma_{\partial q}[\sigma_{AB}, \text{boundary channels}]$$

would describe genuine substrate physics: absorption, partial reflection, relaxation, entropy, or a moving-boundary stress. Its variation may contain a boundary stress and a response conjugate to the limiting capacity, but neither its form nor its spectrum follows from Einstein–Hilbert reduction. The static exterior can be solved without inventing this term; claims about excision, infalling evolution, or finite reflectivity cannot.

F.4 Static spherical vacuum exterior

In static spherical vacuum, the result follows immediately. On \(\mathcal{M}_q\), the bulk equations are the vacuum Einstein equations. The unique asymptotically flat static spherical solution is the Schwarzschild exterior,

$$ds^2 = -\left(1 - \frac{2GM}{c^2 r}\right)c^2 dt^2 + \left(1 - \frac{2GM}{c^2 r}\right)^{-1}dr^2 + r^2 d\Omega^2,$$

so the spherical geometric representative of the capacity variable is

$$q_{\text{geo}}(r) = N^2(r) = 1 - \frac{2GM}{c^2 r}, \quad r > r_h,$$

with

$$r_h = \frac{2GM}{c^2}.$$

This also agrees with the weak-field capacity deficit:

$$\frac{\delta S(r)}{S_\infty} = \frac{2GM}{c^2 r}, \quad q(r) = 1 - \frac{\delta S(r)}{S_\infty}.$$

The exterior \(r > r_h\) is exactly the standard Schwarzschild exterior, and \(q_{\text{geo}} = 0\) at \(r = r_h\). The geometric invariant becomes negative in the trapped region. Interpreting that sign change as exhaustion of a nonnegative substrate capacity motivates restricting the capacity EFT to \(q_{\text{geo}} \geq 0\), but this restriction is not implied by the Einstein equations. The absence of a physical classical interior is therefore a conditional substrate claim, not an action-level theorem.

The exterior-domain result does not decide what an infalling observer experiences at the \(q_{\text{geo}} = 0\) surface. Whether a capacity-exhaustion boundary is smooth, dissipative, anomalous, or absent requires the open functional \(\Gamma_{\partial q}\), not merely the exterior Schwarzschild solution.

Geometric identification of the capacity variable. The spherical reduction supplies a normalization-independent gradient invariant on the orbit space,

$$q_{\text{geo}} = h^{ab}\partial_a R\,\partial_b R = |\nabla R|^2,$$

which in Schwarzschild gives

$$q_{\text{geo}} = 1 - \frac{2GM}{c^2 r} = N^2,$$

which can be identified with the substrate fraction \(q = S_{\text{ent}}/S_\infty\) on the exterior. In spherical dynamical collapse with Misner–Sharp mass \(M_{\text{MS}}(R, t)\),

$$q_{\text{geo}}(R, t) = 1 - \frac{2GM_{\text{MS}}(R, t)}{c^2 R}.$$

The marginal-trapped-surface condition \(\theta_+\theta_- = 0\) coincides with \(q_{\text{geo}} = 0\) independently of null normalization: positive, zero, and negative \(q_{\text{geo}}\) label untrapped, marginal, and trapped spherical regions. Standard collapse in the metric parent can therefore produce the geometric zero. Equating that zero with substrate saturation and refusing the negative branch are the additional bounded-capacity interpretation to be tested by the transport and boundary theory.

Rotating and charged stationary exteriors. The baseline metric parent also admits Kerr, Reissner–Nordström, and Kerr–Newman exteriors when the appropriate conserved gauge sector is present. Their standard horizon entropy

$$S = \frac{k_B A}{4L_*^2}$$

retains its form with \(A\) the appropriate horizon area, and the Hawking temperature follows from the surface gravity as usual,

$$T_H = \frac{\hbar\kappa_{\text{sg}}}{2\pi k_B c}.$$

A nonspherical covariant capacity scalar whose zero selects the outer horizon and whose sign defines a physical domain has not yet been derived. The absence of inner or trapped regions therefore cannot be inferred from the spherical invariant alone.

F.5 Horizon thermodynamics, channel area, and the Immirzi normalization

Because the exterior geometry is unchanged, semiclassical quantities depending only on the exterior near-horizon saddle are unchanged. The Euclidean continuation is used here as an exterior-saddle calculation: the resulting periodicity depends on regularity of the near-horizon exterior geometry, not on adopting the bounded-domain interpretation past \(q_{\text{geo}} = 0\). The Euclidean regularity argument therefore gives the standard Hawking temperature [13, 14],

$$T_H = \frac{\hbar c^3}{8\pi GM k_B}.$$

For the GR exterior saddle, the same calculation gives the Bekenstein–Hawking area law [12, 13],

$$S_{\text{BH}} = \frac{k_B A}{4L_P^2}.$$

On the matched branch \(L_P(G_*) = L_*\). The factorized \(j = 3\) completion supplies a definite map between one capacity cut and physical geometric area. The semiclassical area law provides the continuum normalization for that map.

Projected-area form of the quarter. When a coarse spatial horizon is represented as the boundary of a convex body in Euclidean three-space, Cauchy's surface-area formula gives [8]

$$\bar{A}_{\text{proj}} = \frac{1}{4\pi}\int_{S^2}A_{\text{proj}}(\hat{u})d\Omega = \frac{A}{4}.$$

The matched entropy law can therefore be written as

$$\boxed{\frac{S_{\text{BH}}}{k_B} = \frac{\bar{A}_{\text{proj}}}{L_*^2}.}$$

This identity gives the factor \(1/4\) a geometric interpretation: it converts total surface area into mean orthogonal projection area. It does not determine the microscopic density. Assigning one nat to each projected area \(L_*^2\) is equivalent to the continuum normalization already supplied by the Bekenstein–Hawking law. The formula does not by itself extend this record interpretation to a nonconvex or dynamical horizon.

One cut carries one capacity bit while the geometric boundary face remains. If the bounded-capacity domain terminates at \(q_{\text{geo}} = 0\), the exterior cell has no capacity partner across the outward face. The missing capacity pairing is the same elementary fermionic exclusion used by the mass–entropy postulate, so the canonical cut increment is

$$\boxed{\Delta S_f = \ln 2.}$$

The factorized construction of Appendix Q changes an important older reading of this statement. The absent partner removes the capacity pairing; it does not erase the geometric boundary datum. The open boundary still carries the geometric \(V_3^{\text{geom}}\) representation required for spin-network/spin-foam gluing, while the corresponding capacity \(V_3^{\text{cap}}\) has lost its partner across the cut. Thus a horizon puncture is a geometric \(j = 3\) face decorated by one binary capacity-cut defect, not an additional sevenfold thermodynamic degeneracy.

Accordingly the entropy used here is conditional on the macroscopic boundary geometry,

$$S_{\text{hor}} = S_{\text{cap}}(\text{cut} \mid g_\partial),$$

so the geometric spin/intertwiner labels specify the area sector and are not multiplied as an independent entropy factor. Multiplying a separate isolated-horizon spin degeneracy by the capacity degeneracy would double count the geometry–capacity degree of freedom in this framework; such a construction would be a different microscopic theory, not an extra factor in the present one.

Channel-to-area normalization within the selected completion. For a generic LQG surface puncture the kinematical area spectrum is [113]

$$A_{\text{LQG}}(j) = 8\pi\gamma_{\text{BI}}L_*^2\sqrt{j(j + 1)}.$$

The present branch, however, is built from spin coherent faces and maximal coherent fusion. The area variable that enters the associated Regge geometry is the coherent flux magnitude, linear in \(j\); for a coarse face made from parallel small facets, the large-face area is the sum of the small \(j_f\) and the coarse spin is \(J = \sum_f j_f\) [114]. Appendix B therefore fixes the physical coarse area operator on the maximal-fusion ray to

$$A_{\text{phys}}(J) = 8\pi\gamma_{\text{BI}}L_*^2 J,$$

or equivalently

$$A_{\text{phys}}(J) = Z_A(J)A_{\text{LQG}}(J), \quad Z_A(J) = \frac{J}{\sqrt{J(J + 1)}}.$$

For \(N\) primitive transmitted faces, \(J = 3N\) and hence

$$Z_A(N) = \sqrt{\frac{3N}{3N + 1}}, \quad A_N = 24\pi\gamma_{\text{BI}}NL_*^2.$$

This operator renormalization is fixed by additive coherent area; it is not a new coupling. At one primitive face \(Z_A(1) = \sqrt{3}/2\), and \(Z_A \to 1\) as the block enters the ordinary large-spin Regge regime.

A patch of \(N\) cut capacity channels carries

$$\frac{S_{\text{cap}}}{k_B} = N\ln 2.$$

Requiring this same geometric patch to obey the already-established Einstein/Bekenstein–Hawking normalization gives

$$N\ln 2 = \frac{A_N}{4L_*^2} = 6\pi\gamma_{\text{BI}}N.$$

Therefore

$$\boxed{\gamma_{\text{BI}} = \frac{\ln 2}{6\pi} = 0.0367726000254\ldots}$$

and the physical area carried by one microscopic capacity cut is

$$\boxed{a_{\text{ch}} = \frac{A_N}{N} = 4\ln 2\, L_*^2, \quad \rho_{\text{ch}} = \frac{1}{4\ln 2\, L_*^2}.}$$

The result is exact conditional on the selected factorized \(j = 3\) embedding, the one-cell relational lock, the coherent/Regge additive-area prescription, and the existing one-bit cut rule. It fixes the embedding normalization. The Bekenstein–Hawking law and \(L_P(G_*) = L_*\) supply the continuum normalization, so this is not independent evidence for that law. Within the embedding, the normalization fixes the Immirzi factor.

Bare and coarse area operators. If one equated \(\ln 2\) directly to the unrenormalized \(j = 3\) Casimir area, one would obtain \(\gamma = \ln 2/(4\pi\sqrt{3})\). That assignment treats a single bare puncture eigenvalue as the macroscopic area variable and fails to preserve area under the exact maximal-fusion map. The coherent/Regge branch instead carries the renormalized operator above. Indeed, for every \(N\),

$$Z_A(N)\,8\pi\gamma_{\text{BI}}L_*^2\sqrt{3N(3N + 1)} = 4N\ln 2\, L_*^2,$$

so fine-channel additivity, maximal fusion, the spin-foam Regge area, and the horizon entropy all use one normalization. The two numerical values are therefore not competing Immirzi predictions: the smaller value belongs to a bare operator convention that is not the coarse observable selected by this branch.

Independent bulk-response identity. The graph response supplies a separate normalization check and should not be interpreted as the puncture count. Since

$$\mathcal{G}^{ab}_{\text{loc}} = \frac{G_{\text{tet}}(0)}{3}\delta^{ab}, \quad G_\perp = \frac{2}{3}G_{\text{tet}}(0),$$

the angular response number is

$$n_{\text{hor}} = 4\pi\frac{G_\perp}{\ln 2} = \frac{8\pi G_{\text{tet}}(0)}{3\ln 2}.$$

It is dimensionless and is not the literal number of \(j = 3\) punctures. Using the cell-normalized capacity baseline

$$S^{\text{cell}}_\infty = \frac{3\ln 2}{32\pi G_{\text{tet}}(0)}$$

still gives the exact response identity

$$\boxed{n_{\text{hor}}S^{\text{cell}}_\infty = \frac{1}{4}.}$$

The microscopic puncture theorem above and this bulk Green-response identity are two descriptions of the same normalized continuum coefficient, but they count different objects and must not be multiplied together.

Status. Within the selected factorized \(j = 3\) geometric completion and coherent/Regge area prescription, the channel-to-area normalization is therefore closed:

$$\Delta S_f = \ln 2, \quad j_{\text{geom}} = 3, \quad Z_A(N) = \sqrt{\frac{3N}{3N + 1}}, \quad \gamma_{\text{BI}} = \frac{\ln 2}{6\pi}, \quad a_{\text{ch}} = 4\ln 2\, L_*^2.$$

What remains in the strong-field sector is dynamical rather than kinematical: the saturation/front evolution, boundary Hamiltonian, relaxation spectrum, reflectivity, stretched-layer corrections, and nonspherical capacity map.

F.6 Absorption, ringdown, and echoes

The exterior Regge–Wheeler/Zerilli operators are unchanged. With \(q_{\text{geo}} = 0\) at the marginal surface, the tortoise coordinate \(r_* \sim r_h\ln q_{\text{geo}}\) sends the surface to \(r_* \to -\infty\), and the near-horizon wave equation reduces to \((c^{-2}\partial_t^2 - \partial_{r_*}^2)\psi \simeq 0\). If the usual GR future-horizon regularity condition is retained, it selects

$$\psi \sim e^{-i\omega(t+r_*/c)},$$

and therefore

$$\mathcal{R} = 0.$$

The standard greybody factors and quasinormal spectrum then follow [15, 16]. But if the substrate EFT truly terminates at the marginal surface, future-horizon regularity is a boundary choice, not a consequence of the exterior differential operator alone. The open functional \(\Gamma_{\partial q}\) must determine whether that choice is correct.

If the microscopic boundary has finite reflectivity or lies on a stretched layer

$$q = \epsilon > 0,$$

the region between the exterior potential barrier and that layer behaves as a cavity. A typical echo delay then scales as

$$\Delta t_{\text{echo}} \sim \frac{2r_h}{c}|\ln\epsilon| + \tau_{\text{ch}},$$

where \(\tau_{\text{ch}}\) is a boundary-channel relaxation time. Thus \(\mathcal{R} = 0\) is the GR-matching boundary condition, while \(\mathcal{R}(\omega)\) is a boundary observable to be calculated rather than assumed.

F.7 Dynamical formation as a free-boundary problem

Standard collapse in the metric parent can form a marginal surface \(q_{\text{geo}} = 0\). It does not prove that a substrate capacity field saturates there or that the physical evolution terminates. That identification requires a bounded causal transport law for a capacity variable \(q_{\text{cap}}\), together with a constitutive relation showing \(q_{\text{cap}} = q_{\text{geo}}\) in the regime of overlap. A candidate test system is

$$\partial_t q_{\text{cap}} + D_i J^i = -\Gamma(q_{\text{cap}})\Sigma[T_{\mu\nu}],$$ $$\tau_J(\partial_t + \mathcal{L}_v)J^i + J^i = -D(q_{\text{cap}})D^i q_{\text{cap}}.$$

Here \(J^i\) is the capacity flux, \(\Sigma[T_{\mu\nu}]\) is a positive depletion source built from the collapsing stress-energy, \(D(q_{\text{cap}})\) is a bounded mobility, \(\Gamma(q_{\text{cap}})\) is a bounded depletion rate, and \(\tau_J > 0\) is a relaxation time. Eliminating \(J^i\) gives a telegrapher-type equation with finite characteristic speed

$$v_{\text{cap}} \sim \sqrt{\frac{D_0}{\tau_J}}.$$

Choosing trial constitutive functions such as

$$D(q_{\text{cap}}) = D_0 q_{\text{cap}}(1 - q_{\text{cap}}), \quad \Gamma(q_{\text{cap}}) = \Gamma_0 q_{\text{cap}}$$

makes \(q_{\text{cap}} = 0\) and \(q_{\text{cap}} = 1\) invariant sets, preventing overshoot. These functions are examples, not a derived action.

If \(q_{\text{cap}}\) first reaches zero on a two-surface and the microscopic theory licenses domain termination, that surface becomes a moving boundary

$$\partial\mathcal{M}_q(t) = \{x \mid q_{\text{cap}}(t, x) = 0\}.$$

The level-set kinematics are fixed by differentiating \(q_{\text{cap}}(t, X(t)) = 0\) along the moving surface:

$$V_n = -\frac{\partial_t q_{\text{cap}}}{|\nabla q_{\text{cap}}|}\bigg|_{q_{\text{cap}}\to0^+}.$$

In spherical symmetry this becomes

$$\frac{dr_f}{dt} = -\frac{\partial_t q_{\text{cap}}}{\partial_r q_{\text{cap}}}\bigg|_{r=r_f(t)}.$$

During continued infall the exterior should be Vaidya-like with a slowly varying mass parameter, settling to the Schwarzschild exterior after the front stabilizes. This is a well-posed program, but not yet a closed derivation: the transport coefficients, boundary action, and channel relaxation spectrum must be computed from the graph ensemble or constrained by simulation.

The dynamical system is presumed to preserve the usual covariant conservation of the combined matter-plus-capacity stress-energy, with any local matter depletion balanced by flux, boundary work, or capacity-sector stress. Showing that this conservation structure follows from a graph-derived transport action, rather than imposing it as a constitutive condition, is part of the dynamical closure work.

The concrete closure tests are correspondingly specific. A spherical collapse simulation should show formation of the first \(q = 0\) surface without overshoot into \(q < 0\). A coupled matter-plus-capacity run should approach a Vaidya exterior during accretion and a Schwarzschild exterior after settling while satisfying the combined conservation law. Exterior perturbation simulations with an absorbing boundary should reproduce standard Schwarzschild greybody factors and ringdown, while partial-reflectivity runs should produce controlled echo delays. Finally, a microscopic boundary-action calculation or a graph-ensemble Monte Carlo of saturated boundary channels should reproduce the channel-counting rule that yields \(n_{\text{hor}}S^{\text{cell}}_\infty = 1/4\). These are not new fit knobs; they are the numerical and microscopic tests that would close the dynamical and boundary sectors.

F.8 Weak-field boundary and closure statement

In the weak-field Solar-System regime, the metric-only parent yields

$$\gamma_{\text{PPN}} = \beta_{\text{PPN}} = 1,$$

with the remaining standard PPN coefficients vanishing under the usual assumptions. This follows because the ordinary longitudinal capacity functional is a reduced representation of Einstein gravity. In the carrier-resolved galactic EFT, localized Solar-System sources are absent from the transverse source map, so the leading local solution remains on this Einstein branch. Nonlinear preferred-frame terms still require the covariant projector audit of Appendix N.8.

The weak-field capacity coordinate ceases to be adequate when

$$\frac{|\Phi|}{c^2} = O(1), \quad \frac{\delta S}{S_\infty} = O(1),$$

which is the regime where the spherical invariant \(q_{\text{geo}}\) provides the controlled nonlinear description.

Appendix F therefore closes two precise strong-field statements. The spherically reduced Einstein action makes \(q_{\text{geo}} = (\nabla R)^2 = 1 - 2GM_{\text{MS}}/(c^2 R)\) a composite first integral and returns the Schwarzschild exterior. Separately, the factorized coherent \(j = 3\) boundary construction fixes the microscopic channel area and Immirzi normalization, while the bulk Green-response product remains an exact \(1/4\) cross-check. The bounded capacity interpretation of \(q_{\text{geo}}\), exclusion of the negative branch, and physical boundary dynamics remain additional hypotheses. Rotating and charged solutions belong to the baseline metric parent, but their capacity-variable completion is also open.

Appendix G: Many-Pasts, Operational Closure, Branch Realization, and the Arrow of Time

Appendix G separates the three probability objects used by Many-Pasts: amplitudes for unresolved alternatives, probabilities for decoherent record histories, and conditional probabilities for histories compatible with one present record. This repairs the ambiguity in the earlier notation \(P(H \mid P) \propto e^{-D(H,P)}\). The finite operational construction derived in H.11 has the standard decoherent-histories form [93]; Many-Pasts supplies its record-conditioned ontology.

G.1 What is a history of the entanglement network?

A coarse projective history \(h = (\alpha_1, \ldots, \alpha_n)\) is a sequence of alternatives at ordered substrate times. The alternative \(\alpha_k\) is represented by a projector \(\Pi^{(k)}_{\alpha_k}\). With unitary evolution \(U_{k,k-1}\) between times, its class operator is

$$C_h = \Pi^{(n)}_{\alpha_n}U_{n,n-1}\Pi^{(n-1)}_{\alpha_{n-1}}\cdots U_{2,1}\Pi^{(1)}_{\alpha_1}U_{1,0}.$$

This operator retains amplitudes. It is defined before any classical probability is assigned to the individual history.

Given an initial state \(\rho_0\), H.11 identifies the decoherence functional with the Gram kernel of these branch states:

$$\mathcal{D}(h, h') = \text{Tr}\left(C_h\rho_0 C^\dagger_{h'}\right).$$

A family admits ordinary probabilities when its off-diagonal terms are negligible at the required accuracy,

$$\mathcal{D}(h, h') \simeq 0 \quad (h \neq h').$$

The diagonal entries \(p(h) = \mathcal{D}(h, h)\) are then nonnegative and additive under coarse-graining. When alternatives do not decohere, their class operators must be added before the probability is evaluated. For \(C_A = \sum_{h\in A}C_h\),

$$p(A) = \text{Tr}\left(C_A\rho_0 C^\dagger_A\right),$$

which retains the interference terms. Many-Pasts places no classical distribution over unresolved fine-grained paths.

G.2 What is the present coarse configuration \(P\)?

The present \(P\) is a macroscopic record represented by a final projector \(\Pi_P\). Let \(\mathcal{H}_P\) be a decoherent family of histories whose final alternatives refine that record. Exhaustiveness and decoherence give

$$p(P) = \sum_{h\in\mathcal{H}_P}p(h) = \text{Tr}(\Pi_P\rho_{\text{now}}).$$

The conditional Many-Pasts measure is

$$\boxed{p(h \mid P) = \frac{p(h)}{p(P)}, \quad h \in \mathcal{H}_P.}$$

The common measure over all possible records is normalized first; conditioning on the realized record comes afterwards. Normalizing a new set of histories separately for each already-selected present would leave the probabilities of the alternative presents undefined.

G.3 What is the distance \(D(H, P)\)?

For a decoherent history ending in \(P\), define

$$D(h, P) = -\ln p(h),$$

with \(D = +\infty\) when \(p(h) = 0\). Then

$$p(h \mid P) = \frac{e^{-D(h,P)}}{\sum_{h'\in\mathcal{H}_P}e^{-D(h',P)}}.$$

The distance notation is shorthand for the diagonal decoherence-functional weight, not a second probability law. The earlier expression \(-\ln\text{Tr}(\Pi_P\rho_{H\to\text{now}})\) is recovered when \(H\) already denotes a decohered preparation history and only the final record remains unresolved.

G.4 The derived finite Born branch

A laboratory setting \(x\) is represented by the quantum instrument \(\{\mathcal{M}_{a|x}\}_a\) obtained by reversible coupling to a retained record and then retaining or forgetting its alternatives (H.11). Each map is completely positive and trace non-increasing, while \(\sum_a\mathcal{M}_{a|x}\) is trace preserving. For an initial state \(\rho\),

$$p(a \mid x) = \text{Tr}\left[\mathcal{M}_{a|x}(\rho)\right].$$

If \(C_{h,a|x}\) refines the histories ending in record \(a\), then

$$p(h \mid a, x) = \frac{\text{Tr}(C_{h,a|x}\rho C^\dagger_{h,a|x})}{p(a \mid x)}$$

for a decoherent refinement. The record marginal is the Born probability because the branch weight is the derived squared Hilbert norm. Reversible internal equivalence makes the weight depend only on that norm, and finite additivity under coarse-graining of orthogonal physical records fixes the dependence to be linear. The remaining premises are ordinary probability normalization and nonnegativity, together with record completeness for the decompositions used. Gleason's theorem and generalized-record results remain independent consistency checks [87, 88]. The global state, continuum/Fock realization, and which cosmological histories actually decohere are not fixed by this finite theorem.

G.5 No-signaling

Let Alice and Bob act locally on \(\rho_{AB}\) with instruments \(\{\mathcal{M}^A_{a|x}\}_a\) and \(\{\mathcal{N}^B_{b|y}\}_b\). Their joint record probability is

$$p(a, b \mid x, y) = \text{Tr}\left[(\mathcal{M}^A_{a|x} \otimes \mathcal{N}^B_{b|y})(\rho_{AB})\right].$$

Summing over Bob's record gives

$$p(a \mid x, y) = \sum_b p(a, b \mid x, y) \tag{25}$$ $$= \text{Tr}\left[(\mathcal{M}^A_{a|x} \otimes \mathcal{N}^B_y)(\rho_{AB})\right] \tag{26}$$ $$= \text{Tr}\left[\mathcal{M}^A_{a|x}(\rho_A)\right] = p(a \mid x), \tag{27}$$

where \(\mathcal{N}^B_y = \sum_b\mathcal{N}^B_{b|y}\) is trace preserving. Alice's marginal is independent of \(y\), and the same calculation applies to Bob. Global conditioning on a joint present record does not create a controllable signaling channel because the unconditioned local marginals remain those of standard quantum mechanics.

G.6 Branch realization

The postulate contains one realized present with its records. Its multiplicity lies among the admissible pasts of that present, with no forward branching into co-real macroscopic worlds and no collapse event selecting among them. An outcome is specified by the records that obtain in the present, and the Born weights give their statistics. This account of branch realization adds neither many worlds nor a collapse dynamics.

The present record algebra fixes the resolution at which past alternatives are physically distinguished. Let \(\{R_i\}_{i\in I}\) be mutually orthogonal fine records that a later physical channel maps to one coarse record \(\bar{R} = \sum_{i\in I}R_i\). For a decoherent joint refinement,

$$p(h, \bar{R}) = \sum_{i\in I}p(h, R_i), \quad p(h \mid \bar{R}) = \frac{\sum_{i\in I}p(h, R_i)}{\sum_{h'}\sum_{i\in I}p(h', R_i)}.$$

This is the push-forward of the same joint quantum measure. Record loss needs no additional rule and does not alter the earlier unitary dynamics. It reduces the distinctions represented in the realized present. The history register of H.8 is one microscopic carrier of such distinctions; environmental and macroscopic records are others. The paper does not assume an inaccessible second archive after every physical carrier of a distinction has been erased.

The conditioning also has no forward dynamical role. The class operator \(C_h\), the state, and the quantum instruments determine \(p(h, P)\) before the ratio \(p(h \mid P)\) is formed. A later record permits retrodictive conditioning once it exists; the conditional probability never appears as a force in the preceding evolution. Many-Pasts therefore preserves the causal and no-signaling content of the finite channel structure derived in H.11.

This record-retention reading removes one apparent cosmological alternative under a precise condition. If a state has a trivial macroscopic record algebra with respect to an earlier era, two descriptions that differ only by whether that era occurred before the record-free state are represented by the same present state. A physically meaningful cycle count would require a record surviving the interval. The framework neither supplies such a meta-register nor derives that a record-free cosmological state is reached. The result is an ontological equivalence and makes no claim about cosmic recurrence.

G.7 Arrow of time from conditional typicality

The proposed arrow-of-time extension requires a further typicality statement. Let \(h = \{M_t\}_{t_i\leq t\leq t_0}\) be a decoherent macrohistory conditioned on present records \(M_{t_0}\). A coarse Markov description would assign

$$P(h \mid M_{t_0}) \propto \mu_i(M_{t_i})\prod_{t_i\leq t<t_0}T(M_{t+\Delta t} \mid M_t),$$

where \(\mu_i\) is a boundary measure and \(T\) is the conditional macro-transition probability obtained only after summing the microscopic transitions compatible with each pair of macrostates. Multiplying a separate factor \(e^{S(M_t)}\) at every time would generally count the same microscopic multiplicity twice. A neutral transition law is also insufficient: conditional counting without a low-entropy boundary condition is dominated by high-entropy pasts and Boltzmann-fluctuation histories. The required theorem must show that \(\mu_i\) and \(T\) exponentially suppress those histories strongly enough for ordinary entropy-increasing histories to dominate. The boundary measure now has a forced support. Many-Pasts posits no meta-time and no unrecorded archive: absent a surviving record, "a prior era occurred and all traces vanished" and "no prior era occurred" are not distinct physical states, so the fundamental boundary is the minimal-record state, and a heavily recorded state cannot be initial because its record content constitutes a past (Appendix P.4). \(\mu_i\) is therefore defined on the first physically distinguishable record-bearing state, and the arrow is conditional typicality from that anchor; what the quotient removes is the freedom to place the boundary at an arbitrary high-entropy state. The needed substrate transition law has still not been derived here. This equation defines the target without claiming a completed thermodynamic arrow.

The minimal missing boundary condition can be stated directly. A Substrate Past Hypothesis would require

$$\text{supp}\,\rho_0 \subseteq \mathcal{H}_{M_{\text{low}}}, \quad t_0 - t_i \ll \min(t_{\text{relax}}, t_{\text{rec}}),$$

where \(\mathcal{H}_{M_{\text{low}}}\) is a low-entropy macro-subspace and the elapsed time is short compared with equilibration and recurrence. Under a mixing substrate dynamics, standard large-deviation counting would then favor entropy growth away from that boundary. The paper has not derived this hypothesis or the required mixing estimate. The saturated phase of Section 20 supplies the capacity-sector half of the hypothesis by construction — the pinned state is a near-zero-entropy configuration of the capacity sector — while the matter-sector clause is supplied, for the canonical defect class, by the visibility–source contrapositive of Appendix P.3: a state with no elementary massive defects carries no defect records, so matter-sector simplicity at the anchor is a named conditional rather than a free assumption. The quantitative companion is the well-foundedness bound of Appendix P.4, which prices a false durable record of content \(K\) at \(e^{\ln N_{\text{trials}}-K}\) with \(\ln N_{\text{trials}} \sim 5.6 \times 10^2\) nats, at the conditional grade it inherits from Section 20. Naming these premises isolates the remaining arrow-of-time problem from the already-closed operational probability branch.

The faithful-history-resolution theorem of Appendix H does not supply the missing boundary. Its stationary kernel satisfies

$$p(b)K_*(b, b') = p(b)p(b') = p(b')K_*(b', b),$$

so it is exactly detailed-balanced and time-reversal symmetric. The quantum dilation is globally reversible as well. Local export distinguishes a present register from a history register within the chosen update description, but the thermodynamic orientation still comes from the boundary condition and finite-time typicality theorem above.

G.8 Memoryless dressing and the connection to \(L_*\)

The dressing kernel's temporal reading has a spatial twin. For the canonical elementary defect, the rate at which the exported history becomes distinguishable from vacuum history and the scalar source insertion of Appendix C.5 are the same likelihood ratio, with the one-bit defect statistic sufficient; the identity, its isotropy benchmark \(\ln(7/6)\), and its exact decomposition are established in Appendix P.3. Postulate III and the weak-field source theorem, separable modules until that identity, consume one shared constant.

The history measure does not imply memorylessness merely because it is written as an exponential. The foundational faithful full-support principle selects it when applied to paths at the fixed admissibility marginal; maximum caliber is only the conventional name for that application. For any stationary dressing kernel with one-time marginal \(p_{\eta_*}\),

$$H(B_{t+1} \mid B_t) = g_{\text{share,eff}} - I(B_t; B_{t+1}) \leq g_{\text{share,eff}}.$$

Equality holds only when consecutive configurations are independent, which fixes \(K(b, b') = p_{\eta_*}(b')\) on the support. More generally, the entropy rate of any stationary process with this one-time marginal is bounded by \(g_{\text{share,eff}}\), with equality only when the present is independent of its complete past. The quantum replacement channel transfers the old local information into a dilation register rather than destroying it. Many-Pasts gives that register its history-space interpretation. The finite marked event is a separate statement from this renewal theorem and is supplied by the decorated vertex in H.9.

G.9 Relation to many-worlds, collapse, hidden variables, and decoherence

Many-Pasts is a form of history-space realism built on the decoherent-histories formalism. It posits one realized record-bearing present and no stochastic collapse term. Its probabilities are the diagonal entries of the finite history Gram kernel, conditioned on that present only after the alternative records have been normalized. The distinctive claim lies in the ontology assigned to this conditional measure and in its proposed substrate realization. On finite retained sectors, H.11 derives the Born norm, record instruments, and no-signaling from the real quadratic carrier, reversible record coupling, and locality. The continuum/Fock realization, cosmological initial state, actual decohering histories, and durable global record capacity remain open.

G.10 Familiar quantum examples: double-slit, EPR/Bell, and measurement

In a double-slit experiment the two path alternatives remain combined in one class operator while no which-path record exists. The probability of a detection event therefore includes their interference term. A durable which-path record defines a decoherent refinement, after which the path histories admit separate conditional probabilities and the fringe term is suppressed.

In an EPR or Bell experiment the record is joint, so its history measure carries the standard nonclassical correlations. Summing over the remote record invokes a trace-preserving local map and returns a marginal independent of the remote setting, as shown in G.5. Conditioning on the observed joint record does not alter that prior no-signaling marginal.

Measurement creates a durable record and identifies a decoherent family. Operationally the calculation is ordinary quantum mechanics. The additional claim is that the realized present is supported by the conditional ensemble of compatible past histories.

These examples introduce no new laboratory predictions. They show where positive history probabilities are legitimate and where amplitudes must remain combined.

Appendix H: Microscopic Realization and Coarse-Graining

This appendix develops a microscopic realization of the scalar stiffness and defect ontology. The weak-field coefficient chain does not depend on this realization. Section H.9a gives the complete ordinary-gravity derivation in one sequence; the other sections establish the finite action, prove its component results, identify countermodels, and test the geometric embedding.

H.1 GFT condensate realization and coarse-graining

The candidate microscopic realization is a GFT condensate of bosonic tetrahedral quanta coupled to fermionic defects. Denote the corresponding fields by \(\phi(g_1, \ldots, g_4)\) and \(\psi\). In the condensate regime, write the coarse field as

$$\sigma(x) = \sqrt{n(x)}\, e^{i\theta(x)}.$$

The hydrodynamic identity

$$|\nabla_\mu\sigma|^2 = \frac{(\nabla_\mu n)^2}{4n} + n(\nabla_\mu\theta)^2$$

shows that if

$$S_{\text{ent}}(x) = S_0 + \alpha\ln\frac{n(x)}{n_{\text{bg}}},$$

then the coarse action contains a positive scalar stiffness

$$\gamma \sim \frac{Z_\sigma n_{\text{bg}}}{2\alpha^2} > 0.$$

The coarse source channel arises from fermionic face exclusion. A localized condensate defect appears macroscopically as matter, and the surrounding reduction of available occupancy becomes the long-wavelength EFT field. The continuum kinetic term and source channel can therefore arise from the proposed substrate realization. Deriving every inhomogeneous continuum coefficient from the underlying kernel remains open.

This condensate argument supplies a microscopic realization compatible with the EFT ontology and sign choices. The coefficient derivation remains the one given earlier.

The condensate picture addresses the emergence of continuum geometry. The remainder of this appendix derives the defect dynamics behind faithful sector resolution from the admissibility ensemble of Part II together with the Many-Pasts postulate.

H.2 The dressing Hamiltonian and the lightest seven-channel branch

The faithful sector-resolution relation of Appendix D.4 was written there as a support-scale identity on the history space \(\mathcal{H}_{\text{hist}} = \bigotimes_{m=-3}^3\mathcal{H}^{(m)}_B\), using the refresh kernel \(P_{\eta_*}(b, b') = p_{\eta_*}(b')\) on each sector layer. Its dynamical content is carried by an explicit defect Hamiltonian, which we now write down so that the choice of that kernel can be derived.

The seven face-label channels of the admissibility-closed ensemble define the orthogonal projectors \(E_m\), \(m = -3, \ldots, 3\), of the face algebra \(\mathcal{A}_7\). By Postulate II the elementary defect is fermionic, so each channel carries an occupation number \(n_m \in \{0, 1\}\) with fermionic creation and annihilation operators \(c^\dagger_m, c_m\) and \(n_m = c^\dagger_m c_m\). When channel \(m\) is occupied it is dressed by a cloud state described by a density operator \(\rho_m\) on the boundary ensemble \(\mathcal{H}_B\). The defect Hamiltonian is

$$H = \underbrace{\sum_m\left(\varepsilon_0 n_m - n_m F_m(\rho_m)\right)}_{\text{single-channel terms}} + \underbrace{\sum_{m<m'}V_{mm'}(\rho_m, \rho_{m'})}_{\text{inter-channel coupling}},$$

with \(\varepsilon_0\) the bare cost to occupy a channel, \(F_m\) the free energy released by dressing channel \(m\) with its cloud, and \(V_{mm'}\) the residual coupling between the clouds of distinct channels. Two properties of the ground state of \(H\) supply the two ingredients of the sector-resolution relation.

One-pass occupation and the exponent seven. A channel is energetically occupied when its dressing gain exceeds the fixed bare cost, \(F_m(\rho_m) > \varepsilon_0\). Symmetry makes this inequality the same for all seven channels, so an all-bound configuration is a possible symmetric ground-state branch when the inequality holds. The static Hamiltonian alone cannot identify that branch with the lightest charged particle, because its ordering depends on the undetermined balance between \(\varepsilon_0\) and the dressing free energy.

The adopted recurrence mass functional supplies the missing ordering. Let \(k = \sum_m n_m\) be the number of resolved channels. Fermionic exclusion gives \(0 \leq k \leq 7\), and with the admissibility-closed marginal fixed on every occupied layer the support relation generalizes to

$$\frac{\lambda_k}{L_*} = \frac{2}{3}\exp(k g_{\text{share,eff}} - \Delta_k), \quad m_k = \frac{3\hbar}{2cL_*}\exp(\Delta_k - k g_{\text{share,eff}}),$$

where \(\Delta_k \geq 0\) is the total correlation among the occupied layers. For each \(k\), the minimum occurs at \(\Delta_k = 0\). Between those minima,

$$\frac{m_{k+1}}{m_k} = e^{-g_{\text{share,eff}}} \simeq 5.99 \times 10^{-4},$$

so every missing channel raises the minimum mass by \(e^{g_{\text{share,eff}}} \simeq 1.67 \times 10^3\). The unique lightest resolved one-bit defect in this support-to-length branch therefore has \(k = 7\) and \(\Delta_7 = 0\). The factor seven and the product cloud are selected together, without separate one-pass and factorization postulates. This theorem holds at fixed marginals \(p_{\eta_*}\). Optimizing the cloud over a different marginal would define a different ultraviolet ensemble.

Remark: finite marked corrections and channel occupation. Write \(H = g_{\text{share,eff}}\) and \(f(x) = -\ln(1 - x)\). At a common proper-time cadence, a resolved \(k\)-channel branch in the additive-generator class has

$$E_k = \frac{3\hbar}{2\tau_*}A_k f(e^{-S_k}), \quad S_k \leq kH.$$

Here \(S_k\) is the joint readout entropy and \(A_k\) is its positive motif sum. Retaining the unit unmarked motif gives \(A_k \geq 1\). The selected seven-channel product has \(E_7 = (3\hbar/2\tau_*)Z_e f(e^{-7H})\). Monotonicity and convexity of \(f\), with \(f(0) = 0\), give

$$\frac{E_k}{E_7} \geq \frac{f(e^{-kH})}{Z_e f(e^{-7H})} \geq \frac{e^{(7-k)H}}{Z_e}, \quad 1 \leq k < 7.$$

The second inequality follows from \(f(cx) \geq cf(x)\) for \(c \geq 1\) and \(cx < 1\). Its weakest bound is \(e^H/Z_e = 1659.8886\ldots\) at \(k = 6\). Thus the finite marked correction preserves the occupation ordering even for correlated lower-channel clouds, provided their energy uses this same survival prescription and their motif sum retains the unit term. Correlations among seven occupied channels are treated at fixed vertex in H.5.

Factorized cloud and the additive support. The dressing dimension multiplies across channels precisely when the joint cloud state is a product, \(\rho = \bigotimes_m\rho_m\); the seven contributions \(g_{\text{share,eff}}\) then add. The inter-channel term \(V_{mm'}\) determines whether the static ground state of \(H\) has this form. At fixed marginals, the recurrence mass functional independently selects the seven-channel product as the lightest branch. Each selected channel samples its full ensemble, and the minimizing readout has no inter-layer correlation.

The static Hamiltonian identifies the channel and cloud variables but does not order all of their branches. The next subsections prove the entropy ceiling, the unique memoryless kernel that reaches it, and the joint \(k = 7\), \(\Delta_7 = 0\) minimum of the adopted recurrence mass functional.

H.3 Slot coupling, layer factorization, and the additivity theorem

Two distinct correlation structures appear in the closure data, and separating them is essential: conflating them leads to the false conclusion that the cloud cannot factorize.

The closure invariant is a pure pair coupling. Expanding \(S^2 = \left(\sum_i m_i\right)^2 = \Sigma^2 + 2\sum_{i<j}m_i m_j\) in \(K^2(b) = 48 - \frac{1}{3}(S^2 - \Sigma^2)\) cancels the self-terms exactly and leaves the identity

$$K^2(b) = 48 - \frac{2}{3}\sum_{i<j}m_i m_j.$$

The admissibility weight \(e^{-\eta_* K^2(b)}\) therefore contains only cross-terms between distinct face slots of a single boundary state; it does not factorize over those four slots. This is the exact origin of the residual correlation between face slots on the closed ensemble,

$$I(\text{slot}_0; \text{slot}_1) = 0.1545\ \text{nats}, \quad \frac{I}{H(\text{slot})} = 0.079,$$

so the four faces of one tetrahedron are about eight percent correlated, a structural feature of the closure invariant.

Slot coupling does not obstruct layer factorization. The factorization the support relation requires is over the seven sector layers \(b^{(-3)}, \ldots, b^{(3)}\) of \(\mathcal{H}_{\text{hist}}\), each layer being a full boundary state drawn from the entire 1680-state ensemble. The slot coupling \(-\frac{2}{3}m_i m_j\) lives inside a single layer's boundary state: it relates the four faces of that one tetrahedron and never couples layer \(m\) to layer \(m'\). The two structures act on different objects:

The substrate's intrinsic correlation, the most natural candidate obstruction to factorization, therefore acts at the wrong level to obstruct it: it is internal to a layer, not between layers.

Additivity of the decoherent preparation overlap. Appendix G.3 defines \(D(h, P) = -\ln p(h)\) from the diagonal decoherence-functional probability. For the special case in which \(H\) already denotes a decohered preparation and only the final record remains unresolved, this reduces to \(D(H, P) = -\ln\text{Tr}(\Pi_P\rho_{H\to\text{now}})\). If that preparation state and the resolution projector factorize over channels, \(\rho = \bigotimes_m\rho_m\) and \(\Pi_P = \bigotimes_m\Pi_m\), the trace factorizes and the logarithm converts the product into a sum,

$$\text{Tr}(\Pi_P\rho) = \prod_m\text{Tr}(\Pi_m\rho_m) \implies D = \sum_m\left(-\ln\text{Tr}(\Pi_m\rho_m)\right) = \sum_m D_m.$$

The overlap distance is therefore additive in this special factorized preparation. The effective-support statement used downstream follows independently from Shannon subadditivity: a product cloud with seven equal marginals has joint entropy \(7g_{\text{share,eff}}\) and perplexity \(e^{7g_{\text{share,eff}}}\). Whether the electron's dressing is that product is decided by the two conditions in the next subsections.

The ceiling and the two conditions. Subadditivity bounds the support from above. With each channel marginal fixed to the admissibility weight, \(H(B_m) = g_{\text{share,eff}}\), the joint entropy of a single readout obeys

$$H(B_{-3}, \ldots, B_3) \leq \sum_{m=-3}^3 H(B_m) = 7g_{\text{share,eff}},$$

and the deficit

$$\Delta = \sum_m H(B_m) - H(B_{-3}, \ldots, B_3) \geq 0$$

measures the total correlation among the seven channels. The full support \(\dim_{\text{eff}} = e^{7g_{\text{share,eff}}}\) is reached precisely when each channel carries its full entropy \(g_{\text{share,eff}}\) and the channels are mutually independent, \(\Delta = 0\). The first is a condition on each layer in substrate time; the second is a condition on the seven layers at a single readout. The next two subsections establish the unique faithful-resolution kernel and the minimal-mass factorized branch, while separating those results from the still-open microscopic implementation.

H.4 Maximum caliber: complete renewal is uniquely selected

The closure calculation fixes the one-time marginal \(p_{\eta_*}\). Applying faithful full-support resolution to entire paths maximizes their entropy rate, a condition conventionally called maximum caliber [89]. This is the temporal form of the existing foundational requirement. For an arbitrary stationary process, its entropy rate is

$$h_\mu := \lim_{n\to\infty}H(B_0 \mid B_{-1}, \ldots, B_{-n}) \leq H(B_0) = g_{\text{share,eff}}.$$

Equality holds only when \(B_0\) is independent of its complete past. Stationarity then makes the process independent and identically distributed, with every conditional distribution equal to \(p_{\eta_*}\). Maximum path entropy therefore selects complete renewal uniquely. Finite capacity alone does not imply renewal. The selection follows from faithful full-support resolution, whose temporal formulation is maximum caliber.

The one-parameter family below illustrates the cost of retained memory:

$$K_a(b, b') = a\,\delta(b, b') + (1 - a)p_{\eta_*}(b'), \quad a \in [0, 1],$$

which repeats the current state with probability \(a\) and otherwise redraws from the stationary weight. Every member has \(p_{\eta_*}\) as its stationary distribution and the same single-time marginal, so the equilibrium ensemble cannot say which one governs the dressing. The per-channel conditional entropy \(H(b' \mid b) = \sum_b p_{\eta_*}(b)H\left(K_a(b, \cdot)\right)\), evaluated on the exact 1680-state ensemble, nonetheless slides with the memory \(a\):

Only the memoryless endpoint \(a = 0\) — the replacement kernel \(K_*(b, b') = p_{\eta_*}(b')\) of Appendix D.4 — returns the full \(g_{\text{share,eff}}\). For any stationary Markov kernel with that marginal,

$$H(B_{t+1} \mid B_t) = H(B_{t+1}) - I(B_t; B_{t+1}) \leq g_{\text{share,eff}},$$

and equality holds if and only if \(I(B_t; B_{t+1}) = 0\). The Markov corollary is therefore

$$\boxed{K_*(b, b') = p_{\eta_*}(b').}$$

Single-label local moves conserve slot ordering, fracturing each parity copy into twenty-four sectors of thirty-five states, so no such local kernel reaches the full ensemble at any parameter value (Appendix D.4). The native-vertex construction of H.8 implements the selected nonlocal-in-label-space replacement as one operation on the complete local tetrahedral register.

The lightest-defect branch selects the same endpoint independently within the adopted support map. If a per-pass temporal mutual information \(I_t\) reduces the fresh entropy from \(g_{\text{share,eff}}\) to \(g_{\text{share,eff}} - I_t\), then at fixed \(L_*\)

$$\lambda(I_t) = \lambda(0)e^{-I_t}, \quad m(I_t) = m(0)e^{I_t}.$$

Every retained temporal correlation raises the recurrence mass, so electron lightness again selects \(I_t = 0\). Faithful full-support resolution selects the vacuum history process; mass minimization independently selects the same process in the defect sector.

A dimensionless mixing test shows that the selected kernel is dynamically available. Take any ergodic generator \(Q\) (a symbol used only within this mixing test; the survival complement of H.6 is a distinct operator) with \(p_{\eta_*}\) as its detailed-balance stationary weight. The test uses nonlocal transpositions of two occupied labels, not the injectivity-preserving single-label shifts excluded in the preceding paragraph. A continuous-time swap generator built from the same admissibility weights serves. The kernel \(e^{Q\tau}\) loses endpoint mutual information,

$$I(\tau=0.1) \approx 3.4\ \text{nats}, \quad I(\tau=1) \approx 0.030, \quad I(\tau=2) \approx 1 \times 10^{-4},$$

so the replacement kernel is the late-\(\tau\) limit of a broad class of dimensionless mixing dynamics. Faithful full-support resolution selects this endpoint but leaves the approach to it unspecified. H.8 derives the quantum replacement channel and reversible update; H.9 supplies the fresh amplitude, marked event, and update interaction. Durable history capacity and the stable geometric GFT embedding remain open.

H.5 Independent channels: the electron as the lightest defect

The second condition is that the occupied channels factorize at one readout, so that the deficit \(\Delta_k\) of H.3 vanishes. Appendix H.2 already showed that electron lightness selects both the maximum occupation \(k = 7\) and \(\Delta_7 = 0\) once the support-to-length dictionary is adopted. The calculation here isolates the factorization part of that joint minimum.

Keep the deficit explicit in the baseline support relation. With each channel at its full entropy, the joint entropy is \(7g_{\text{share,eff}} - \Delta\), and

$$\frac{\lambda}{L_*^{(0)}} = \frac{2}{3}e^{7g_{\text{share,eff}}-\Delta}.$$

At a fixed substrate scale \(L_*\), a one-bit charged defect whose dressing carries correlation \(\Delta\) has spatial support \(\lambda \propto e^{-\Delta}\), and through \(m = \hbar/(c\lambda)\) a mass

$$m \propto e^\Delta.$$

Correlation therefore contracts the support and raises the recurrence mass. Because \(\Delta \geq 0\), the lightest one-bit charged defect is the product dressing \(\Delta = 0\); correlated dressings describe heavier branches within the same recurrence/support map. Independence is not inferred from the absence of an interaction term. It is forced by mass minimization in the theory's own definition of the electron anchor.

Correlated clouds at fixed marked vertex. The finite correction preserves the product minimum within the specified vertex. Hold the seven marginals at \(p_{\eta_*}\), all occupations at \(n_m = 1\), and the internal routing of H.9 fixed. The marked action then contains only the constants \(g_{\text{share,eff}}\), \(\eta_*\), and the occupied pair projectors. Its auxiliary records retain their specified contractions. A change in the joint cloud readout law \(P\) changes its entropy, while the internal motif operator is unchanged. Consequently

$$Z_e[P] = (1 + \zeta_*)(1 + 7\zeta_*^2) = Z_e.$$

With \(H = g_{\text{share,eff}}\), the fixed-marginal KL identity gives

$$\Delta = 7H - H(P) = D\left(P\|p_{\eta_*}^{\otimes 7}\right) \geq 0, \quad r(P) = e^{-H(P)} = re^\Delta.$$

For the selected additive generator,

$$\boxed{\frac{E(P)}{E(p_{\eta_*}^{\otimes 7})} = \frac{f(re^\Delta)}{f(r)} \geq e^\Delta, \quad f(x) = -\ln(1 - x).}$$

Equality with the minimum requires \(\Delta = 0\), hence \(P = p_{\eta_*}^{\otimes 7}\). This establishes the unique product minimum over the fixed-marginal readout laws of the same marked vertex. A model that retains only \(Z_\mu\) has the weaker lower bound \(e^\Delta/(1 + 7\zeta_*^2)\) and leaves \(\Delta < \ln(1 + 7\zeta_*^2) = 1.83741 \times 10^{-4}\) nats unexcluded. Such a model changes the specified electron return block. A coupling between the cloud correlations and the auxiliary records would likewise require a different vertex.

Faithful full support fixes memoryless refresh. The occupation bound in H.2 and the fixed-vertex result above select \(k = 7\) and \(\Delta_7 = 0\) within the marked-transfer prescription. H.6 gives the baseline survival scale and H.9 its finite dressing:

$$L_* = -\frac{3}{2}Z_e\lambda_e\ln\left(1 - e^{-7g_{\text{share,eff}}}\right), \quad G_* = \frac{c^3 L_*^2}{\hbar},$$

with no gravitational quantity used as input.

The decorated action realizes the nonlocal replacement, fresh state, and charged marked event. In the factorized completion those operations live on the finite capacity/record factor and need not be re-derived from the geometric magnetic indices. A specific many-body GFT realization must still determine how the decoration couples to inhomogeneous matter and transverse response; it need not re-select memorylessness, the clock conversion, or \(\Delta = 0\).

H.6 Mass from the positive survival transfer operator

Faithful full-support resolution fixes the replacement process on history space but not a dimensional duration. The duration comes from the paper's existing electron anchor once the already-declared effective-support event is represented in the transfer space. The electron supplies the dimensional anchor; the determinant-survival and marked-generator prescriptions specify its dynamical readout.

The marked seven-channel transfer. On the factorized lightest branch, likelihood multiplication on each renewed cloud has determinant \(e^{-g_{\text{share,eff}}}\). The product determinant is

$$r := \Delta_{\tau_p}^{\otimes 7}(R^{\otimes 7}) = e^{-7g_{\text{share,eff}}}.$$

This is the scalar charged determinant line of the decorated transfer, not the probability of an arbitrarily chosen seven-layer microstate. The selected one-bit survival action assigns the positive no-loop transfer

$$T^{(0)}_{\text{surv}} = 1 - r,$$

equivalently the one-dimensional Grassmann determinant \(\int d\bar{c}\, dc\, e^{-\bar{c}(1-r)c} = 1 - r\). One may therefore define

$$H^{(0)}_{\text{surv}} = -\frac{\hbar}{\tau^{(0)}_*}\ln T^{(0)}_{\text{surv}},$$

with exact raw gap

$$\boxed{E^{(0)}_{\text{raw}} = -\frac{\hbar}{\tau^{(0)}_*}\ln(1 - r).}$$

The unit mixing gap on the renewed probability space is a different operator and is not identified with this exponentially small charged gap.

Electron calibration and exact cell length. Before the finite marked response is included, the transverse export gives \(H^{(0)}_e = (3/2)H^{(0)}_{\text{surv}}\). The baseline electron calibration is

$$m_e c^2 = -\frac{3\hbar}{2\tau^{(0)}_*}\ln(1 - r).$$

It fixes

$$\boxed{\tau^{(0)}_* = -\frac{3}{2}\tau_e\ln(1 - r), \quad L^{(0)}_* = c\tau^{(0)}_* = -\frac{3}{2}\lambda_e\ln(1 - r).}$$

For \(r = e^{-7g_{\text{share,eff}}}\),

$$L^{(0)}_* = -\frac{3}{2}\lambda_e\ln\left(1 - e^{-7g_{\text{share,eff}}}\right) = \frac{3}{2}\lambda_e e^{-7g_{\text{share,eff}}}\left[1 + \frac{1}{2}e^{-7g_{\text{share,eff}}} + O(e^{-14g_{\text{share,eff}}})\right].$$

The relative logarithmic correction to the leading expression is \(1.4 \times 10^{-23}\). The baseline induced scale is

$$G^{(0)}_* = \frac{9}{4}\frac{\hbar c}{m_e^2}\left[\ln\left(1 - e^{-7g_{\text{share,eff}}}\right)\right]^2,$$

whose leading form is proportional to \(e^{-14g_{\text{share,eff}}}\). H.9 derives the finite response and gives \(\tau_* = Z_e\tau^{(0)}_*\), \(L_* = Z_e L^{(0)}_*\), and \(G_* = Z_e^2 G^{(0)}_*\).

Why this supplies the phase bridge. A positive Euclidean transfer operator defines an energy spectrum. Analytic continuation supplies the Lorentzian phase frequency of the same charged mode. An independent identification of an integer renewal count with a condensate winding is unnecessary. H.9 realizes the determinant recurrence and marked response in the decorated charged vertex.

The temporal recurrence gives \(\tau_e = N_{\text{eff}}\tau_*\). Defining the associated causal length by \(L_* = c\tau_*\) then gives \(\lambda_e = N_{\text{eff}}L_*\). This is the scale convention used in the static graph calculation. It does not identify one graph displacement with \(L_*\). The physical displacement per update must follow from the coupled spatial transfer, and a geometric tetrahedron may carry a different length fixed by its embedding.

Internal locality and the native vertex. The tested single-label move cannot implement replacement. Injectivity-preserving shifts fracture the 1680 states into

$$48\ \text{components} = 4!\ \text{orderings} \times 2\ \text{orientations}, \quad 35\ \text{states per component}.$$

H.8 constructs a one-layer reversible dilation when the complete tetrahedral register is the native local vertex. This establishes circuit depth one, not the physical duration; the duration is fixed spectrally by the electron equation above. H.9 decorates that gate with the fresh amplitude and marked charged event.

H.7 Conditional dressing functionals and the condensate-spectrum lemma

The dressing Hamiltonian named in Section 23,

$$H = \sum_m\left(\varepsilon_0 n_m - n_m F_m(\rho_m)\right) + \sum_{m<m'}V_{mm'}(\rho_m, \rho_{m'}),$$

carries three named functionals \(\varepsilon_0, F_m, V\). A proposed mean-field condensate gives a conditional reconstruction of these functionals. The required condensate has not yet been derived from a specified GFT action. The two residuals of H.6 also remain separate outputs of the missing fluctuation calculation.

Mean-field reduction. In a GFT condensate the boundary-data order parameter is \(\langle\hat{\varphi}(b)\rangle = \sigma(b)\). The admissibility calculation fixes a normalized modulus profile \(p_{\eta_*}(b)\), so a candidate condensate may be written

$$|\sigma(b)|^2 = \bar{n}\, p_{\eta_*}(b).$$

This fixes neither the overall density \(\bar{n}\) nor any of the phases. A general 1680-component condensate has a common phase and relative-phase directions, and their physical status is determined by the kinetic and interaction kernels of the specified GFT action. Likewise, \(\eta_*^{-1}\) may be used as a formal closure-temperature parameter, but the identity

$$\eta_*\langle K^2\rangle_{\eta_*} = \frac{3}{2}$$

is the stationarity condition generated by the three-component determinant weight, not a literal equipartition theorem for the bounded discrete spectrum. Conditional on a condensate realization, \(T_* = 1/\eta_* = 33.48\) may nevertheless be read as the dimensionless substrate closure temperature, with \(\langle K^2\rangle_{\eta_*} = \frac{3}{2}T_* = 50.22\); this is an interpretation of the same stationary point and introduces no new number.

The refresh projector. The memoryless transition matrix is \(K_*(b, b') = p_{\eta_*}(b')\). Let \(D_p = \text{diag}(p_{\eta_*})\) and define the detailed-balance symmetrization

$$\tilde{K}_* = D_p^{1/2}K_* D_p^{-1/2}.$$

With \(v_b = \sqrt{p_{\eta_*}(b)}\), one obtains

$$\tilde{K}_* = |v\rangle\langle v| \equiv P, \quad \tilde{L}_{\text{mix}} = I - \tilde{K}_* = P_\perp.$$

The classical replacement process has one stationary direction in the weighted probability space and removes every orthogonal probability mode after one discrete update. The vector \(|v\rangle\) belongs to the symmetrized classical transfer representation; it is not automatically the quantum state of the cell. It does not follow that \(P_\perp\) is a Hermitian mass matrix in a closed Lorentzian GFT. The Markov update, a quantum density operator, and the coherent fluctuation Hessian are different objects.

Conditional refresh–Hessian bridge. The minimal reversible bridge uses the positive refresh Dirichlet form as the internal Euclidean quadratic cost,

$$\mathcal{K}^{(2)}_{\text{int}} = \Delta_{\text{ref}}P_\perp, \quad \Delta_{\text{ref}} > 0.$$

A fixed-\(j\) Euclidean effective action realizing this bridge is

$$\Gamma_E[\sigma] = \int d^4x_E\left[Z\,\partial_\mu\sigma^\dagger\partial^\mu\sigma - \mu^2\sigma^\dagger\sigma + \frac{\lambda}{2}(\sigma^\dagger\sigma)^2 + \Delta_{\text{ref}}\sigma^\dagger P_\perp\sigma\right].$$

It has \(\sigma_0 = \sqrt{\bar{n}}\, e^{i\theta_0}v\) with \(\bar{n} = \mu^2/\lambda\). With canonically normalized real fluctuations, its Euclidean quadratic operator contains a radial singlet with \(m_h^2 = 2\mu^2/Z\), a common phase with \(m_\pi^2 = 0\) when number U(1) is exact, and 1679 complex relative modes with \(m_\perp^2 = \Delta_{\text{ref}}/Z\). Lorentzian propagation requires the usual analytic continuation and the global \((-, +, +, +)\) convention of Appendix A. The exclusion source couples at linear order to the density fluctuation \(h\) because \(\delta(\sigma^\dagger\sigma) = 2\sqrt{\bar{n}}\, h + \cdots\). In the constant-phase realization displayed here, the common phase is derivatively coupled and has no linear zero-frequency overlap with that static density source. This is not a structural decoupling theorem for a relational GFT condensate. If \(\sigma = \sqrt{n}\, e^{i\theta}\) and the background carries relational phase current \(\partial_\chi\theta_0 \neq 0\), the Madelung term contains

$$n(\partial_\chi\theta)^2 \supset 2(\partial_\chi\theta_0)\delta n\,\partial_\chi\delta\theta,$$

so density and phase can mix. The complete coherent Hessian and defect/source vertex must determine whether a phase mode is gapped, remains light but source-orthogonal, or mediates an additional response. Simplicial interactions may break number U(1) and gap it.

An open-system completion may instead implement renewal as the quantum replacement channel

$$\mathcal{E}_*(\rho) = \rho_*\text{Tr}\,\rho.$$

The classical closure ensemble fixes the diagonal probabilities of \(\rho_*\) in the boundary basis. H.9 adopts the diagonal maximum-entropy completion and constructs its reversible update. A dissipative representation with Lindblad generator \(\dot{\rho} = \Gamma_{\text{ref}}[\mathcal{E}_*(\rho) - \rho]\) gives every traceless input perturbation a relaxation gap. A geometric condensate embedding must determine how this transfer description appears in its coherent fluctuation spectrum.

Selection, mixing, and marked survival are distinct operators. The combinatorial Hamiltonian selects the admissibility profile. The mixing generator \(\tilde{L}_{\text{mix}} = P_\perp\) controls decay toward that profile. The exponentially small electron scale comes from a third object: the scalar determinant transfer of the seven-channel likelihood operator,

$$r := \Delta_{\tau_p}^{\otimes 7}(R^{\otimes 7}) = e^{-7g_{\text{share,eff}}},$$

whose one-bit survival transfer is \(T^{(0)}_{\text{surv}} = 1 - r\). Over a baseline proper-time step \(\tau^{(0)}_*\) its exact spectral gap is

$$\text{gap}(H^{(0)}_{\text{surv}}) = -\frac{1}{\tau^{(0)}_*}\ln(1 - r), \quad m_e c^2 = \frac{3}{2}\hbar\,\text{gap}(H^{(0)}_{\text{surv}}).$$

Solving gives \(\tau^{(0)}_* = -(3/2)\tau_e\ln(1 - r)\) and \(L^{(0)}_* = c\tau^{(0)}_*\). H.9 enumerates \(Z_e\) and uses its additive-generator prescription to obtain \(L_* = Z_e L^{(0)}_*\). The unit mixing gap and charged survival gap are not identified.

\(\varepsilon_0\), the bare cost. \(\varepsilon_0\) is the condensate chemical potential \(\mu_0 = \partial E/\partial N\), the energy to commit one cell against the mean field, of order \(E_* = \hbar c/L_*\). Energies in \(H\) are measured in units of \(E_*\), and \(T_* = 1/\eta_*\) is dimensionless. The rest mass is read from the positive marked-survival operator \(H_{\text{surv}}\), not from the static spectrum of \(H\) or the unit mixing gap of \(L_{\text{mix}}\).

\(F_m\), the dressing free energy. A bound channel carries a cloud \(\rho_m\) over the boundary ensemble. The proposed released free energy is \(F_m(\rho_m) = T_* S[\rho_m]\). At fixed mean closure its maximum is the Gibbs state \(p_{\eta_*}\) with entropy \(g_{\text{share,eff}}\). On the selected lightest branch the seven clouds carry total entropy \(7g_{\text{share,eff}}\). The faithful state-weighted determinant of their likelihood operator is exactly \(e^{-7g_{\text{share,eff}}}\). This is a quenched multiplicative transfer rate, not a raw single-draw Gibbs probability.

Two energies, kept distinct. The separation cost and rest energy are different quantities. The separation cost is the entropic free energy required to dissolve a defect into the substrate; in this construction it is large and linear in \(g_{\text{share,eff}}\). The rest energy is \(m_e c^2 = \hbar\Gamma_e\), with \(\Gamma_e\) set by the rare joint recurrence and therefore exponential in \(-7g_{\text{share,eff}}\). Reading the separation cost as the mass would place the electron near tens of \(E_*\) and remove the required suppression.

\(V\), the inter-channel coupling. The permutation-invariant singlet

$$|s\rangle = \frac{1}{\sqrt{7}}\sum_m |m\rangle$$

is the natural channel through which a scalar capacity fluctuation couples uniformly to the seven sectors. Within the minimal constant-phase refresh–Hessian completion displayed above, the radial density fluctuation is the only linearly source-coupled singlet at static order and the relative sector is gapped. Integrating out that density mode gives

$$V_{mm'} = v\langle m|s\rangle\langle s|m'\rangle = \frac{v}{7},$$

a rank-one outer product. The per-incidence overlap is \(1/7\); the separate orientation doubling gives the \(2/7\) used in Appendix I.1. The implication is

$$\text{one source-coupled density singlet} \implies \text{rank}(V) = 1.$$

This result is derived inside the minimal bridge, not from the closure ensemble alone. A microscopic interaction that mixes \(P\) and \(P_\perp\), closes the relative gap, or sources the common phase would add exchange channels and deform the uniform structure.

Positive-transfer phase and the exchanged mode. The projector calculation separates the static source from the transfer clock. The static exclusion source couples to the radial density singlet. Independently, the positive Euclidean survival operator defines the electron energy, and ordinary analytic continuation gives the Lorentzian phase frequency \(\omega_e = E_e/\hbar\) of that same charged mode. The decorated vertex supplies the finite charged routing, and the additive-generator prescription specifies its energy readout. The remaining transverse question concerns the microscopic origin of the carrier source projector and metric contact: the interaction must preserve the established projection and must not create an additional sourced scalar pole beyond the Einstein constraint response.

The remaining microscopic lemma. The effective projector model narrows the missing calculation without requiring geometry–capacity mixing to vanish. For a specified fixed-\(j\) simplicial GFT action admitting \(\sigma_b = \sqrt{\bar{n}p_{\eta_*}(b)}e^{i\theta}\), compute its complete Bogoliubov or Schwinger–Keldysh kernel. After the within-tick evolution \(\mathcal{U}^\dagger\), exact renewal gives

$$\mathcal{T}^\dagger = P\mathcal{U}^\dagger = \begin{pmatrix}A & B\\0 & 0\end{pmatrix} \quad \text{on } P\mathcal{K} \oplus P_\perp\mathcal{K},$$

so the replaceable sector cannot carry an autonomous nonzero transfer pole even when \(B \neq 0\); H.9a proves the spectral statement. The remaining calculation is instead to show that the retained block \(A\) has only the Einstein constraint and transverse–traceless gravitational support at low energy, while any additional persistent relative mode is gapped, relaxational, gauge/constraint, or source-orthogonal. The decorated transfer vertex of H.9 fixes the charged recurrence and its edge Hessian. A geometric GFT must still determine whether its retained condensate realization produces a coherent mass, a relaxation gap, or both, and verify that no additional source-coupled light field survives. The closure weight fixes \(P\) exactly, while the refresh–Hessian bridge for the geometric condensate remains a selected effective completion.

H.8 Faithful history resolution and reversible Many-Pasts renewal

Four distinct statements fall under "refresh": the classical history process, the quantum channel, its reversible implementation, and the conversion of a positive charged spectrum into proper time. They are treated separately below. Section H.9 supplies the charged marked vertex. Maximum-caliber terminology and the clock construction add no foundational premise.

Finite-record entropy identity. For a joint law \(P_n\) of \(n\) records with fixed marginal \(p_*\) on each record,

$$D_{\text{KL}}(P_n\|p_*^{\otimes n}) = nH(p_*) - H(P_n) \geq 0.$$

Equality holds precisely at \(P_n = p_*^{\otimes n}\). Faithful resolution, defined here as maximal joint entropy subject to the fixed marginals, therefore selects independent layer records. Applied to every finite temporal block it selects independent successive records as well. The stationary entropy-rate statement below gives the same endpoint.

Maximum path entropy. Let \(\mathcal{B}\) be the finite admissible boundary space and let \(p_*(b) = p_{\eta_*}(b)\) be its fixed stationary marginal. For any stationary process \(\{B_n\}\) on \(\mathcal{B}\),

$$h_\mu = \lim_{N\to\infty}\frac{1}{N}H(B_1, \ldots, B_N) = H(B_0 \mid B_{-1}, B_{-2}, \ldots) \leq H(p_*).$$

Equality holds exactly when \(B_0\) is independent of its complete past. Applying the same statement after every time translation factorizes every finite joint distribution,

$$P(B_1 = b_1, \ldots, B_N = b_N) = \prod_{n=1}^N p_*(b_n).$$

The unique maximum-caliber history process is therefore i.i.d. For a Markov representation its transition kernel is

$$\boxed{K_*(b, b') = p_*(b').}$$

Here "maximum caliber" is terminology for faithful full-support resolution on history space [89]. It is not a sixth theory-defining input. The foundational requirement already says that an admissible resolution carries the full available entropy compatible with its fixed data; applying that same requirement to a path gives the equality case above. Finite capacity alone would not imply renewal, but the paper's stronger faithful full-support premise does.

The selected process carries no thermodynamic orientation. It obeys

$$p_*(b)K_*(b, b') = p_*(b)p_*(b') = p_*(b')K_*(b', b),$$

so its stationary path measure is exactly invariant under time reversal. Faithful history resolution derives local renewal, not a low-entropy past and not a physical cadence.

Proper time from the positive marked-transfer spectrum. The renewal theorem fixes the discrete process but says nothing about seconds per update. Proper time comes from the determinant survival operator of H.6,

$$r = \Delta_{\tau_p}^{\otimes 7}(R^{\otimes 7}) = e^{-7g_{\text{share,eff}}}, \quad T^{(0)}_{\text{surv}} = 1 - r.$$

Defining the baseline transfer over \(\tau^{(0)}_*\) gives

$$H^{(0)}_{\text{surv}} := -\frac{\hbar}{\tau^{(0)}_*}\ln T^{(0)}_{\text{surv}}, \quad E^{(0)}_{\text{raw}} = -\frac{\hbar}{\tau^{(0)}_*}\ln(1 - r)$$

on its nonzero support. The baseline transverse identification \(m_e c^2 = (3/2)E^{(0)}_{\text{raw}}\) therefore yields

$$\boxed{\tau^{(0)}_* = -\frac{3}{2}\tau_e\ln(1 - r), \quad L^{(0)}_* = c\tau^{(0)}_* = -\frac{3}{2}\lambda_e\ln(1 - r).}$$

The leading form is \(L^{(0)}_* = (3/2)\lambda_e r[1 + O(r)]\). H.9 supplies the finite marked factor \(Z_e\) and hence the physical \(L_* = Z_e L^{(0)}_*\). No update diameter, throughput maximization, or geometric conversion factor enters. The complete tetrahedron being a native gate is a locality and circuit-depth statement only; its physical embedding size is a separate consistency question.

A positive Euclidean transfer spectrum also supplies, by analytic continuation, the Lorentzian phase frequency. The clock and phase use the same charged spectral mode and require no independent winding postulate. H.9 derives the recurrence as the state-weighted determinant and implements its finite marked dressing.

Quantum replacement theorem. Let \(\mathcal{H}_A\) be the replaceable closure-register Hilbert space and \(\mathcal{E} : A \to A'\) a trace-preserving quantum channel. It does not include the retained position and internal fiber of a persistent charged defect. Complete local renewal means that the output contains no information about the input, including information visible through an arbitrary reference \(R\):

$$(\text{id}_R \otimes\mathcal{E})(\rho_{RA}) = \rho_R \otimes \rho_* \quad \text{for every } \rho_{RA}.$$

Taking product inputs first shows that every normalized input has the same output \(\rho_*\). Linearity then gives the unique channel on all operators,

$$\boxed{\mathcal{E}_*(X) = \rho_*\text{Tr}\,X.}$$

Every traceless perturbation is annihilated after one update. As a superoperator, \(\mathcal{E}_*\) has one stationary direction and zero on the traceless operator subspace. The closure ensemble fixes \(\langle b|\rho_*|b\rangle = p_*(b)\) in the boundary basis. H.9 selects the diagonal maximum-entropy completion for the decorated transfer action; a different coherent GFT state would define a different microscopic completion and must be tested against that vertex. H.11 proves that this statement is compatible with particle interference only when the renewed output excludes the persistent marked-position fiber and the discarded free-renewal record is branch-independent.

Reversible history export. Local replacement is compatible with global unitarity by Stinespring dilation [90]. Introduce the old present register \(A\), a fresh register \(F\), its purifier \(R\), and a history register \(H\). For

$$|\Psi_*\rangle_{FR} = \sum_{b\in\mathcal{B}}\sqrt{p_*(b)}|b\rangle_F|b\rangle_R$$

in the diagonal closure-state realization, a register permutation can act as

$$|\psi\rangle_A|\Psi_*\rangle_{FR}|0\rangle_H \longmapsto |\Psi_*\rangle_{AR}|0\rangle_F|\psi\rangle_H.$$

Tracing out \(R, F, H\) leaves \(\rho_*\) on the new closure present and removes every dependence on the old replaceable state. The inverse unitary recovers that state from \(H\), so no global information has been destroyed. Postulate III supplies the interpretation of the exported register as history structure; the dilation theorem alone calls it an environment. For an unmeasured freely propagating defect, the state exported by this map may not encode the cell address of the mark. The marked fiber is transported in the retained defect system, and only a genuine interaction may correlate it with a discarded environmental record. Repeating the construction indefinitely requires fresh and history capacity or a controlled recycling mechanism, which the present paper has not derived.

The tetrahedral circuit. For

$$\mathcal{B} = \{(s; m_1, m_2, m_3, m_4) : s = \pm, m_i \in \{-3, \ldots, 3\}, m_i \neq m_j\}, \quad |\mathcal{B}| = 1680,$$

the register permutation above is one circuit layer if the complete tetrahedron is a native local gate and the fresh admissible register has already been prepared. Neither injectivity nor the collective function \(K^2(m_1, m_2, m_3, m_4)\) then requires a sequence of face repairs; the gate replaces one admissible four-face object by another.

A depth-one layer of disjoint two-face gates cannot test all six injectivity relations. The direct pair-comparison construction uses the three perfect matchings

$$\{(1, 2), (3, 4)\}, \quad \{(1, 3), (2, 4)\}, \quad \{(1, 4), (2, 3)\},$$

and therefore has depth three. This proves the stated depth for the direct comparator and excludes depth one in a pairwise architecture without a jointly coupled ancilla. It is not a lower-bound proof for every possible ancilla-assisted state-preparation algorithm. H.9 takes the complete tetrahedron as the native decorated gate; a geometric GFT embedding must realize that locality and sustain the prepared state dynamically.

Consequences and remaining gap. The foundational faithful full-support condition closes the classical replacement kernel when applied to histories. Detailed-balance symmetrization then closes \(P = |\sqrt{p}\rangle\langle\sqrt{p}|\) and \(P_\perp = I - P\) in probability space. Reference decoupling closes the quantum replacement-channel form, and the native tetrahedral architecture supplies a one-layer reversible dilation. H.9 adds the marked recurrence, after which positivity and the electron anchor close the clock, length, and phase-frequency conversion. These results remove an arbitrary transition matrix, geometric clock factor, and separate winding assumption while showing how local memorylessness can coexist with a globally retained past.

They do not by themselves close the coherent GFT Hessian. The exclusion sufficiency lemma of C.5 proves that one scalar statistic contains all local classical information distinguishing the canonical defect from the vacuum. The replacement process removes orthogonal probability memory. A microscopic spectrum must still show that no additional conserved or coherent light field survives and that the source couples only to the capacity singlet. The bridge from \(P_\perp\) to a Euclidean mass or Schwinger–Keldysh relaxation pole therefore retains its conditional grade. The stationary reset also leaves the macroscopic arrow-of-time problem in G.7 unchanged.

H.9 Decorated simplicial marked-transfer vertex

The renewed marginal, lightest seven-channel branch, determinant recurrence, and reversible present/history exchange determine a finite charged vertex with a fully specified marked sector. In the selected factorized completion of Section 23 and Appendix Q, this vertex acts on the capacity/record factor; an EPRL-class simplicial factor carries the vacuum geometry. The equivariant \(V_3\) frame lock relates the two factors without identifying their magnetic indices. On the native tetrahedron, the vertex realizes faithful full-support history resolution, and the established one-bit fermionic defect supplies the hard-core mark. The field space and routing are physical choices: the countervertices below fail the finite audits.

The marked fiber from primitive fusion. The nine-state marked space also has an independent geometric derivation. A shared face begins with two primitive fermionic slots of spin \(j_0 = 3/2\). Maximum-capacity fusion selects the unique maximal channel,

$$V_{3/2} \otimes V_{3/2} = V_3 \oplus \underbrace{(V_0 \oplus V_1 \oplus V_2)}_Q, \quad \dim Q = 9.$$

The selected \(V_3\) is the seven-state effective bulk link used by the closure ensemble. The nonmaximal information is the representation \(Q = V_0 \oplus V_1 \oplus V_2\). The marked response was obtained from a different construction: the closure response is a vector \(V_1\), and one response excitation on each of the present and history strands gives

$$\mathcal{H}_{\text{mark}} = V_1^P \otimes V_1^H = V_0 \oplus V_1 \oplus V_2.$$

Thus there is an \(SU(2)\) intertwiner

$$\mathcal{I} : Q \longrightarrow \mathcal{H}_{\text{mark}}, \quad \mathcal{I}|(j_0 j_0); J, M\rangle = |(1_P 1_H); J, M\rangle, \quad J = 0, 1, 2,$$

with \(\mathcal{I}^\dagger\mathcal{I} = P_Q\) and \(\mathcal{I}\mathcal{I}^\dagger = I_9\). Each irrep occurs once, so the intertwiner is unique up to one phase on each \(J\) block.

The match is more selective than the dimension count. For a response strand of spin \(s\),

$$V_s \otimes V_s = \bigoplus_{J=0}^{2s}V_J,$$

whereas the complement of maximal fusion for two primitive spin-\(j\) slots is

$$Q(j) = (V_j \otimes V_j) \ominus V_{2j} = \bigoplus_{J=0}^{2j-1}V_J.$$

The two representations agree if and only if \(j = s + 1/2\). The independently fixed closure-vector value \(s = 1\) therefore selects \(j = 3/2\). Appendix B proves the representation theorem and shows that unitarity of the selected sharp reversible split fixes its retained output to \(Q\). Since the complete H.9 one-mark carrier also has dimension nine, the complement map is an isometric carrier-space identification. This proves the incidence used here without deriving every microscopic detail: the equal weights, flags, relative block phases, hard-core realization, and routing remain selected structure of the explicit vertex below.

Fresh-state amplitude from the closure action. The Gaussian response uses an auxiliary vector whose norm includes the quantum variance in Appendix B.2. Define

$$c(b) = \sum_i m_i\hat{n}_i, \quad C(b) = \sqrt{\frac{K^2(b)}{|c(b)|^2}}\, c(b).$$

Injectivity ensures \(|c(b)|^2 = (4\Sigma^2 - S^2)/3 > 0\), so the definition is regular on the full ensemble. It gives \(C_a C_a = K^2\) and transforms as a vector under a common frame rotation. The coordinates \(C_a\) are a selected auxiliary realization of the scalar closure weight; their definition does not identify three commuting measurements of the quantum closure operator. The diagonal renewed state has amplitude

$$A_*(b) = Z^{-1/2}\exp\left[-\frac{\eta_*}{2}K^2(b)\right], \quad |A_*(b)|^2 = p_*(b).$$

For a standard three-component Gaussian auxiliary \(\xi_a\),

$$\exp\left[-\frac{\eta_*}{2}C^2\right] = \int\frac{d^3\xi}{(2\pi)^{3/2}}\exp\left[-\frac{1}{2}\xi^2 + i\sqrt{\eta_*}\,\xi_a C_a\right].$$

The amplitude-level closure incidence is therefore \(\sqrt{\eta_*}\). The selected charged decoration reuses this auxiliary strand with the same incidence. This fixes the response coupling within the chosen vertex. Faithful full-support resolution is being applied on a marked record space with no further joint constraint. Adding a defect-only invariant would define a nonminimal countervertex; it is allowed as a different theory and is one of the falsifiers of the construction, not an extra premise needed by the selected action.

The reversible update has independent present and history Gaussian strands. Their first normalized Hermite excitations obey

$$\langle a|c\rangle_P = \delta_{ac}, \quad \langle b|d\rangle_H = \delta_{bd}.$$

The ordered products \(|a, b\rangle = |a\rangle_P \otimes |b\rangle_H\) form nine orthonormal states,

$$\langle a, b|c, d\rangle = \delta_{ac}\delta_{bd}.$$

They carry \(\mathbb{R}^3 \otimes \mathbb{R}^3 = 1 \oplus 3 \oplus 5\). Introduce hard-core marked states \(d^\dagger_{ab}|0\rangle\) and the stranded vertex

$$V_Q = \sum_{a,b=1}^3 d^\dagger_{ab}\xi^P_a\xi^H_b + \text{h.c.}$$

Its reduced Gram matrix is \(I_3 \otimes I_3 = I_9\). Equal marked weights follow from this two-strand contraction, not from rotational symmetry or the number nine alone. Trace and epsilon contractions would split the three rotation sectors and falsify the vertex.

Pair contraction and canonical dilation. For each unordered occupied pair \(e = (m, m')\), the two directed scalar returns are orthogonal alternatives. Define

$$R_e = q\left(\langle m\to m'| + \langle m'\to m|\right), \quad q = \frac{2}{7},$$

and \(B_e = \sqrt{\eta_*}R_e\). Then

$$B_e B^\dagger_e = 2\eta_* q^2 = \frac{8\eta_*}{49} \equiv u, \quad 0 < u < 1.$$

The canonical unitary dilation of this contraction is

$$U_e = \begin{pmatrix}\sqrt{I - B^\dagger_e B_e} & -B^\dagger_e\\B_e & \sqrt{I - B_e B^\dagger_e}\end{pmatrix}.$$

On the occupied scalar-event support its no-event amplitude is

$$\alpha = \sqrt{1 - u} = \sqrt{1 - \frac{8\eta_*}{49}}.$$

This identifies the origin of every entry: \(\eta_*\) comes from the same closure amplitude that prepares the vacuum, \(2q^2\) is the norm of the directed return pair, and the square root follows from reversible dilation.

For an auxiliary Grassmann representation, introduce \(\Gamma_e = (\gamma^P_e, \gamma^H_e)^T\) and \(J = \begin{pmatrix}0 & 1\\-1 & 0\end{pmatrix}\). With \(n_m = c^\dagger_m c_m\) and \(n_d = \sum_{ab}d^\dagger_{ab}d_{ab}\), the marked action is

$$\boxed{\begin{aligned}S_{\text{mark}} = &\ g_{\text{share,eff}}n_d + \Lambda_d n_d(n_d - 1)\\&+ \frac{1}{2}\sum_{m<m'}\Gamma^T_{mm'}[1 + n_d n_m n_{m'}(\alpha - 1)]J\Gamma_{mm'}, \quad \Lambda_d \to +\infty.\end{aligned}}$$

The quadratic coefficient is \(\alpha = \sqrt{1 - u}\), not the probability \(1 - u\). The Grassmann integral returns a Pfaffian amplitude, so inserting \(1 - u\) in each \(2 \times 2\) Majorana block would count the no-event probability twice and produce \((1 - u)^{21}\) instead of the required amplitude \((1 - u)^{21/2}\). The Majorana variables represent the antisymmetric present/history amplitude; they are not additional propagating particles. When the marked fiber or either incident channel is absent, the reference block is \(J\) and has Pfaffian one. On the established \(n_m = 1\) seven-channel branch, all \(\binom{7}{2} = 21\) blocks become \(\alpha J\), so

$$Z_{\text{edge}} = \alpha^{21} = \left(1 - \frac{8\eta_*}{49}\right)^{21/2}.$$

These event records do not belong to the physical composite space \(\Lambda^2\mathbb{C}^7\). The natural rank-one singlet lift on that composite space has the exact \(6 + 15\) split derived in the audit; it does not act on the separately stranded internal present/history response records used here.

This distinction is why all 21 pair records survive even though the scalar source couples through one rank-one singlet. The singlet projector acts on physical channel amplitudes. The 21 Majorana blocks label mutually distinguishable internal present/history incidences, so quotienting them by the physical singlet would identify different incidences and violate faithful resolution. Their edge labels are internal to the marked response and carry no spatial cell address under the free vertex.

Complete decorated vertex. Let \(\mathcal{G}_v\) denote the geometric fixed-spin gluing tensor and \(W_*\) the native update that prepares \(A_*(b)\) and exports the old replaceable closure register. The position and nine-state marked fiber belong to the retained defect system, not to a cell-addressed environment chain. Under free transport \(W_*\) must therefore be translation-covariant and must leave the discarded renewal output independent of the defect's spatial branch. This is the no-which-path condition derived in H.11; it restricts the realization of the already-required quantum vertex and adds no foundational postulate. The incidence \(V_Q\) above identifies the nine marked basis vectors with the ordered Gaussian response products. The transfer space also contains the unmarked vacuum. Its projector is

$$\Pi_Q = |0\rangle\langle 0| + \sum_{a,b=1}^3 |a, b\rangle\langle a, b|.$$

For \(e = (m, m')\), define

$$P_e = n_d n_m n_{m'}, \quad \tilde{U}_e = (I - P_e) + P_e U_e.$$

The complete one-step decorated transfer is

$$\boxed{\mathcal{A}_v = \mathcal{G}_v W_*\Pi_Q\exp[-g_{\text{share,eff}}n_d - \Lambda_d n_d(n_d - 1)]\prod_{m<m'}\tilde{U}_{mm'}.}$$

The empty state therefore has unit weight, while each of the nine one-mark states has fugacity \(e^{-g_{\text{share,eff}}}\). The factors \(\tilde{U}_e\) act on distinct edge-history strands and commute. The Grassmann action above is the Pfaffian trace representation of these controlled pair factors; it is not an additional operator multiplied into \(\mathcal{A}_v\). Faithful full-support resolution on this internal response space, which has no further joint edge constraint, selects the tensor product of its one-edge marginals. Taking its finite internal trace at fixed position branch gives

$$\boxed{\zeta_* = 9e^{-g_{\text{share,eff}}}\left(1 - \frac{8\eta_*}{49}\right)^{21/2} = 0.005123584484947.}$$

Complete edge-Hessian audit. On the 21-dimensional edge-label space, two distinct edges either share one endpoint or are disjoint. Let \(A_1\) and \(A_0\) be their adjacency matrices. The Johnson edge algebra has spectra

$$\text{spec}(A_1) = \{-2^{\times14}, 3^{\times6}, 10^{\times1}\},$$ $$\text{spec}(A_0) = \{-4^{\times6}, 1^{\times14}, 10^{\times1}\}.$$

A generic permutation-covariant edge kernel can therefore carry three eigenvalues. The conditioned vertex instead gives

$$\mathcal{K}^{(2)}_{\text{edge}} = I_{21} \otimes \alpha J, \quad \text{spec}_{\text{edge}} = \{\alpha^{\times21}\},$$

with zero off-diagonal edge norm. Coherently averaging the induced edge permutation matrix over all \(7! = 5040\) channel relabelings gives \(\mathbf{1}\mathbf{1}^T/21\), of rank one. That operation would collapse the result to a shared singlet. It is not the charged gluing used here: \(m = -3, \ldots, 3\) are physical spin weights, \(K^2\) is not invariant under arbitrary relabeling, and the defect Hamiltonian contains diagonal occupations and density interactions rather than hopping terms \(c^\dagger_m c_{m'}\).

Primitive routing enumeration. Write \(w = e^{-g_{\text{share,eff}}}\alpha^{21}\), so \(\zeta_* = 9w\). The one-step transfer block contains three finite state spaces. The universal sector has one empty state and nine marked states, hence

$$Z_\mu = 1 + 9w = 1 + \zeta_*.$$

The electron feedback sector has one empty state and states \(|m; I_L, I_R\rangle\), with seven persistent charged labels and two independently renewed nine-state endpoints. Its dimension is

$$1 + 7 \times 9 \times 9 = 568,$$

and its trace is

$$Z_{e,\text{return}} = 1 + 7 \times 81w^2 = 1 + 7\zeta_*^2.$$

The second-shell sector has one empty state and nine marked states with one scalar passage \(q = 2/7\), giving

$$Z_{\tau,2} = 1 + 9qw = 1 + \frac{2}{7}\zeta_*.$$

The direct product contains \(10 \times 568 \times 10 = 56,800\) configurations. Exhaustive enumeration reproduces the analytic product exactly. The finite routing factors are therefore

$$Z_\mu = 1 + \zeta_*, \quad Z_e = (1 + \zeta_*)(1 + 7\zeta_*^2), \quad Z_{\tau,2} = 1 + \frac{2}{7}\zeta_*.$$

One native update applies each controlled gate once. Hard-core unit charge restricts the marked return to vacuum plus one return motif; the charged label persists diagonally while the response endpoints renew; and the shell grading terminates after \(N = 2\). These restrictions implement structures already present elsewhere in the paper: one native renewal pass, the one-bit fermionic anchor, the physical spin-weight labels, and the three-shell closure ladder. They add no separate foundational premise. Relaxing them gives explicit controls. A shared endpoint polarization changes the electron factor to \(1+7\zeta_*^2/9\) and moves \(G\) to \(-14.6\sigma\); off-diagonal charged propagation gives \(1 + 49\zeta_*^2\) and moves it to \(+98.1\sigma\).

Charged generator and proper-time matching. The internal enumeration gives positive dimensionless motif weights \(w_i\). For the electron their values and multiplicities are

$$\sum_i w_i = (1 + 9w)(1 + 567w^2) = Z_e.$$

The dynamical completion assigns these weights additively to the charged Euclidean generator. Write \(r = e^{-7g_{\text{share,eff}}}\), \(a = -(3/2)\ln(1 - r) > 0\), and let \(n_e\) project onto the retained electron sector. For dimensionless marked activity \(s\), the selected transfer obeys

$$T(s + t) = T(s)T(t), \quad T(0) = I, \quad T(1) = e^{-a n_e}.$$

Continuity on the occupied scalar block fixes \(T(s) = e^{-as n_e}\). Assigning activity \(w_i\) to each motif gives

$$T_e = e^{-aZ_e n_e}, \quad H_e = -\frac{\hbar}{\tau_*}\ln T_e = -\frac{3\hbar Z_e}{2\tau_*}\ln(1 - r)n_e.$$

The vacuum eigenvalue is one and the charged eigenvalue lies strictly between zero and one. This specifies the positive effective charged transfer after the internal records have been eliminated. The equality of motif weights and generator activities is a constitutive rule of the selected vertex. A derivation from a native microscopic evolution would have to reproduce this matching. Multiplying a transfer by a partition trace would instead add \(-(\hbar/\tau_*)\ln Z_e\) to its energy; that operation is absent from the present prescription.

A finite recurrence completion. The same motif weights admit an explicit quadratic contraction that decorates a single charged recurrence. Let \(M = 5680\) be the electron motif count, let \(v_i = \sqrt{w_i}\), and introduce auxiliary Grassmann variables for those \(M\) alternatives and one charged determinant variable. Choose the dimensionless quadratic kernel

$$\mathcal{K}_{\text{evt}} = \begin{pmatrix}1 & -\sqrt{r}\, v^\dagger\\-\sqrt{r}\, v & I_M\end{pmatrix}, \quad v^\dagger v = \sum_i w_i = Z_e.$$

The auxiliary variables represent internal recurrence alternatives. Their uncoupled determinant is one. Eliminating them gives the charged Schur complement and normalized Grassmann determinant

$$K_{\text{charged}} = 1 - rv^\dagger v = 1 - rZ_e, \quad \det\mathcal{K}_{\text{evt}} = 1 - rZ_e.$$

The spectrum of \(\mathcal{K}_{\text{evt}}\) consists of \(1 \pm \sqrt{rZ_e}\) and \(M - 1\) unit eigenvalues, so it is positive when \(rZ_e < 1\). With \(n_e\) the charged projector, the positive effective transfer and its occupied energy are

$$T_{\text{evt}} = (I - n_e) + (1 - rZ_e)^{3/2}n_e, \quad E^{\text{evt}}_e = -\frac{3\hbar}{2\tau_*}\ln(1 - rZ_e).$$

The electron anchor would then give

$$L^{\text{evt}}_* = -\frac{3}{2}\lambda_e\ln(1 - rZ_e).$$

The coupling \(\sqrt{r}\, v_i\) specifies how the motifs attach to the charged determinant line. It defines this alternative finite action without a fitted coefficient; the isolated internal trace does not select that attachment. Its difference from the additive-generator energy is

$$\frac{E^{\text{evt}}_e}{E_e} - 1 = \frac{f(rZ_e)}{Z_e f(r)} - 1 = \frac{Z_e - 1}{2}r + O(r^2) = 7.36676 \times 10^{-26}.$$

At fixed \(m_e\) the fractional length change is the same, and the induced \(G_*\) changes by \(1.47335 \times 10^{-25}\) relatively. The displayed numerical precision is unchanged. For a fixed-vertex correlated cloud, \(r \mapsto re^\Delta\) gives \(E^{\text{evt}}_e(P)/E^{\text{evt}}_e(P_*) \geq e^\Delta\) by the same convexity argument as H.5. The fixed-marginal Shannon bound \(H(P) \geq H\) ensures \(Z_e r(P) \leq Z_e e^{-H} < 1\) throughout this readout class.

The coefficient chain below uses the additive-generator prescription \(E_e = (3\hbar/2\tau_*)Z_e f(r)\). The finite recurrence contraction supplies a second fully specified action. Selection between their exact spectra requires a physical rule for attaching the marked response to the charged evolution.

The independent-factor completion gives another check with the same weights. Its positive transfer

$$T_{\text{ind}} = \prod_i(1 - rw_i)^{(3/2)n_e}$$

has occupied energy \(-(3\hbar/2\tau_*)\sum_i\ln(1 - rw_i)\). Relative to \(H_e\), its fractional difference is \(7.32 \times 10^{-26}\) at the stated ensemble. This estimate applies to this specified alternative. Positivity alone supplies no universal bound on other generator prescriptions.

Predictions and scope. The comparisons use CODATA 2022 for \(m_e\), \(G\), and \(m_\mu/m_e\), with the exact SI definitions of \(h\) and \(c\). The PDG tau mass is \(1776.93 \pm 0.09\) MeV; the selected ratio gives \(m_\tau c^2 = 1776.9188\) MeV and a residual \(-0.125\sigma\) [1, 2]. These residuals use experimental standard uncertainties and do not include a probability distribution over alternative microscopic prescriptions.

The physical scale and charged-lepton ratios are

$$L_* = Z_e\left[-\frac{3}{2}\lambda_e\ln(1 - e^{-7g_{\text{share,eff}}})\right], \quad G_* = \frac{c^3 L_*^2}{\hbar},$$ $$\frac{m_\mu}{m_e} = 720\frac{2}{7}Z_\mu, \quad \frac{m_\tau}{m_e} = 720^2\left(\frac{2}{7}\right)^4 Z_\mu Z_{\tau,2}.$$

They evaluate to \(G_* = 6.6742890813 \times 10^{-11}\ \text{m}^3\text{kg}^{-1}\text{s}^{-2}\), \(m_\mu/m_e = 206.768280237\), and \(m_\tau/m_e = 3477.343310\). The residuals are \(-0.073\sigma\), \(-0.535\sigma\), and \(-0.125\sigma\).

The finite marked weights and charged transfer prescriptions are specified; the numerical chain uses the displayed additive-generator prescription. Its mass minimum holds at fixed marginals and marked vertex, while its ordinary gravitational normalization uses the physical matching of C.5. The alternative kernels remain useful falsifiers: a collective edge mode, a physical pair-composite interpretation, persistent endpoint polarization, additional defect-only coupling, or off-diagonal charged propagator changes the answer. The geometric branch is also more specific than a "stable condensate" requirement. Appendix B supplies exact \(J = 3\) phase selection and maximal-fusion blocking; Appendix Q supplies a separate gauge-covariant capacity factor with an equivariant frame lock; and EPRL/Regge constructions supply controlled two-helicity block witnesses. H.9a gives the full ordinary-gravity chain. H.10 proves the massless-TT infrared theorem on the selected metric-Regge branch, constructs an exact fixed-\(j = 3\) perfect-action realization, closes the factorized capacity audits, and proves the clock-test theorem. It also shows that the capacity premises do not fix the geometric gluing tensor or measure. That geometric selection problem, durable history storage, the thermodynamic arrow, and the microscopic origin of the carrier-resolved transverse contact remain open.

Reproduction. The supplementary scripts reproduce the ensemble invariants, state-weighted determinant, \(48 \times 35\) locality fracture, Gaussian closure amplitude, unitary pair dilation, nine-state Gram matrix, 21-block edge Hessian, and 56,800-state routing trace. They also verify the dressed scale and corrected charged-lepton ratios of Appendix I.1. Appendix R contains the self-contained numerical-spine script; the complete Hessian and graph enumerations remain in the accompanying audit files.

H.9a From the finite cell to the Einstein action

Purpose and logical structure. The ordinary-gravity claim is not that the 1680-state capacity code secretly contains the Einstein action. Geometry and capacity are distinct operator factors in the selected realization, and each step below has its own logical grade. The derivation is

$$\text{finite capacity cell} \longrightarrow \text{selected relational quantum geometry} \longrightarrow \text{metric Regge branch}$$ $$\longrightarrow \text{Einstein gravity in the infrared}.$$

The representation theory and maximal-fusion ray are exact once the factorized geometric completion is selected. Metricity, block-scale Lorentz invariance, and membership in the appropriate semiclassical phase are the named geometric premises. The capacity action fixes the channel, record dynamics, source, and dimensional scale; it does not by itself choose the complete ultraviolet geometric propagator.

1. Relational frame geometry. In the selected spatial-frame completion, each tetrahedral cell \(c\) carries a local frame \(R_c\), while only relative transport between adjacent cells is observable:

$$g_{cc'} = R_c U_{cc'}R_{c'}^{-1} \in SU(2).$$

Here \(U_{cc'}\) is genuine link transport expressed in reference frames, with \(U_{c'c} = U_{cc'}^{-1}\). A change of local frame, \(R_c \mapsto h_c R_c\), acts by

$$g_{cc'} \mapsto h_c g_{cc'}h_{c'}^{-1}.$$

A loop based at \(c\) has holonomy \(R_c U_{\text{loop}}R_c^{-1}\) and can carry curvature. Thus the selected relational description has the standard local \(SU(2)\) gauge redundancy. If \(J_{cf}\) is the flux through face \(f\) in the frame of cell \(c\), gauge invariance imposes the Gauss constraint

$$G_c = \sum_{f\subset\partial c}J_{cf} = 0.$$

The geometric state space of a primitive tetrahedron with four spin-three faces is therefore

$$\mathcal{H}^{\text{geom}}_c = \text{Inv}_{SU(2)}\left(V_3^{\otimes 4}\right).$$

Since

$$V_3 \otimes V_3 = \bigoplus_{J=0}^6 V_J,$$

each intermediate \(J = 0, \ldots, 6\) can be paired with the same \(J\) in the other pair to form a singlet. Hence

$$\dim\text{Inv}_{SU(2)}(V_3^{\otimes 4}) = 7.$$

This seven-dimensional intertwiner space is geometric. It must not be identified with the independent seven-state capacity carrier. The cell space remains

$$\mathcal{H}_{\text{cell}} = \mathcal{H}_{\text{geom}} \otimes \mathcal{H}_{\text{cap}} \otimes \mathbb{C}^2_{\text{orient}},$$

with the precise factorization and gauge action given in Appendix Q.4. The equality of two dimensions does no physical work.

Classically one writes the face flux as

$$\mathbf{E}_f = A_f\mathbf{n}_f, \quad \sum_f A_f\mathbf{n}_f = 0.$$

For nondegenerate outward normals, the closure data reconstruct a convex tetrahedron up to translation. The dictionary is therefore

$$j_f \leftrightarrow A_f, \quad \iota_c \leftrightarrow \text{tetrahedral shape}, \quad g_{cc'} \leftrightarrow \text{relative connection}.$$

This is the first gravity step: the selected geometric factor carries the usual holonomy–flux data of relational three-geometry. It is a selected completion of the finite substrate, not a consequence of the 1680 count alone.

2. Carrier selection and the exact coarse ray. The primitive positive mismatch operator acts in the capacity fusion factor:

$$Q_f = 6P_0 + 5P_1 + 3P_2, \quad \ker Q_f = V_3^{\text{cap}}.$$

Consequently

$$F_3 e^{-sQ_f}F^\dagger_3 = I_{V_3^{\text{cap}}} \quad (s > 0).$$

The equivariant lock of Appendix Q.5 then pairs this selected carrier with a geometric copy,

$$L_3 : V_3^{\text{cap}} \longrightarrow V_3^{\text{geom}}, \quad L_3 U_{\text{cap}}(g) = U_{\text{geom}}(g)L_3.$$

Thus \(Q_f\) selects the capacity carrier and the lock selects the matching geometric representation. Neither operation supplies curvature:

$$\boxed{\begin{array}{c}Q_f \text{ and } L_3 \text{ select and relate the face data;}\\\text{the geometric host supplies their propagation and curvature.}\end{array}}$$

For a block of \(N\) aligned primitive faces, the highest-spin irrep occurs once in \(V_3^{\otimes N}\). There is therefore a unique normalized equivariant fusion map up to phase,

$$F_N : V_3^{\otimes N} \longrightarrow V_{3N}, \quad J_N = 3N.$$

On spin coherent states,

$$F_N|3, \mathbf{n}\rangle^{\otimes N} = |3N, \mathbf{n}\rangle,$$

and coherent transport fuses exactly:

$$\langle 3, \mathbf{n}|D^3(u)|3, \mathbf{n}'\rangle^N = \langle 3N, \mathbf{n}|D^{3N}(u)|3N, \mathbf{n}'\rangle.$$

The semiclassical parameter is therefore the block size, not a running microscopic spin:

$$j_{\text{primitive}} = 3\ \text{fixed}, \quad N \to \infty, \quad J = 3N \to \infty.$$

The fusion identity is exact on the maximal coherent ray. It is not a proof that every interacting refinement trajectory flows onto that ray; the inter-cell dynamical refinement remains a host-theory test.

2a. The locked ray has an exact collective phase space. The number and phase of a pre-glued spin-three face mode form the collective phase space of the coarse spin. For coherent amplitude

$$\alpha = \sqrt{\bar{N}}\, e^{-i\theta}$$

and transported endpoint normals \(\mathbf{n}_L, \mathbf{n}_R\), the Berry one-form is

$$\vartheta_B = \bar{N}\, d\theta + 3\bar{N}\left[\mathcal{A}(\mathbf{n}_L) - \mathcal{A}(\mathbf{n}_R)\right].$$

With

$$J = 3\bar{N}, \quad \xi = \frac{\theta}{3},$$

this becomes

$$\vartheta_B = J\, d\xi + J\left[\mathcal{A}(\mathbf{n}_L) - \mathcal{A}(\mathbf{n}_R)\right],$$

the twisted-geometry symplectic potential on \(T^*SU(2)\). The identification is exact on the locked ray. Because every allowed spin is \(J = 3N\), the twist fiber obeys

$$(J, \xi) \in 3\mathbb{N} \times \mathbb{R}/(2\pi/3)\mathbb{Z} \simeq (N, \theta) \in \mathbb{N} \times S^1.$$

The resulting \(\mathbb{Z}_3\) quotient is structural, not a coordinate artifact.

The same ray has an exact overcomplete family. With

$$c_N(\alpha) = e^{-|\alpha|^2/2}\frac{\alpha^N}{\sqrt{N!}}, \quad |\alpha, \mathbf{n}\rangle\rangle := \sum_{N=0}^\infty c_N(\alpha)\sqrt{6N + 1}|3N, \mathbf{n}\rangle,$$

Glauber orthogonality and the spin-coherent resolution give

$$\boxed{\int\frac{d^2\alpha}{\pi}\int\frac{d\Omega(\mathbf{n})}{4\pi}|\alpha, \mathbf{n}\rangle\rangle\langle\langle\alpha, \mathbf{n}| = \bigoplus_{N=0}^\infty I_{V_{3N}}.}$$

Thus the collective number, twist, and normals provide a genuine coherent-state path-integral chart, with relative uncertainties \(\Delta J/\bar{J} = O(\bar{J}^{-1/2})\). This construction requires a physical phase: the selected GFT field must be real or its global U(1) phase must be broken. It is stated for a pre-glued link because the twist is relational between its two ends.

2b. Lorentzianization is applied after fusion. The representation fixes the ordering. On the lowest \(SU(2)\) block \(V_j\) of the principal-series representation \((\rho, k) = (\gamma j, j)\), rotational covariance and \(K\cdot L = \rho k\) give

$$P_j K_i P_j = \frac{\gamma j}{j + 1}L_i.$$

Applying the Lorentzian Y map separately to primitive \(j = 3\) facets therefore gives the coefficient \(3\gamma/4\). Fusing first to \(J = 3N\) and applying Y only at block scale gives instead

$$\frac{\gamma J}{J + 1} = \gamma\frac{3N}{3N + 1} \longrightarrow \gamma.$$

The two orderings differ by 25% at primitive spin. The area normalization below was fixed from the large-\(J\) law with \(J = 3N\), so its consistent implementation is

$$\boxed{\text{exact } SU(2) \text{ fusion first, block-scale } Y_{\gamma_*} \text{ second}.}$$

Facet-first Lorentzianization would require a different Immirzi normalization and would define a different theory.

2c. The orientation doublet is renewed and geometrically inert; its fermion audit remains open. The binary orientation factor cannot be a hidden persistent register. If a uniformly populated lift label \(\sigma \in \{+, -\}\) survived from one renewal step to the next, then the renewed boundary data would obey

$$I(B_t; B_{t+1}) \geq H(\sigma) = \ln 2,$$

contradicting the full-support memoryless kernel proved in H.4. The doublet is therefore refreshed rather than transported as an additional geometric degree of freedom.

On the selected pure-gravity branch this causes no loss of geometric information. The two lifts \((\nu, \omega) = (+, +)\) and \((-, -)\) have the same Regge-sector label \(\mu = \nu\omega = +1\), determine the same metric, and carry the same \(e^{+iS_R}\) asymptotic phase. The proper vertex may consequently be extended as

$$\mathcal{G}_v = A^{\text{prop}}_{v,\mu=+} \otimes I_{\text{orient}},$$

so that gravity depends only on the metric-equivalence class and the refreshed bit is inert.

That conclusion does not automatically extend to fermions, which couple to a tetrad rather than to the metric alone. A matter completion must verify

$$\boxed{\mathcal{A}_{\text{matter}}[e, \psi] = \mathcal{A}_{\text{matter}}[e', \psi'] \quad \text{under the simultaneous lift flip at fixed } g_{\mu\nu},}$$

with the corresponding spinor transformation included. In practice this means auditing the Dirac term and any parity-odd Holst– or Nieh–Yan-type fermion coupling for residual lift dependence. If the equality failed, preserving a lift would conflict with the renewal result above. The tetrad-sign check is therefore an open consistency audit on the matter completion; none of the pure-geometric results below depends on its outcome.

3. One channel fixes one area increment on the matched branch. The physical area on the coherent maximal-fusion ray is linear in the additive flux,

$$A(J) = 8\pi\gamma_* L_*^2 J.$$

A boundary cut removes one capacity partner and contributes

$$S_N/k_B = N\ln 2.$$

Matching this microscopic cut to the Bekenstein–Hawking normalization on the Einstein branch gives

$$N\ln 2 = \frac{A_N}{4L_*^2} = \frac{8\pi\gamma_* L_*^2(3N)}{4L_*^2},$$

and hence

$$\boxed{\gamma_* = \frac{\ln 2}{6\pi}, \quad A_N = 4\ln 2\, L_*^2 N, \quad a_{\text{ch}} = 4\ln 2\, L_*^2.}$$

This is a normalization match, not an independent derivation of the black-hole area law; Appendix Q.11 gives the finite-\(N\) area-operator statement and that caveat in full.

The dimensional scale is not left free here. Appendix D.4 and Sections H.6–H.9 use the electron recurrence and the marked-transfer factor to obtain

$$\boxed{L_* = -\frac{3}{2}Z_e\lambda_e\ln\left(1 - e^{-7g_{\text{share,eff}}}\right), \quad G_* = \frac{c^3 L_*^2}{\hbar}.}$$

Thus the area relation fixes the geometric normalization in units of \(L_*\), while the independent charged-transfer calculation fixes the numerical length and therefore the numerical Newton scale. The two statements should not be conflated.

4. Metricity is a selected physical subspace. Gauge-closed tetrahedra alone span a twisted-geometry space: adjacent cells can agree on a face area while assigning different intrinsic shapes to that face. The ordinary one-metric branch additionally imposes shape matching,

$$q^{(c)}_{ab}\big|_f = q^{(c')}_{ab}\big|_f.$$

On this support, neighboring tetrahedra glue into a single piecewise-flat metric geometry. Therefore

$$\boxed{\text{Gauss closure + shape matching} \implies \text{Regge geometry}.}$$

Shape matching is not generated by \(Q_f\). It is part of the selected metric completion and must be preserved by the geometric vertex and by its coupling to the capacity decoration.

4a. Equal face areas force disphenoid cells; the face-wise lock forces regular cells. Two facts narrow the selected subspace. First, every primitive face carries \(j = 3\), so on the coherent ray every face has the same area \(a_{\text{ch}} = 4\ln 2\, L_*^2\) (paragraph 3). A tetrahedron whose four faces have equal area is a disphenoid: opposite edges are equal and the four faces are congruent. Primitive cells are therefore disphenoids exactly, with no selection, and the only residual shape freedom is the shape of the congruent face triangle, two parameters per cell. Second, the capacity-closure polynomial of Appendix Q.7, \(|\sum_i b_i v_i|^2 = 4\sum_i b_i^2 - (\sum_i b_i)^2\), is written on the vertex directions \(v_i\) of the regular tetrahedron, and Appendix Q.4 implements Postulate I by the equivariant lock on the two relational frames. If that lock is read face-wise, so that capacity port \(i\) and geometric face \(f_i\) are one face of one substrate and \(\mathbf{n}_{f_i} = R_c v_i\) for a single rotation \(R_c\), then the four normals form a regular tetrahedron and the cell is regular. Every face is then an equilateral triangle of area \(a_{\text{ch}}\), adjacent cells agree on face shape identically, and

$$\boxed{\text{face-wise lock} \implies \text{shape matching holds kinematically}.}$$

Twisted configurations are excluded rather than selected, and the native spatial geometry is the equilateral-triangulation class with fixed edge

$$a^2 = \frac{16\ln 2}{\sqrt{3}}L_*^2 = 6.403 L_*^2, \quad a = 2.530 L_*.$$

With time-like edges set by the renewal cadence and equal by homogeneity, the spacetime building blocks are the two causal-dynamical-triangulation simplex types with one asymmetry \(\tilde{\alpha} = \ell_t^2/a^2\): the native geometric host is the CDT class with Regge weight \((\ln 2/2\pi)\sum_h N_h\Theta_h\) and \(G = G_*\). The first fact is a theorem. The second is a reading of Postulate I that the present text does not derive; without it the cells are disphenoids with a two-parameter shape freedom and metricity returns to the selected status of paragraph 4. Appendix J.10 draws the consequences: the CDT bare couplings become predictions rather than tuned inputs, and the phase of the native point becomes a definite question.

Capacity transport cannot replace this reading. Every capacity observable has the form \(I_{\text{geom}} \otimes O\) (Appendix Q.4), while shape mismatch is a geometric datum, so no condition built from bandwidth transport, face labels, or face holonomies can enforce shape matching.

5. Local composition fixes the Regge form, not its coefficient. In a piecewise-flat metric geometry, curvature is concentrated on triangular hinges \(h\) with area \(A_h\) and deficit or boost angle \(\Theta_h\). Let the leading local hinge amplitude be a continuous phase \(W(A, \Theta) \in U(1)\). Invariance under subdivision of a hinge and composition of successive curvature wedges requires

$$W(A_1 + A_2, \Theta) = W(A_1, \Theta)W(A_2, \Theta),$$ $$W(A, \Theta_1 + \Theta_2) = W(A, \Theta_1)W(A, \Theta_2).$$

The two continuous Cauchy equations imply

$$\boxed{W(A, \Theta) = e^{i\alpha A\Theta}}$$

for a real constant \(\alpha\). Local composition therefore fixes the bilinear Regge form,

$$S_{\text{geom}} = \hbar\alpha\sum_h A_h\Theta_h,$$

but it does not determine \(\alpha\). Matching this branch to the already-fixed Newton normalization sets

$$\boxed{\alpha = \frac{c^3}{8\pi\hbar G_*} = \frac{1}{8\pi L_*^2}.}$$

The resulting action and its channel form are

$$\boxed{S_R = \frac{c^3}{8\pi G_*}\sum_h A_h\Theta_h,}$$ $$\boxed{\frac{S_R}{\hbar} = \frac{\ln 2}{2\pi}\sum_h N_h\Theta_h = \gamma_*\sum_h J_h\Theta_h.}$$

The last equation is the central microscopic normalization identity. Each cut channel supplies \(\ln 2\) of boundary entropy, and bending its associated area through \(\Theta\) supplies the phase \((\ln 2/2\pi)\Theta\) on the matched Einstein branch. Proper-EPRL asymptotics provide a controlled witness of the same Regge phase; they do not turn the composition argument into an independent derivation of its coefficient.

6. Smooth metric limit. For a sequence of Regge geometries approaching a smooth metric,

$$2\sum_h A_h\Theta_h = \int d^4x \sqrt{-g}\, R + O\left(a^2\int d^4x \sqrt{-g}\,\mathcal{R}^2\right),$$

where \(a\) is the coarse lattice spacing and \(\mathcal{R}^2\) denotes the appropriate curvature-squared combination. Hence

$$S_R = \frac{c^3}{16\pi G_*}\int d^4x \sqrt{-g}\, R + S_{\text{higher}}.$$

Allowing the volume term gives

$$\boxed{S_{\text{grav}} = \frac{c^3}{16\pi G_*}\int d^4x \sqrt{-g}(R - 2\Lambda) + S_{\text{higher}}.}$$

The fixed-measure argument of Appendix O determines how \(\Lambda\) enters the ordinary branch; it is logically separate from the local curvature derivation above.

The stronger infrared uniqueness statement is conditional. If the selected phase has one metric, locality at leading derivative order, diffeomorphism covariance, block-scale local Lorentz invariance, and only two massless helicity-2 modes, then the two-derivative nonlinear action is Einstein–Hilbert plus a cosmological term. The cited Han-type and proper-EPRL results witness the Regge/Einstein and two-helicity sectors in their controlled regimes. They are not one theorem for the full decorated action. In particular,

$$\boxed{\begin{array}{c}\text{local Lorentz invariance at block scale is selected and witnessed,}\\\text{not derived from the primitive } SU(2) \text{ cell}.\end{array}}$$

The ordinary infrared logic and the capacity-side compatibility theorem do not require a proof of every spin-foam refinement. A claim that a particular proper or causal EPRL vertex is the faithful block image does, however, retain the open inter-cell fusion and host-convergence tests of H.10.

The uniqueness statement stops at two derivatives. Gauge-invariant, parity-even quasilocal operators suppressed by powers of \(kL_{\text{block}}\) leave the infrared coefficient chain unchanged, and the finite 1680 count does not fix their Wilson coefficients. The corresponding TT kernel has the schematic expansion

$$K_{\text{eff}}(k) = Z_G k^2 P_{\text{TT}} + c_4 L_{\text{block}}^2 k^4 + \cdots,$$

so the controlled Einstein regime is \(kL_{\text{block}} \ll 1\) and the first ordinary-branch ultraviolet deviations are expected at the block scale defined in H.10, not automatically at one primitive length. In four dimensions, Lovelock uniqueness and massless spin-two self-coupling select Einstein–Hilbert only after locality, one-metric covariance, and the two-helicity Lorentzian phase are supplied [129, 130]. Therefore

$$\boxed{S_{\text{higher}} = \sum_i c_i\mathcal{O}_i, \quad c_i \text{ not fixed by the present postulates}.}$$

Pure-gravity curvature-squared terms are field-redefinition or topological at this order, while matter moves the corresponding freedom into contact operators. Predictions at the holonomy-turnover or other Planckian scales are consequently conditional on the stated minimal completion \(c_i = 0\) beyond the lock-generated tower; the infrared Einstein universality class is not.

7. The transfer defect becomes a metric source. Let a stable localized defect have transfer energy

$$E_d = m_d c^2.$$

Over a proper-time interval \(\Delta\tau\), its Lorentzian transfer phase is

$$\mathcal{A}_d(\Delta\tau) = \exp\left(-\frac{iE_d\Delta\tau}{\hbar}\right).$$

Composition along a worldline therefore gives

$$\boxed{S_d[z, g] = -m_d c^2\int d\tau = -m_d c\int ds.}$$

At leading derivative order, this is also the unique local reparametrization-invariant scalar action built from one metric and the worldline tangent. Its metric variation produces

$$\boxed{T^{\mu\nu}_d(x) = \frac{m_d c}{\sqrt{-g}}\int d\tau\, u^\mu u^\nu\delta^{(4)}\left(x - z(\tau)\right).}$$

The ordinary action is consequently

$$\boxed{S_{\text{ord}} = \frac{c^3}{16\pi G_*}\int d^4x \sqrt{-g}(R - 2\Lambda) - \sum_d m_d c^2\int d\tau_d + \cdots,}$$

and variation of the same metric gives

$$\boxed{G_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G_*}{c^4}T_{\mu\nu}.}$$

For propagating charged leptons, the field-theory completion of the same ordinary branch uses the shell-grade multiplet \(\Psi = (\psi_0, \psi_1, \psi_2)^T\) and the positive mass operator of Appendix I.1:

$$\boxed{S_{g\Psi} = \frac{c^3}{16\pi G_*}\int d^4x \sqrt{-g}(R - 2\Lambda) + \int d^4x \sqrt{-g}\,\bar{\Psi}\left(i\hbar c\gamma^a e_a^\mu D_\mu - c^2\hat{M}\right)\Psi.}$$

Here \(D_\mu\) is the spin- and gauge-covariant derivative defined by the same tetrad and charged connection. The kinetic operator is diagonal in shell grade at this order. With the global \((-, +, +, +)\) signature and the displayed \(i\gamma^\mu D_\mu\) convention, the Clifford algebra is \(\{\gamma^\mu, \gamma^\nu\} = -2g^{\mu\nu}I\). In a local inertial frame, a mass eigenstate obeys

$$(\gamma^\mu p_\mu - m_N c)\psi_N = 0, \quad E_N^2 = c^2\mathbf{p}^2 + m_N^2 c^4,$$

where the second equation follows by multiplying by \(\gamma^\nu p_\nu + m_N c\) and using the Clifford relation. Metric variation of this same action gives the Dirac stress tensor in the Einstein equation above. The eigenvalue \(m_N\) therefore enters both the inertial dispersion relation and the stress tensor obtained by metric variation. The localized WKB limit is the worldline action \(S_d\).

This gives a single leading-order dynamics for the displayed lepton spectrum, relativistic propagation, and ordinary gravity on the selected Lorentzian branch. Its covariant Dirac kinetic term is an infrared completion. The native decorated vertex must still generate that term, reproduce its speed \(c\), pass the tetrad-sign audit of paragraph 2c, and preserve the precision mass ratios under its radiative corrections.

Appendix O's fixed covariant event measure supplies a conditional microscopic reading of the proper-time composition. For a defect worldtube of invariant four-volume \(V_4 = A_\perp L\), the premise \(V_4 = v_* N_{\text{evt}}\) gives, at fixed transverse profile,

$$N_{\text{evt}} \propto L = c\tau.$$

Equal transfer phase per event then produces

$$\prod_{n=1}^{N_{\text{evt}}}e^{-i\delta\varphi_d} = \exp\left(-\frac{i}{\hbar}m_d c^2\tau\right).$$

For a Poisson covariant count,

$$\mathbb{E}[N_{\text{evt}}] = \frac{A_\perp L}{v_*}, \quad \text{Var}\, N_{\text{evt}} = \mathbb{E}[N_{\text{evt}}], \quad \frac{\delta\tau}{\tau} = O(N_{\text{evt}}^{-1/2}).$$

This explains how microscopic event cadence can count invariant proper time, conditional on the fixed-measure premise. Covariance and one-metric coupling fix the worldline form; the event measure is its proposed microscopic realization, not an additional proof of Lorentz invariance. Variation with respect to the path gives \(u^\nu\nabla_\nu u^\mu = 0\), while variation of the same functional with respect to the metric gives \(T^{\mu\nu}_d\). One functional therefore fixes both inertial and gravitational mass.

8. Newtonian limit and the weak equivalence principle. For

$$ds^2 = -\left(1 + \frac{2\Phi}{c^2}\right)c^2 dt^2 + \left(1 - \frac{2\Psi}{c^2}\right)d\mathbf{x}^2,$$

a slowly moving defect obeys

$$d\tau = dt\left[1 + \frac{\Phi}{c^2} - \frac{v^2}{2c^2} + O(c^{-4})\right].$$

Therefore

$$S_d = \int dt\left[-m_d c^2 + \frac{1}{2}m_d v^2 - m_d\Phi + \cdots\right].$$

The same transfer coefficient multiplies the inertial and gravitational terms:

$$\boxed{m_{\text{inertial}} = m_{\text{gravitational}} = E_d/c^2.}$$

There is no independent gravitational charge to tune. The linearized 00 equation yields

$$\boxed{\nabla^2\Phi = 4\pi G_*\rho,}$$

and ordinary matter with negligible anisotropic stress gives

$$\boxed{\Phi = \Psi}$$

at leading order. These are the Newtonian force law and baseline no-slip lensing derived earlier in Appendix D, now displayed as consequences of the common metric action.

The three-dimensional commutant \(\text{End}_{SU(2)}(V_0\oplus V_1\oplus V_2) \simeq \mathbb{C}^3\) does not introduce three gravitational charges. That space describes the internal marked record fiber; the source tensor structure is fixed instead by the covariant worldline functional. Its \(V_0 \oplus V_1 \oplus V_2\) channel content can dress the response, but cannot replace the single coefficient \(m_d\) in \(T^{\mu\nu}_d\).

9. Renewal cannot add a second ordinary gravitational pole. Let \(\Pi_*\) be the stationary projector on the renewed capacity operator space and define

$$P = I_{\text{geom}} \otimes \Pi_*, \quad Q = I - P.$$

For an arbitrary bounded within-tick geometry–capacity evolution \(\mathcal{U}^\dagger\), the renewal step is

$$\mathcal{T}^\dagger = P\mathcal{U}^\dagger.$$

On \(P\mathcal{K} \oplus Q\mathcal{K}\) it has the exact block form

$$\boxed{\mathcal{T}^\dagger = \begin{pmatrix}A & B\\0 & 0\end{pmatrix}, \quad (\mathcal{T}^\dagger)^n = \begin{pmatrix}A^n & A^{n-1}B\\0 & 0\end{pmatrix}.}$$

Consequently

$$\boxed{\text{spec}(\mathcal{T}^\dagger) \setminus \{0\} = \text{spec}(A) \setminus \{0\}.}$$

Within-tick capacity fluctuations may dress the retained evolution through \(B\), but a replaceable capacity mode has no autonomous persistence after renewal. This is stronger and cleaner than requiring geometry–capacity mixing to vanish. A conservation-protected collective variable already contained in \(A\) is not excluded; that is why the infrared galactic response is graded separately from ordinary vacuum gravity.

10. Closure status. The derivation can now be read in one line:

$$\boxed{\begin{aligned}\text{selected relational cells} &\Rightarrow SU(2)\text{ flux geometry},\\j_{\text{primitive}} = 3 &\Rightarrow J = 3N,\\N\ln 2 &\Rightarrow A_N = 4\ln 2\, L_*^2 N,\\\text{shape matching} &\Rightarrow \text{piecewise-flat metric},\\\text{local composition} + G_* &\Rightarrow S_R = \frac{c^3}{8\pi G_*}\sum_h A_h\Theta_h,\\\text{smooth Lorentzian two-helicity phase} &\Rightarrow S_{\text{EH}},\\E_d + \text{proper-time composition} &\Rightarrow S_d = -m_d c^2\int d\tau,\\\delta(S_{\text{EH}} + S_d) = 0 &\Rightarrow G_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G_*}{c^4}T_{\mu\nu},\\\text{renewal triangularity} &\Rightarrow \text{no additional replaceable capacity pole}.\end{aligned}}$$

Exact results in this chain include the representation decompositions, the selected equivariant lock, maximal-channel fusion, the locked coherent phase space, the channel–area algebra after continuum normalization, the transfer scale, and renewal triangularity. The selected completion supplies the geometric factor, shape-matched one-metric support, and ultraviolet propagator. The one-metric requirement and local composition fix the leading Regge/Einstein universality class; the channel area and \(G_*\) fix its normalization. Proper-EPRL, Han-type, and area-Regge results independently witness that the selected coarse data realize this class. H.10 then proves the TT infrared limit and gives an exact fixed-\(j = 3\) perfect-action pullback. The unresolved question is which additional geometric principle selects a unique native gluing tensor and measure from the many hosts compatible with the capacity premises.

H.10 Gaussian GFT–geometric-relative-entropy bridge and reduced embedding audits

At any finite regulator, positive dressed and reference Hessians on a common gauge-reduced support determine a unique normalized Gaussian determinant and positive relative-covariance functional. This is an algebraic result. In the factorized completion, the finite capacity sector fixes the equilibrium projector, records, defects, and coefficient chain. The geometric spin-foam/GFT sector supplies the gauge-reduced spin-2 geometry. The selected maximal-fusion ray and the Han-type refinement results of Appendix B and Section 23 provide a controlled semiclassical witness. Proper-EPRL one-sector asymptotics supply a separate compatibility check. The exact operator identification for the relative-information functional, the coupled geometry–capacity audits below, and the all-state nonperturbative Lorentzian continuation remain conditional.

A. Physical Hessians and relative covariance. Let \(\mathcal{H}'_{\text{phys}}\) be a finite-dimensional regulated real fluctuation space after gauge fixing and removal of the same exact zero modes in the equilibrium and dressed backgrounds. For a complex GFT field this means its real Nambu form, with the conjugate components included in the physical trace. On that common support, take

$$S^{(2)}_0[\varphi] = \frac{1}{2}\langle\varphi, \mathbb{K}_0\varphi\rangle, \quad S^{(2)}[\varphi] = \frac{1}{2}\langle\varphi, \mathbb{K}\varphi\rangle, \quad \mathbb{K}_0 > 0, \quad \mathbb{K} > 0.$$

Whitening by the equilibrium Hessian defines

$$\Theta = \mathbb{K}_0^{-1/2}\mathbb{K}\mathbb{K}_0^{-1/2} > 0, \quad \mathcal{G} = \Theta^{-1} = \mathbb{K}_0^{1/2}\mathbb{K}^{-1}\mathbb{K}_0^{1/2} > 0.$$

Equivalently, with \(C_0 = \mathbb{K}_0^{-1}\) and \(C = \mathbb{K}^{-1}\),

$$\mathcal{G} = C_0^{-1/2}CC_0^{-1/2}.$$

Every prime below means this identical gauge and zero-mode prescription. Equality of the two physical supports is a hypothesis to be tested in the microscopic spectrum; it is not supplied by the determinant identity.

B. Finite-regulator determinant theorem. The equilibrium-normalized Gaussian functional is

$$\boxed{\hat{Z}_G = \frac{\int_{\mathcal{H}'_{\text{phys}}}D\varphi\, e^{-\frac{1}{2}\langle\varphi, \mathbb{K}\varphi\rangle}}{\int_{\mathcal{H}'_{\text{phys}}}D\varphi\, e^{-\frac{1}{2}\langle\varphi, \mathbb{K}_0\varphi\rangle}} = (\det'\Theta)^{-1/2} = (\det'\mathcal{G})^{1/2}.}$$

Consequently,

$$\boxed{2\ln\hat{Z}_G = -\text{Tr}'\ln\Theta = \text{Tr}'\ln\mathcal{G}.}$$

The proof is just the finite Gaussian integral

$$Z[\mathbb{K}] = (2\pi)^{N_{\text{phys}}/2}(\det'\mathbb{K})^{-1/2}.$$

Taking the ratio cancels the measure normalization, while multiplicativity of the determinant under the congruence defining \(\Theta\) gives the displayed result. The proof requires neither \([\mathbb{K}_0, \mathbb{K}] = 0\) nor simultaneous diagonalization. Positivity is needed to define the real Gaussian measure and the principal operator logarithm. The real/Nambu convention fixes the exponent \(-1/2\); a complex-coordinate calculation gives the same result only after its conjugate variables and Jacobian are counted consistently in \(N_{\text{phys}}\) and \(\text{Tr}'\).

C. Exact quadratic-source Legendre transform and Gaussian KL divergence. Write the whitened dressed Hessian as

$$\Theta = I - U > 0, \quad U = U^\dagger,$$

and define the normalized connected functional

$$W[U] = \ln\hat{Z}_G[U] = -\frac{1}{2}\text{Tr}'\ln(I - U).$$

Its matrix derivative is

$$2\frac{\delta W}{\delta U} = (I - U)^{-1} = \mathcal{G}.$$

The convex conjugate with the corresponding quadratic-source normalization is

$$\Gamma_{\text{rel}}[\mathcal{G}] = \sup_{\substack{U=U^\dagger\\I-U>0}}\left[\frac{1}{2}\text{Tr}'(\mathcal{G}U) - W[U]\right].$$

Stationarity gives \(U = I - \mathcal{G}^{-1}\). Substitution therefore yields the exact identity

$$\boxed{\Gamma_{\text{rel}}[\mathcal{G}] = \frac{1}{2}\text{Tr}'[\mathcal{G} - I - \ln\mathcal{G}].}$$

If \(g_i > 0\) are the eigenvalues of \(\mathcal{G}\), then \(g_i - 1 - \ln g_i \geq 0\), with equality only at \(g_i = 1\). Hence

$$\Gamma_{\text{rel}} \geq 0, \quad \Gamma_{\text{rel}} = 0 \iff \mathcal{G} = I \iff C = C_0$$

on the common support. In probabilistic language this is precisely

$$\Gamma_{\text{rel}} = D_{\text{KL}}(\mathcal{N}(0, C)\|\mathcal{N}(0, C_0)).$$

Thus the positive mismatch functional in Sections 25 and 29.9 is not an inserted potential. It is the convex conjugate of the normalized Gaussian trace logarithm.

D. Conditional relation to geometric relative entropy. Bianconi's geometric-relative-entropy construction uses a positive metric-response operator \(G_B = \Theta_B^{-1}\) and the same operator-convex mismatch form [28]. If a specified decorated GFT gluing tensor derives the operator identification

$$\Theta_B = \Theta, \quad G_B = \mathcal{G},$$

then the Bianconi trace-log and Legendre functionals are the Gaussian effective functionals of the dressed GFT covariance relative to equilibrium. Under that still-conditional identification, \(G_B\) can be read in this theory as a composite covariance or susceptibility ratio, slaved to the condensate, metric, matter, and relational boundary data. The functional identity proved above is exact; the microscopic gluing and operator map are not. Bianconi's independent derivative dynamics are not imported into the metric-only ordinary branch, and this paper does not attribute the composite reading to her construction itself.

E. Euclidean sign and continuation. The Euclidean one-loop contribution obtained by integrating out the Gaussian fluctuation is

$$\boxed{\Delta\Gamma_{E,\text{eff}} = -\ln\hat{Z}_G = \frac{1}{2}\text{Tr}'\ln\Theta = -\frac{1}{2}\text{Tr}'\ln\mathcal{G}.}$$

This sign is not the sign of the positive Legendre/KL mismatch. The logarithm of a multiplicity or partition ratio and the Euclidean effective free-energy contribution carry opposite signs, while the convex conjugate measures the nonnegative cost of changing the covariance. A common Lorentzian parent must fix the analytic continuation and the sign of its retarded influence kernel. The determinant theorem and positive Legendre identity do not depend on that unresolved continuation.

F. Conditional local derivative expansion. Suppose the regulated operator is quasilocal in a geometric cell basis,

$$\Theta_{vv'} = \delta_{vv'}\Theta_v + \mathcal{R}_{vv'},$$

with \(\mathcal{R}\) derivative-suppressed on fields varying slowly compared with \(L_*\). Only under that assumption may the exact trace be organized as

$$2\ln\hat{Z}_G = \nu_*\int d^4x\sqrt{|g|}\left[-\text{tr}_{\text{phys}}\ln\Theta(x)\right] + O\left(L_*\nabla\Theta, L_*^2\nabla^2\Theta\right).$$

Here \(\text{tr}_{\text{phys}}\) is the trace over the local gauge-reduced geometric fiber and \(\nu_*\) is its continuum block density. Neither is fixed by the 1680-state capacity probability distribution. In the factorized completion they follow from the geometric gluing and blocking calculation, while the equilibrium capacity factor contributes its rank-one stationary transfer projector in probability space, and matter/record insertions couple through the equivariant lock. The finite determinant is exact before locality is assumed; applying it as a local continuum density remains conditional on the quasilocal derivative expansion.

G. Low-curvature and static Einstein corollary. If the covariant operator supplied by the embedding has the small-response form

$$\Theta = I - \tilde{U}, \quad \tilde{U} = \beta\tilde{R} - \alpha\tilde{M} + O(R^2, RM, M^2, \nabla^2),$$

where \(\tilde{R}\) and \(\tilde{M}\) are the dimensionless curvature and matter insertions on the physical fiber, then

$$-\text{Tr}'\ln(I - \tilde{U}) = \text{Tr}'\,\tilde{U} + \frac{1}{2}\text{Tr}'\,\tilde{U}^2 + \cdots.$$

For the controlled geometric branch the EPRL/Regge refinement results of Appendix B supply the Einstein–Hilbert spin-2 support independently of this trace-log expansion. The extra question here is whether the relative-covariance operator itself inherits the same covariant fiber trace, Ward identity, and normalization when capacity defects dress that geometry. The already-controlled longitudinal reduction fixes its static target:

$$I^{\text{static}}_{\text{cap}} = \int dt\, d^3x\left[-\frac{\gamma}{2}(\nabla\delta S)^2 + \kappa\rho\,\delta S\right], \quad \delta S = -\frac{2S_\infty}{c^2}\Phi,$$ $$G = \frac{c^2\kappa}{8\pi\gamma S_\infty}, \quad \boxed{I^{\text{static}}_{\text{cap}} = Z_S I_{\text{Newton}}.}$$

The ordinary longitudinal capacity response is not a propagating scalar pole. Under the capacity–metric dictionary it is the Hamiltonian-constraint combination of the Einstein metric, so its static Green function may be unscreened without adding independent Cauchy data. A microscopic condensate realization therefore passes the ordinary-gravity audit when every scalar fluctuation beyond that constraint response is gapped, relaxational, pure gauge/constraint, or exactly source-orthogonal in the physical retarded kernel. The radiative gauge-reduced support must remain transverse-traceless. This spectrum statement follows from the geometric completion, not from the trace-log identity by itself.

H. Marked-record support-preservation lemma. The pair contraction and the seven-channel recurrence are different quantities. From H.9,

$$u = \frac{8\eta_*}{49} = 0.00487621949\ldots, \quad \alpha = \sqrt{1 - u} = 0.9975589108\ldots,$$ $$\alpha^{21} = 0.9499693634\ldots, \quad r = e^{-7g_{\text{share,eff}}} = 2.77557 \times 10^{-23}.$$

The positivity of the marked edge factor follows from \(0 < u < 1\), not from the exponential smallness of \(r\). On the established seven-channel one-mark branch, integrating out the 21 internal present/history Majorana blocks multiplies the remaining bosonic geometric transfer by the strictly positive scalar \(\alpha^{21}\). Taking the branch-independent internal trace of the nine-state hard-core marked fiber gives

$$\zeta_* = 9e^{-g_{\text{share,eff}}}\alpha^{21} = 0.00512358448\ldots > 0.$$

Therefore the integrated marked-record sector, by itself, neither removes nor creates a bosonic geometric mode. This is a decoration-side support lemma. On the factorized vacuum branch the geometric TT support is fixed by the controlled spin-foam continuum; for non-equilibrium defect backgrounds one must still verify that dressing does not introduce new gauge-reduced zero modes, alter the constraints, or create an additional sourced pole.

I. Exact internal-fiber identity and limited ADM reading. The primitive fusion identity of H.9 is

$$V_{3/2} \otimes V_{3/2} = V_3 \oplus V_0 \oplus V_1 \oplus V_2.$$

Removing the selected maximal channel gives

$$Q = (V_{3/2} \otimes V_{3/2}) \ominus V_3 = V_0 \oplus V_1 \oplus V_2, \quad \dim Q = 1 + 3 + 5 = 9.$$

Independently, the marked response fiber obeys

$$\mathcal{H}_{\text{mark}} = V_1^P \otimes V_1^H = V_0 \oplus V_1 \oplus V_2.$$

The intertwiner already constructed in H.9 therefore proves equivalence of the two internal fibers, with one copy of each irrep. Under spatial rotations, the ADM shift and symmetric spatial metric decompose as

$$h_{0i} \in V_1, \quad h_{ij} \in V_0 \oplus V_2.$$

The nine marked states have the spatial-rotation content of the three shift components and six spatial-metric components. The lapse \(h_{00}\) remains a separate scalar constraint multiplier. This \(SU(2)\) identity does not make the marked fiber the metric. It supplies the blockwise representation target for the equivariant geometry–capacity lock. Inter-cell covariance, the lapse constraint, the diffeomorphism Ward identity, and TT propagation remain in the geometric factor; the marked factor can only dress those responses.

J. Equilibrium capacity normalization and physical support. The stationary distribution has two related but distinct representations. In the detailed-balance/symmetrized probability space,

$$|v\rangle = \sum_{b\in\mathcal{B}}\sqrt{p_{\eta_*}(b)}|b\rangle, \quad P_{\text{stat}} = |v\rangle\langle v|$$

is the rank-one stationary projector. The actual renewed quantum register is instead the diagonal density operator already fixed in H.8,

$$\rho_* = \sum_b p_{\eta_*}(b)|b\rangle\langle b|, \quad \text{Tr}\,\rho_* = 1,$$

and the replacement channel is \(\mathcal{E}_*(X) = \rho_*\text{Tr}\,X\). Here \(P_{\text{stat}}\) is rank one in transfer/probability space, whereas \(\rho_*\) has the full support required by faithful renewal.

After the complete homogeneous capacity free energy is centered as in Section 25, the normalized homogeneous equilibrium reference contributes unit capacity weight. Thus for that reference state,

$$\boxed{\mathcal{A}^{\text{ref}}_{\text{full}} = \mathcal{A}^{\text{ref}}_{\text{geom}}}$$

up to the already-divided gauge/measure normalization. For a generic empty geometry \(T\), however,

$$\mathcal{A}_{\text{full}}[T] = \mathcal{A}_{\text{geom}}[T]\hat{Z}_{\text{cap}}[T],$$

where centering removes the source-independent homogeneous extensive term. Geometry-dependent finite contributions need not equal unity. The capacity Hilbert space is also not one-dimensional. Instead, the renewed local closure register has one stationary superoperator direction: \(\mathcal{E}_*\) annihilates every traceless perturbation of that replaceable register. The renewed register therefore cannot furnish an additional coherent pole. Any gapless capacity pole must lie in a retained or conservation-protected collective sector, such as the transport carrier tested in Section 24. Persistent marked fibers must satisfy the source-overlap conditions above.

Schematically, on the controlled linearized witness branch, the combined continuum results imply a physical Hessian of the form

$$\mathbb{K}_{\text{phys}}(k) \sim Z_G k^2 P_{\text{TT}} + O(a^2 k^4) + O(\Lambda_{\text{sf}}^{-1}k^2),$$

where the separate terms summarize the Einstein spin-2 limit and the known derivative/large-spin suppressions rather than quote one theorem for a single decorated proper-EPRL model. The first explicit length-metric correction in the area-Regge continuum is \(O(a^2 C^2)\) [125]. The independent area-metric RG analysis of Ref. [131] supports the decoupling of nonmetric shape-mismatch modes in the infrared, but it is not used here as a theorem about the full EPRL flow. Its failure to make parity generic is instead handled by the exact microscopic orientation exchange imposed in Appendix Q.

K. The selected geometric hosts and why their specification matters. The capacity construction fixes the \(j = 3\) carrier, its maximal-fusion ray, and the infrared normalization; it does not fix a unique ultraviolet geometric propagator. Two distinct selected completions are therefore kept separate.

Covariant fixed-facet host. Every microscopic geometric face is a spin-three facet with its own relational incidence identity and its own dual loop. A coarse face is a partition into \(N\) such facets. Refinement creates new interfaces; it never replaces one boundary cycle by many indistinguishable copies carrying a common holonomy variable. Schematically,

$$\boxed{f_{\text{coarse}} = \bigsqcup_{a=1}^N f_a, \quad j(f_a) = 3, \quad g_{f_a}\ \text{retained separately}.}$$

This distinction answers the known stack-localization objection. Repeating the same face on one cycle produces

$$\omega_h \sim [\lambda\tau_k(g_h)]^{p_h}, \quad \text{Re}\, S_h = p_h\ln\frac{|\tau_k(g_h)|}{d_k^2} \leq 0,$$

with equality only at flat holonomy. It therefore localizes onto flat \(SU(2)\) connections as \(p_h \to \infty\), even at fixed \(k\) [115]. A coarse-label host does not evade the problem merely by restricting \(J\) to multiples of three:

$$\sum_{J\in3\mathbb{N}}d_J\chi_J(\phi) = \frac{1}{3}\sum_{r=0}^2\sum_J d_J\chi_J(\phi)e^{2\pi irJ/3},$$

which is supported on flat or \(\mathbb{Z}_3\)-twisted-flat class angles. By contrast, a relational facet partition has weight \(\prod_a w(g_{f_a})\) on distinct loops and contains no power of a single \(\tau_k(g)\), so that specific mechanism does not directly apply. This is not a proof that every RG flow avoids a topological phase; the non-topological clause in the convergence condition below remains essential.

Han's many-small-spin construction supplies only the single-vertex half of the desired blocking statement: a generalized vertex with parallel fixed-spin facets has the same large total spin \(J = \sum_a j_a = 3N\) and the corresponding Euclidean Regge critical data [114]. It does not show that many primitive four-simplices coarse-grain into one block vertex. The grades are therefore

$$\begin{array}{ll}F_N|3, \mathbf{n}\rangle^{\otimes N} = |3N, \mathbf{n}\rangle & \text{exact kinematics},\\\text{parallel facets within one vertex} & \text{controlled Euclidean witness},\\\text{inter-cell vertex fusion} & \text{open dynamics}.\end{array}$$

The parallel-normal restriction is exactly the maximal channel \(V_{3N} \subset V_3^{\otimes N}\). In the amplitude realization

$$\mathcal{A}_{\text{fine}} = \mathcal{A}_N F_N, \quad \|Q_N\psi\|_{\text{phys}} = 0,$$

so the nonmaximal geometric complement is null rather than a second set of propagating face modes. The marked \(V_0 \oplus V_1 \oplus V_2\) records remain entirely in \(\mathcal{H}_{\text{cap}}\), consistent with the factorization of Appendix Q. Proper and causal EPRL amplitudes are then block-scale witnesses on \(V_{3N}\), not primitive definitions; their face dimension \(6N + 1\) is the fused image of the primitive \(d_3 = 7\) carrier. Their grade remains conditional on the open inter-cell line above.

Canonical compression freezes the area spectrum. Let \(q(\Psi) = \sum_I\|C_I\Psi\|^2\) be a closable positive master form and let \(\mathcal{H}_*\) be the closure of spin networks whose edge spins lie in \(3\mathbb{N}\). Restricting and closing the form defines a positive self-adjoint compression \(\mathcal{M}_*\) with

$$\langle\Psi, \mathcal{M}_*\Psi\rangle = \sum_I\|C_I\Psi\|^2, \quad \ker\mathcal{M}_* = \ker\mathcal{M} \cap \mathcal{H}_*.$$

This is compression in form sense, not the sum of separately projected constraints:

$$\langle\Psi, P\mathcal{M}P\Psi\rangle - \sum_I\|PC_I P\Psi\|^2 = \sum_I\|(I - P)C_I\Psi\|^2 \geq 0.$$

For the standard fundamental-holonomy regularization, \(C_I^\dagger C_I\) changes any edge spin by at most two units. Projection back to \(3\mathbb{N}\) therefore retains only the zero shift. Hence

$$[\mathcal{M}_*, \hat{A}_e] = 0 \quad \text{for every edge } e,$$

and its symbol on the locked coherent ray is independent of the conjugate twist \(\xi\). Areas are frozen: the compressed standard master constraint is excluded as a geometric host, rather than listed as an admissible candidate.

The lock itself fixes the repair. Replace the fundamental trace by the normalized spin-three trace

$$\text{Tr}_{1/2}\left(h[h^{-1}, V]\right) \longmapsto \frac{c_3}{3\cdot4\cdot7}\text{Tr}_3\left(h[h^{-1}, V]\right),$$

with \(c_3\) fixed by the same classical limit. Locked transitions \(0, \pm3\) then survive, and \(\mathcal{M}^{(3)}_*\) carries \(\Delta J = 0, \pm3, \pm6\) with the nontrivial symbol

$$\mathcal{M}^{(3)}_* \supset \frac{2}{9}(1 - \cos 3\xi) = \xi^2 - \frac{3}{4}\xi^4 + O(\xi^6).$$

Thus

$$\langle\mathcal{M}^{(3)}_*\rangle = M[J, \xi, \mathbf{n}] + O(\xi^4) + O(\bar{N}^{-1}),$$

and the canonical alternative is specifically the spin-three-regularized master constraint, not the compression of the standard one. Its fixed holonomy scale and \(2\pi/3\) periodicity are the structural \(\mathbb{Z}_3\) of the locked phase space. Apparent higher-representation zeros at \(\xi + 2\pi/3\) are the same locked state, not additional kernels. This canonical completion and the covariant fixed-facet completion are both admissible, but their equivalence is not assumed.

L. Exact gapped embedding and perfect-action pullback. A one-scale nearest-neighbor alignment does not justify discarding nonmaximal block channels. On a connected primitive graph \(G\),

$$H_G = \kappa\sum_{(ab)\in E(G)}(9 - J_a\cdot J_b)$$

has the maximal-spin ray as its kernel, but on the one-magnon sector

$$H_G = 3\kappa L_G, \quad \Delta_G = 3\kappa\lambda_2(G) \longrightarrow 0$$

for growing local blocks. Nearest-neighbor alignment therefore leaves a gapless spin-wave complement.

The tree fusion map supplies a uniformly gapped parent instead. At each node \(v\) joining spins \(j_L, j_R\), define

$$Q_v = j_L j_R - J_L\cdot J_R, \quad q_J = \frac{1}{2}\left[(j_L + j_R)(j_L + j_R + 1) - J(J + 1)\right].$$

Nested-block Casimirs commute, so

$$\boxed{H_T = \kappa\sum_{v\in T}Q_v, \quad \ker H_T = V_{3N}, \quad \text{gap}\, H_T \geq 6\kappa.}$$

The kernel is independent of the fusion tree by multiplicity one; only the decoupled excited spectrum remembers the tree.

Let \(F_N : V_3^{\otimes N} \to V_{3N}\) be the normalized coisometry, \(F_N F^\dagger_N = I\), and \(Q_N = I - F^\dagger_N F_N\). For any positive self-adjoint coarse operator \(K^{\text{coarse}}_N\), define

$$K^{\text{prim}}_N = F^\dagger_N K^{\text{coarse}}_N F_N + \Delta_N Q_N, \quad \Delta_N \geq 6\kappa.$$

On the physical ray,

$$F_N K^{\text{prim}}_N F^\dagger_N = K^{\text{coarse}}_N, \quad F_N(K^{\text{prim}}_N - z)^{-1}F^\dagger_N = (K^{\text{coarse}}_N - z)^{-1}.$$

Moreover,

$$\text{spec}\, K^{\text{prim}}_N = \text{spec}\, K^{\text{coarse}}_N \cup \{\Delta_N\}_{Q_N}.$$

The construction also preserves coupled matter–geometry dynamics. At a finite regulator let \(\mathcal{H}_d\) carry the persistent defect variables, set

$$\tilde{F}_N = F_N \otimes I_d, \quad \tilde{Q}_N = I - \tilde{F}^\dagger_N\tilde{F}_N,$$

and let \(K^{\text{coarse}}_{N,gd}\) be a positive self-adjoint operator on \(V_{3N} \otimes \mathcal{H}_d\). It may contain metric-dependent defect transport and backreaction, so no geometry–matter factorization is needed. The joint pullback

$$K^{\text{prim}}_{N,gd} = \tilde{F}^\dagger_N K^{\text{coarse}}_{N,gd}\tilde{F}_N + \Delta_N\tilde{Q}_N$$

satisfies

$$\tilde{F}_N(K^{\text{prim}}_{N,gd} - z)^{-1}\tilde{F}^\dagger_N = (K^{\text{coarse}}_{N,gd} - z)^{-1}.$$

For a deparametrized self-adjoint Hamiltonian the same block identity gives

$$\tilde{F}_N e^{-itH^{\text{prim}}_{N,gd}/\hbar}\tilde{F}^\dagger_N = e^{-itH^{\text{coarse}}_{N,gd}/\hbar}.$$

The interacting spectrum, propagation, backreaction, and their correlations are therefore retained together on the embedded physical sector.

The flat complement \(\Delta_N Q_N\) may be replaced by the hierarchical \(H_T\) above; its excited eigenvalues then spread while its kernel and uniform lower gap are unchanged. Poles, residues, the \(i0\) prescription, and the zero fiber therefore pull back exactly. Associativity of maximal fusion,

$$F_{N+M} = F_{(N,M)} \circ (F_N \otimes F_M) \quad \text{up to phase},$$

makes \(I_N := F^\dagger_N\) a projective family. For \(\mathcal{A}_{\text{fine}} := \mathcal{A}_N F_N\) one has \(\mathcal{A}_{\text{fine}} \circ I_N = \mathcal{A}_N\), which is cylindrical consistency at the amplitude level. This construction proves that a positive primitive operator can carry a chosen healthy coarse Einstein sector. It does not show that the strictly local primitive weights dynamically generate that perfect operator. In four dimensions the corresponding perfect action is generically nonlocal [128]; here the nonlocality is confined to the first reliable semiclassical block,

$$L_{\text{block}} := \sqrt{A_0} = \sqrt{a_{\text{ch}}N_0} = \sqrt{4\ln 2\, N_0}\, L_*,$$

where \(N_0\) is the smallest facet number for which \(J_0 = 3N_0\) supports the required proper/causal Lorentzian asymptotics at the stated numerical tolerance. This definition follows from the physical coarse area \(A_0 = a_{\text{ch}}N_0\); \(\sqrt{3N_0}L_*\) would instead confuse the spin label with an area in units of \(L_*^2\).

The present calculation does not derive \(N_0\). For numerical planning it uses the provisional working bracket \(J_0 \sim 10\text{–}30\), hence \(N_0 \sim 4\text{–}10\) and

$$L_{\text{block}} \sim 3.3\text{–}5.3 L_*.$$

Existing analytic and single-vertex numerical asymptotic studies motivate this test range [124, 134], but do not establish it for the decorated vertex. A direct calculation must replace the bracket with a measured tolerance bound. Provisionally, \(kL_* \ll 0.2\text{–}0.3\) is sufficient for \(kL_{\text{block}} \ll 1\), so ordinary-branch ultraviolet corrections should appear at a few \(L_*\) rather than exactly at \(L_*\). This is an estimated locality range, not a closed prediction.

Selected metric-Regge host theorem. The perfect-action construction above permits a stronger statement than an existence witness. Choose the covariant host to be a regular, shape-matched metric-Regge block sequence on the ordinary branch. Four-dimensional linearized Regge calculus has propagating gauge-invariant lattice curvature degrees of freedom [126], and its weak-field limit recovers the standard continuum theory [127]. On the regular centrally subdivided lattice, the modern area-Regge calculation gives the gauge-reduced length-metric symbol [125]

$$K^{\text{TT}}_n(k) = Z_G k^2 P_{\text{TT}} + O(a_n^2 k^4), \quad Z_G > 0,$$

where the metric/shape-matched branch is the length-Regge subspace and does not retain the additional area-metric modes. Here \(a_n = L_{\text{block}}/L_{\text{obs},n} \to 0\) is an infrared scaling parameter; the physical microscopic cutoff \(L_*\) is not sent to zero.

This symbol also completes the operator statement. For \(z \notin \mathbb{R}\), identify the regular-lattice TT spaces with their Fourier interpolation in the continuum TT Hilbert space and write

$$R_n(k; z) = [K^{\text{TT}}_n(k) - z]^{-1}.$$

For every fixed physical momentum,

$$R_n(k; z) \longrightarrow [Z_G k^2 P_{\text{TT}} - z]^{-1}, \quad \|R_n(z)\| \leq \frac{1}{|\text{Im}\, z|}.$$

Dominated convergence first gives resolvent convergence on TT Schwartz wave packets of compact Fourier support, and the uniform bound extends it by density:

$$\boxed{K^{\text{TT}}_n \xrightarrow{\text{s.r.}} Z_G(-\Delta)P_{\text{TT}}.}$$

Equivalently, positive functional calculus gives for the deparametrized Hamiltonians

$$H_n = c\sqrt{K^{\text{TT}}_n/Z_G} \xrightarrow{\text{s.r.}} c|\mathbf{k}|P_{\text{TT}}.$$

Thus \(\inf\sigma(H_{\text{TT}}) = 0\), genuine TT states occur arbitrarily close to zero energy, and the retarded kernel has two massless helicity-2 poles with positive residue. This is the relevant masslessness criterion; a nonzero pointwise density at the spectral threshold is not required.

The exact primitive pullback now carries the result to fixed microscopic spin:

$$K^{\text{prim}}_n = F^\dagger_n K^{\text{TT}}_n F_n + H_{T,n}, \quad H_{T,n}F^\dagger_n = 0, \quad H_{T,n}|_{Q_n} \geq 6\kappa,$$ $$\boxed{F_n(K^{\text{prim}}_n - z)^{-1}F^\dagger_n = (K^{\text{TT}}_n - z)^{-1}.}$$

The physical maximal-fusion ray therefore inherits the Einstein resolvent exactly, while every nonmaximal fusion mode remains uniformly gapped. Denote this closed branch theorem by

$$\boxed{(H_{\text{TT}})\quad \text{metric-Regge branch} \implies \text{massless two-helicity Einstein TT limit}.}$$

Together with the perfect-action pullback, it also proves that a compatible fixed-\(j = 3\) primitive realization exists.

This is not yet a phase theorem for the native theory. The Hessian result is restricted to the selected shape-matched branch, and the perfect operator is constructed from the healthy coarse operator. Neither controls which configurations dominate a nonperturbative sum. Once a definite local geometric tensor and measure have been supplied, the remaining covariant host condition would be

$$\boxed{(H_{\text{phase}})\quad \mu_{\text{native}} \xrightarrow{\text{IR}} \mu_{\text{metric-Regge/Einstein}},}$$

meaning that the resulting measure must concentrate at long distance on shape-matched, non-topological metric configurations with a gapless positive-residue TT sector, rather than on a BF-like, degenerate, bifurcated, or otherwise non-Einstein phase. The underdetermination theorem below shows that the present premises do not yet define \(\mu_{\text{native}}\), so this condition is not presently a well-posed derived calculation. An unrestricted Lorentzian EPRL sum and uniqueness of the selected ultraviolet propagator are not claimed.

Capacity-side closure on the displayed factorized branch. The capacity audits listed above are consequences of the finite decoration and the host identities above, not additional spectral assumptions. After the internal marked fiber is traced,

$$\boxed{\mathcal{G}_{\text{dec}} = \zeta_*\mathcal{G}_{\text{geom+particle}}, \quad \zeta_* = 9e^{-g_{\text{share,eff}}}\alpha^{21} > 0.}$$

Therefore the decoration changes no denominator: its poles are those of geometry plus the worldline particle, and the signs of all residues are preserved. It introduces neither a ghost nor a tachyon. Renewal triangularity separately removes autonomous replaceable-register poles, while the tensor lift below preserves the positive physical inner product and causal boundary value.

There are two equivalent bookkeeping pictures. In the dilated picture the complete update is unitary, with

$$V^\dagger V = P + \frac{1}{9}\sum_{a=1}^9 W^\dagger_a W_a Q = I,$$

so Hermiticity and positivity are explicit. In the traced Euclidean picture, the corresponding requirement is an Osterwalder–Schrader audit of the complete transfer kernel, including its marked subblocks. Vacuum renewal is counted per invariant event and the defect cadence is proper time. The capacity sector contains no vector \(n^\mu\) from which to construct a leading preferred-frame curvature operator. Refresh-first ordering also gives exact trigger independence:

$$(\text{id}_R \otimes\mathcal{E}_*)(\rho_{R,\text{ren}}) = \rho_R \otimes \rho_*, \quad \langle K^2\rangle_{\text{cond}} = \langle K^2\rangle_{\eta_*}.$$

Thus the factorized ordinary branch closes the capacity pole, residue, positivity, causality, preferred-frame, and refresh-order audits. A different nonfactorized ultraviolet realization must reproduce these identities.

Local Einstein-phase stability under finite capacity dressing. The exact factorized theorem above is stronger than what is needed for a general gauge-compatible decoration. Suppose an undecorated host already has the gauge-reduced infrared kernel

$$K_g(k) = Z_G k^2 P_{\text{TT}} + O(k^4 L_*^2), \quad Z_G > 0,$$

with no additional massless gravitational pole. If the capacity sector has no conservation-protected vacuum pole, its static geometric correction is local or quasilocal, and its TT correction is bounded by the host stiffness, then the Ward identity forbids a TT mass term:

$$\delta K_{\text{TT}}(0) = 0, \quad \delta K_{\text{TT}}(k) = \delta Z_G k^2 P_{\text{TT}} + O(k^4 L_*^2).$$

Consequently, for \(|\delta Z_G| < Z_G\),

$$\boxed{K_{\text{full}}(k) = Z_G^{\text{eff}}k^2 P_{\text{TT}} + O(k^4 L_*^2), \quad Z_G^{\text{eff}} = Z_G + \delta Z_G > 0,}$$

and no new gravitational zero mode appears. Thus a finite, pole-free, gauge-compatible, perturbatively bounded decoration is locally stable around an already established Einstein phase. This is not a claim that a positive \(Z_{\text{cap}}[T]\) leaves every saddle unchanged, that it cannot renormalize a curvature coefficient, or that it is phase-neutral globally. The microscopic normalization \(G_*\) fixes the target renormalized coefficient; a coupled realization must still pass the normalization audit below. On the displayed factorized vacuum branch the stronger identity \(\mathcal{G}_{\text{dec}} = \zeta_*\mathcal{G}_{\text{geom+particle}}\) establishes these hypotheses exactly. For a nonfactorized embedding they are explicit acceptance conditions.

Geometric-host underdetermination theorem. The preceding stability result removes the capacity factor from the host question; it does not make the host follow from the capacity premises. Write the complete factorized vertex as

$$\boxed{\mathcal{A}_v[\mathcal{G}] = \mathcal{G}_v\mathcal{D}_v, \quad \mathcal{D}_v = W_*\Pi_Q e^{-g_{\text{share,eff}}n_d - \Lambda_d n_d(n_d - 1)}\prod_{m<m'}\tilde{U}_{mm'}.}$$

The finite factor \(\mathcal{D}_v\) fixes the \(V_3\) carrier, stationary measure, renewal projector, marked transfer, and dimensional coefficient chain. The geometric gluing tensor \(\mathcal{G}_v\) supplies propagation and curvature. If \(\mathcal{G}_v\) is replaced by another gauge-covariant tensor \(\mathcal{G}'_v\) with the same \(V_3\) boundary representation and equivariant-lock interface, every closed capacity invariant is unchanged. This follows because the gauge action factorizes and all capacity operators commute with its geometric factor. In addition, the exact vacuum lift gives

$$\boxed{\mathcal{H}^{\text{full,vac}}_{\text{phys}} \cong \mathcal{H}^g_{\text{phys}},}$$

so on the displayed branch the full vacuum phase is the phase of the chosen geometric tensor.

The manuscript already contains a concrete separation. The sharp primitive object

$$\mathcal{G}^{(0)}_{v,\sharp} = P^{\text{inv}}_v\left[\bigotimes_{(ab)\in E(v)}P^{(ab)}_3\right]P^{\text{inv}}_v$$

enforces the representation and intertwiner support but contains no curvature phase. The selected Einstein completion instead carries

$$\mathcal{G}^E_v \sim \exp\left[\frac{i}{\hbar}\frac{c^3}{8\pi G_*}\sum_h A_h\Theta_h\right].$$

Both admit the same finite capacity decoration. Requiring the geometric factor to admit an EPRL/Regge branch is therefore a host acceptance condition, not a consequence of the capacity construction.

Even the existence of the Regge saddle does not determine phase dominance. At finite regulator, a local positive gauge-invariant factor

$$\exp\left[\lambda\sum_h f(L_*^2\mathcal{R}_h)\right], \quad f(x) = 0 \text{ near } x = 0,$$

can leave the smooth saddle, its perturbative Hessian, \(G_*\), and the two-helicity branch unchanged while exponentially favoring rough configurations at large volume. Hence

$$\boxed{\text{existence of an Einstein saddle} \not\Longrightarrow \text{Einstein-phase dominance}.}$$

It follows that the proposition

$$\boxed{\text{the present finite-capacity premises uniquely determine } \mathcal{G}_v}$$

is excluded. The perfect-action pullback proves that an Einstein-compatible fixed-\(j = 3\) completion exists; it does not remove this selection freedom. A hosted minimal completion may select that operator and inherit \((H_{\text{TT}})\) exactly. A genuinely derived completion requires a new geometric principle that fixes a definite local gluing tensor and measure, followed by a proof that its nonperturbative measure lies in the shape-matched Einstein phase. Until the first step is supplied, \((H_{\text{phase}})\) is not a well-posed consequence of the current premises.

What a nonfactorized geometry–capacity embedding must preserve. The external geometric witness leaves the following interface checks for any alternative fully coupled vertex. They are already satisfied by construction on the displayed factorized branch, but a proposed nonfactorized realization is acceptable only if all conditions hold on the same gauge-reduced background:

1. Spectrum/source overlap: the capacity decoration introduces no additional source-coupled zero pole; beyond the Einstein scalar constraint, scalar fluctuations are gapped, relaxational, pure gauge/constraint, or exactly source-orthogonal. In condensate realizations with a nonzero relational-time phase gradient, density–phase mixing must be checked in the full Hessian rather than assumed to vanish from a zero-gradient truncation.
2. Constraint and Ward structure: coupling the capacity/record factor leaves the geometric constraint algebra and diffeomorphism Ward identity intact, including in defect backgrounds.
3. Normalization: the static source normalization continues to reproduce \(G_* = c^3 L_*^2/\hbar\) without an additional geometric-capacity coefficient.
4. Transfer/physical-inner-product structure: for a Euclidean/GFT transfer realization, the complete decorated transfer—not merely the finite marked subblocks—must satisfy the required positivity/reflection property; self-adjointness alone is not sufficient. For a directly Lorentzian covariant realization, the corresponding requirement is positivity of the rigging form together with consistent causal gluing, rather than Osterwalder–Schrader positivity.
5. Lorentzian continuation: the retarded poles have the correct causal prescription and the capacity dressing does not create ghosts, tachyons, or preferred-frame radiative modes.

These are interface-preservation tests, not a demand that the finite capacity factor rederive the Einstein graviton sector. Failure would reject the alternative realization, not reopen the exact factorized capacity theorem.

M. Exact rigging-map factorization and the Lorentzian endpoint. The controlled spin-foam result is stronger than a purely kinematic embedding and weaker than an exact all-state covariant projector. Refs. [121, 122] give a coupled large-spin/refinement regime in which Regge solutions approach the Einstein sector; in the controlled linearized construction the low-energy excitations give all smooth linearized Einstein solutions and hence exactly the two graviton helicities. This establishes the identity and existence of the ordinary semiclassical gravitational branch.

The unmodified EPRL edge map is not idempotent, so the formal sum over arbitrary foams does not define a projector as written; this obstruction is independent of the detailed spin-foam model [113, 116]. The Riemannian "Warsaw" modification repairs the edge projector, but no corresponding general Lorentzian theorem is available. Ref. [132] derives a rigging-map structure only after assuming the appropriate distributional continuum limit. The raw Lorentzian EPRL state sum is therefore not treated as an exact physical projector here.

Exact capacity-decorated lift theorem. The capacity contribution to the rigging problem can be evaluated exactly when the mixed renewal state is represented on its operator space. Let

$$\omega_*(A) = \text{Tr}(\rho_* A), \quad \rho_* = \sum_b p_{\eta_*}(b)|b\rangle\langle b|$$

be the faithful stationary state of the renewed quantum register and let \(\mathcal{K}_{\text{cap}}\) be the GNS/Hilbert–Schmidt completion of the capacity observable algebra with inner product

$$\langle A, B\rangle_{\omega_*} = \text{Tr}(\rho_* A^\dagger B).$$

The Heisenberg renewal map is

$$\Pi_*(A) \equiv \mathcal{E}^\dagger_*(A) = \omega_*(A)I.$$

It obeys

$$\boxed{\Pi_*^2 = \Pi_*, \quad \Pi^\dagger_* = \Pi_*, \quad \text{Ran}\,\Pi_* = \mathbb{C}I}$$

with the adjoint taken in the \(\omega_*\) inner product: indeed \(\langle A, \Pi_* B\rangle_{\omega_*} = \omega_*(A)\omega_*(B) = \langle\Pi_* A, B\rangle_{\omega_*}\). The rank-one projector \(|\sqrt{p}\rangle\langle\sqrt{p}|\) used in the detailed-balance probability representation is the commutative-coordinate image of this same one-dimensional stationary transfer direction; neither object says that the quantum state \(\rho_*\) is pure.

Let

$$\eta_g : D_g \longrightarrow D_g^*$$

be any positive geometric rigging map. On the empty, unmarked vacuum sector define

$$\boxed{\eta_{\text{full}}(\phi \otimes A)[\psi \otimes B] = \eta_g(\phi)[\psi]\langle A, \Pi_* B\rangle_{\omega_*}.}$$

Sesquilinearity, the constraint-annihilation property, reality, and positivity are inherited from \(\eta_g\) and the orthogonal projector \(\Pi_*\). Quotienting its null space and completing gives

$$\boxed{\mathcal{H}^{\text{full,vac}}_{\text{phys}} \cong \mathcal{H}^g_{\text{phys}} \otimes \text{Ran}\,\Pi_* \cong \mathcal{H}^g_{\text{phys}}.}$$

This theorem is exact and regulator independent for the equilibrium capacity decoration. It says that renewal adds no vacuum constraint solution or physical pole once a geometric rigging map exists.

Renewal is also compatible with an arbitrary coupled dynamics of the persistent sector. Let \(S\) denote the joint geometry–defect system and \(R\) the replaceable closure register. In the Schrödinger picture, \(\mathcal{E}_*(X) = \rho_*\text{Tr}\,X\), so every joint state satisfies

$$(\text{id}_S \otimes\mathcal{E}_*)(\rho_{SR}) = \text{Tr}_R(\rho_{SR}) \otimes \rho_*.$$

For any joint geometry–defect unitary \(U_S\), a renewal step followed by the physical evolution gives

$$(U_S \otimes I_R)(\text{id}_S \otimes\mathcal{E}_*)(\rho_{SR})(U^\dagger_S \otimes I_R) = U_S\text{Tr}_R(\rho_{SR})U^\dagger_S \otimes \rho_*.$$

The replacement changes no reduced state or internal correlation of the persistent geometry–defect system. A microscopic dilation must additionally satisfy the no-which-path condition: its discarded output may not acquire a cell address correlated with a defect trajectory. Thus renewal and coupled propagation coexist exactly once the persistent defect dynamics is supplied; the renewal identity does not derive that dynamics.

Exact finite-regulator group averaging. There is also an exact regulated construction. Parametrized GFT makes the reparametrization constraint explicit; for a self-adjoint regulated geometric Hamiltonian \(H_\Lambda\), introduce an auxiliary clock pair \((\chi, p_\chi)\) and

$$C_\Lambda = p_\chi + H_\Lambda.$$

Then refined algebraic quantization gives

$$\boxed{\eta_\Lambda = 2\pi\delta(C_\Lambda) = \int_\mathbb{R}d\tau\, e^{i\tau C_\Lambda}}$$

as a distributional group average, with the physical inner product obtained by inserting \(\delta(C_\Lambda)\) [117]. At any finite spin/mode/particle-number regulator a Hermitian truncation is self-adjoint, so this construction is exact even with interactions retained inside that finite matrix. Ref. [117] proves the clock-neutral/deparametrized equivalence explicitly for the free GFT; the interacting cutoff removal is not supplied by that paper and is not imported here as a theorem.

The same exact mechanism is available canonically: the Lorentzian LQG master-constraint programme constructs a positive symmetric master constraint with a Friedrichs self-adjoint extension, and its direct-integral zero spectral fiber defines the physical Hilbert space [118, 119]. This is an exact nonperturbative canonical witness for how the geometric rigging map can exist. It is not a proof that the raw Lorentzian EPRL foam sum equals that canonical projector.

Clock-test-space convergence theorem. The earlier three-hypothesis spectral statement is unnecessarily strong. Let \(H_\Lambda\) be self-adjoint on the regulated GFT Hilbert space and define the deparametrized constraint

$$C_\Lambda = p_\chi + H_\Lambda \quad \text{on } L^2(\mathbb{R}_\chi) \otimes \mathcal{H}_{\text{GFT}}.$$

For product test vectors \(\Phi = f \otimes \phi\) and \(\Psi = g \otimes \psi\) with \(f, g \in \mathcal{S}(\mathbb{R})\), group averaging is exactly

$$\boxed{\eta_\Lambda(f \otimes \phi)[g \otimes \psi] = 2\pi\langle\phi, h_{fg}(H_\Lambda)\psi\rangle, \quad h_{fg}(\lambda) = \overline{\hat{f}(-\lambda)}\hat{g}(-\lambda).}$$

Indeed, \(p_\chi\) and \(H_\Lambda\) act on separate tensor factors, and the \(\tau\) integral gives \(2\pi\delta(p + \lambda)\) in clock-momentum space. Since \(f\) and \(g\) are Schwartz, \(h_{fg} \in C_0(\mathbb{R})\). Therefore

$$\boxed{H_\Lambda \xrightarrow{\text{s.r.}} H \implies \eta_\Lambda(\Phi)[\Psi] \longrightarrow \eta(\Phi)[\Psi].}$$

The limiting form is positive,

$$\eta(f \otimes \phi)[f \otimes \phi] = 2\pi\int_\mathbb{R}|\hat{f}(-\lambda)|^2\, d\mu_{\phi\phi}(\lambda) \geq 0,$$

and it annihilates \(C\) on the test space.

The same Schwartz smearing supplies the causal boundary value. The spectral density of \(C_\Lambda\) on these vectors is the convolution

$$\rho_\Lambda(E) = \int\overline{\hat{f}(E - \lambda)}\hat{g}(E - \lambda)\, d\mu_\Lambda(\lambda).$$

It is uniformly bounded, Lipschitz in \(E\), and integrable because the spectral measure is finite and the clock kernel is Schwartz. Strong-resolvent convergence at fixed \(\varepsilon > 0\) followed by the Plemelj limit therefore gives

$$G_F = (C - i0)^{-1} = \text{PV}\, C^{-1} + i\pi\delta(C), \quad \text{Im}\, G_F(\Phi, \Phi) = \pi\rho_{\Phi\Phi}(0) = \frac{1}{2}\eta(\Phi)[\Phi] \geq 0.$$

Thus density domination, the pointwise-\(\rho(0)\) condition, and limiting absorption hold on the Schwartz clock test space and need not be assumed. Only strong-resolvent convergence remains for this particular all-state rigging-map route. It is a sufficient mathematical construction, not a requirement that the physical cutoff be removed.

The theorem applies to deparametrized evolution. The equation \(p_\chi + H = 0\) does not project onto \(\ker H\), and \(\chi\) must be a physical relational matter clock. If \(\chi\) is the usual massless-scalar GFT clock, the low-energy spectrum contains two TT gravitational helicities and the clock scalar. It contains no additional capacity-gravity scalar. A condensate Bogoliubov mode identified with the clock must satisfy \(c_s^2 \to 1\) as \(k \to 0\). A master constraint \(\mathbf{M} \geq 0\) defines a separate zero-fiber problem and cannot replace \(H\) in the clock equation. The masslessness test for the selected branch is

$$\boxed{\begin{array}{c}(H_{\text{TT}})\quad H_n \xrightarrow{\text{s.r.}} H_{\text{Ein}}, \quad \inf\sigma(H_{\text{Ein}}|_{\text{TT}}) = 0,\\H_{\text{Ein}}|_{\text{TT}} = c|\mathbf{k}| + O(k^3 L_{\text{block}}^2), \quad \text{with two positive-residue TT helicities}.\end{array}}$$

The selected metric-Regge theorem above establishes \((H_{\text{TT}})\), and the perfect-action identity pulls it back to an exact fixed-\(j = 3\) realization. It does not establish the native-measure condition \((H_{\text{phase}})\). The distinct canonical completion retains its own open test,

$$\boxed{(H'_{\text{can}})\quad \mathcal{M}^{(3)}_\Lambda \xrightarrow{\text{s.r.}} \mathcal{M}^{(3)} \text{ with a non-topological zero fiber carrying TT Einstein support}.}$$

Equivalence between the covariant and canonical completions is not assumed. Fixed-complex \(q\)-deformed Lorentzian EPRL models supply finite nonzero-\(\Lambda\) regulators [120]; they do not establish the native phase condition or regulate the empty \(\Lambda = 0\) branch universally.

The status is therefore

$$\boxed{\begin{array}{l}\text{Gaussian determinant bridge: exact,}\\\text{quadratic-source Legendre/KL mismatch: exact,}\\\text{marked support preservation: exact on the established branch,}\\\text{maximal-fusion blocking: exact,}\\\text{renewal triangularity: exact,}\\\text{selected-branch Einstein theorem } (H_{\text{TT}})\text{: closed,}\\\text{Einstein-compatible fixed-}j = 3 \text{ completion: exists,}\\\text{capacity premises uniquely fix } \mathcal{G}_v\text{: excluded,}\\\text{derived native phase } (H_{\text{phase}})\text{: undefined until } \mathcal{G}_v \text{ is fixed,}\\\text{capacity-decorated rigging-map lift: exact,}\\\text{finite-regulator group averaging: exact for self-adjoint truncations,}\\\text{clock-test positivity and causal boundary value: exact,}\\\text{canonical } (H'_{\text{can}})\text{: open and not required by the selected covariant host.}\end{array}}$$

The vacuum rigging-map problem contains no unresolved capacity step. The selected metric-Regge branch satisfies the Einstein TT infrared theorem \((H_{\text{TT}})\), and it has a compatible fixed-\(j = 3\) primitive realization. The host itself remains underdetermined: the current premises leave \(\mathcal{G}_v\) and its geometric measure as selected data. A new geometric principle must fix both before \((H_{\text{phase}})\) becomes a well-posed derived theorem about the nonperturbative measure. A canonical–covariant correspondence would be a further result. The raw EPRL sum does not supply that projector.

Candidate selection principle. The face-wise interpretation of the relational lock (Appendix H.9a, paragraph 4a) supplies the one candidate already present in the postulates for the additional geometric principle named above. It does not choose \(\mathcal{G}_v\) among gauge-covariant tensors; it restricts the configuration space on which any such tensor acts to equilateral causal triangulations with fixed edge \(a = 2.530 L_*\), on which the Regge weight is \((\ln 2/2\pi)\sum_h N_h\Theta_h\) with no free coefficient. Under this interpretation \(\mu_{\text{native}}\) is the CDT measure at couplings fixed by the theory, and \((H_{\text{phase}})\) becomes the well-posed numerical question of Appendix J.10: whether that ensemble, with the capacity weighting switched on, is extended with a gapless TT sector at its predicted point. This interpretation remains a premise.

H.11 Coherence under renewal and reconstruction of the retained marked Hilbert sector

The question. The renewed register contains no information about its input, whereas quantum interference is carried by off-diagonal operators. Coherence can survive only if the interfering degrees of freedom remain in the retained system and the discarded complement carries no path record. The calculation below first states this division exactly. It then asks whether the retained nine-state mark must be assigned a complex Hilbert space or whether that structure follows from the displayed fusion and Lorentzian action. It adds no fourth foundational postulate.

Exact renewed-output theorem. For

$$\mathcal{E}_*(X) = \rho_*\text{Tr}\,X,$$

the Heisenberg dual is fixed by

$$\text{Tr}[\mathcal{E}_*(X)O] = \text{Tr}\left[X\mathcal{E}^\dagger_*(O)\right].$$

Substitution gives

$$\boxed{\mathcal{E}^\dagger_*(O) = \text{Tr}(\rho_* O)I.}$$

Every later observable on the renewed output therefore pulls back to a multiple of the identity, and every traceless input operator is annihilated. Equivalently, for any reference \(R\),

$$(\text{id}_R \otimes\mathcal{E}_*)(\rho_{RA}) = \rho_R \otimes \rho_*.$$

The replacement channel is entanglement-breaking on the renewed register. A global dilation can still preserve the old state in its complementary output, as the history register in H.8 does. The theorem excludes coherence from the replaced output only; it does not exclude global coherence.

Complementary-record coherence theorem. Let \(X\) denote the defect position and \(D\) its retained internal marked fiber. For two freely transported branches, write the relevant part of a one-step isometry as

$$V\left(|x\rangle_X|q\rangle_D|0\rangle_E\right) = |F(x)\rangle_X U_D|q\rangle_D|e_x\rangle_E,$$

and similarly for \(y\). The environment \(E\) denotes only what is discarded when the defect state is read. Directly tracing \(E\) gives

$$|x\rangle\langle y| \longmapsto \underbrace{\langle e_y|e_x\rangle}_{\gamma_{xy}}|F(x)\rangle\langle F(y)| \otimes U_D|q\rangle\langle q|U^\dagger_D.$$

Thus

$$\boxed{\mathcal{C}_{xy}(1) = \gamma_{xy}\mathcal{C}_{xy}(0), \quad \gamma_{xy} = \langle e_y|e_x\rangle.}$$

For independent record factors over \(n\) updates,

$$\mathcal{C}_{xy}(n) = \left(\prod_{k=1}^n\gamma^{(k)}_{xy}\right)\mathcal{C}_{xy}(0).$$

The equation separates the two possible microscopic readings. If the vertex leaves a cell-addressed record, \(|e_x\rangle = |x\rangle_E\) and \(|e_y\rangle = |y\rangle_E\), then \(\gamma_{xy} = 0\) for \(x \neq y\): position coherence dies in one update. If the persistent internal fiber remains inside \(X \otimes D\) and the discarded free-renewal output is branch-independent, \(|e_x\rangle = |e_y\rangle\), then \(\gamma_{xy} = 1\) and renewal produces no intrinsic position decoherence. Intermediate overlaps produce ordinary partial decoherence.

Calling a record "worldline-attached" is not by itself sufficient. A spatially localized record that moves along two different worldlines is still a which-path record if it is discarded. In this paper the phrase means the stronger structure: the marked response fiber is retained as part of the coherently transported defect, while the discarded renewal register contains no branch address. Genuine interactions with detectors or an environment may, and ordinarily do, make the discarded states distinguishable.

What \(Q \perp V_3\) does and does not prove. Primitive fusion gives the orthogonal decomposition

$$V_{3/2} \otimes V_{3/2} = V_3 \oplus Q, \quad Q = V_0 \oplus V_1 \oplus V_2,$$

and hence

$$P_Q P_3 = 0, \quad \text{Tr}\, P_Q = 9, \quad \text{Tr}\, P_3 = 7.$$

If a discarded record actually resolves these two sectors, the same overlap theorem sets every \(Q\)–\(V_3\) off-diagonal to zero. This gives exact marked/unmarked sector dephasing under a sector-resolving export. It is compatible with charge-sector superselection, but it is not yet a derivation of the full electric-charge superselection rule. That stronger statement requires the mark to be identified with the generator of a local \(U(1)\) gauge symmetry and requires the physical observable algebra to preserve its Gauss-law sectors. Orthogonality of two representation subspaces alone does not force the environment to measure them.

Forced vertex constraint, not a new premise. The complete microscopic Hilbert space at a marked vertex must therefore be read as

$$\mathcal{H}_v = \mathcal{H}_{\text{ren}} \otimes \mathcal{H}_X \otimes\left(\mathbb{C}|0\rangle \oplus Q\right),$$

where \(\mathcal{H}_{\text{ren}}\) is replaced by \(\rho_*\), while \(\mathcal{H}_X \otimes Q\) is transported coherently. In the free sector the dilation of \(W_*\) must satisfy

$$\boxed{\langle e_y|e_x\rangle = 1 \quad \text{for unmeasured position branches } x, y}$$

after the canonical translation identifying their background renewal states. The exact equality defines the minimal ideal free vertex; electron-interference experiments impose a finite lower bound on the product of overlaps over their actual coherence time [31]. A cell-fixed alternative is not another interpretation of the same action. It is an experimentally excluded countervertex.

Sections 3.3 and 22 already require the operational branch to reproduce ordinary quantum interference. The overlap theorem converts that existing requirement into an explicit microscopic test of \(W_*\), so the condition adds no ontological premise. Electron interference would falsify an action that violates it. The finite calculation verifies the dual-channel identities, the \(Q\)–\(V_3\) projector orthogonality, and the zero, unit, and intermediate record-overlap cases to machine precision.

Fast/slow architecture. The result separates the fast and slow roles within renewal. The fast closure register supplies the objective replacement process and its exported history correlations. The persistent marked fiber carries internal phase and position coherence. The fast output cannot feed old off-diagonal data back into present-cell observables because \(\mathcal{E}^\dagger_*(O)\) is scalar; the slow coherent sector can nevertheless evolve unitarily because it is not the argument of \(\mathcal{E}_*\). Renewal and interference are therefore compatible provided the complete vertex respects the tensor split above.

Spatial propagation remains a coupled-vertex condition. The overlap theorem fixes the coherence requirement on the discarded register, while the charged transfer spectrum fixes the rest cadence \(\tau_*\). Neither result determines the small-momentum coefficient of the retained defect's orbital propagator. A complete matter vertex must therefore transport \(\mathcal{H}_X \otimes Q\) gauge-covariantly, preserve the no-which-path condition, and produce a pole with rest gap \(\epsilon_* = m_e c^2\tau_*/\hbar\) and invariant speed \(c\). The definition \(L_* = c\tau_*\) supplies the associated causal length; it does not identify a graph bond, face area, or hopping distance with \(L_*\).

Complementary outputs of primitive fusion. The two sides of the primitive split have different infrared roles,

$$V_{3/2} \otimes V_{3/2} = \underbrace{V_3}_{\text{matched geometric channel}} \oplus \underbrace{Q}_{\text{retained mismatch record}}.$$

Appendix H.10 proves \((H_{\text{TT}})\) on the maximal \(V_3\) ray: two positive-residue, massless helicity-2 modes with \(E = c|\mathbf{k}|\), together with an exact fixed-\(j = 3\) pullback whose nonmaximal gravitational complement is gapped. The reconstruction below concerns the orthogonal \(Q\) output. It does not replace the graviton theorem; it explains why the retained record acquires quantum state geometry. On the factorized vacuum branch the finite capacity/record decoration adds no graviton pole.

Three nine-state objects and their relation. Three objects that share the same local nine coordinates must not be identified. First, the H.9 field \(d_{ab}\) is a transient response motif: the incidence vertex creates and removes it, and its occupation number is not a conserved charge. Second, the persistent marked defect carries a coherent internal fiber from one renewal event to the next. Third, the marked shells used in the charged-lepton construction are persistent excitations of that defect. The incidence theorem below identifies the response coordinates at one event with the fusion complement \(Q\); it does not make the transient occupation number the persistent object. The Hilbert space belongs to the coherently transported internal fiber. At each renewal event the diagonal persistent label is re-expressed through the response endpoints, while the free vertex passes their relative phases to the next event without exporting a cell address. The lepton shells are structured states carried by that same persistent defect. This is the ontology map used below:

$$\boxed{\begin{array}{c}\text{event response coordinates} \longleftrightarrow Q \longleftrightarrow \text{persistent coherent fiber,}\\n_d \text{ is not the persistent charge}\end{array}}$$

Real carrier space and the incidence theorem. The local reconstruction begins without complex scalar multiplication as a quantum-kinematic premise. Because \(Q = V_0 \oplus V_1 \oplus V_2\) contains only integer-spin irreducible representations, it has the invariant real form

$$Q_\mathbb{R} = V_{0,\mathbb{R}} \oplus V_{1,\mathbb{R}} \oplus V_{2,\mathbb{R}} \simeq \mathbb{R}^9.$$

The complex notation used in the parent GFT action is a convenient complexification; H.11 restricts its quadratic part to \(Q_\mathbb{R}\) before constructing any complex structure.

The sharp reversible split of B.4 has a transmitted map \(F_3 P_3\) and a retained map \(C\). Unitarity gives

$$P_3 + C^T C = I_{16}, \quad \boxed{C^T C = P_Q.}$$

In the selected H.9 incidence vertex the retained carrier is the complete nine-state one-mark space, so \(\dim Q_\mathbb{R} = \dim\mathcal{H}_{d,\mathbb{R}} = 9\) and \(C\) is an isometric identification. Rotational covariance fixes it blockwise up to the multiplicity-one freedom. Denote the resulting real isometry by \(L\):

$$L^T L = P_Q, \quad LL^T = I_9, \quad \boxed{d = L\chi, \quad \chi = P_Q\Phi_\mathbb{R} = L^T d.}$$

This proves the carrier-space incidence within the selected sharp reversible vertex. It does not derive the hard-core implementation, flags, or detailed H.9 routing from fusion; those remain selected parts of the marked vertex.

Quadratic inheritance on the real form. Restrict Q.9 to real \(Q\) fluctuations about the selected \(V_3\) background. At fixed spatial momentum, its positive quadratic operator is

$$\mathcal{H}_Q(k) = Zk^2 I_Q + M_c^2 I_Q + \kappa(6P_0 + 5P_1 + 3P_2).$$

The condensate may shift the common term \(M_c^2\) but cannot remove the relative splitting. Since \(\delta\chi = L^T\delta d\), the chain rule gives

$$\delta^2\Gamma_f = \delta d^T L\mathcal{H}_Q L^T\delta d, \quad \boxed{\mathcal{H}_D = L\mathcal{H}_Q L^T.}$$

An extra one-mark quadratic term would be a new defect-only invariant and would define a different, nonminimal action. The hard-core constraint does not alter the fixed-one-mark quadratic sector.

For a local relative frame, \(L = L(g(x))\), the same substitution gives more than the tangent connection alone. Define

$$A_\mu = L\partial_\mu L^T \in \mathfrak{so}(9), \quad D_\mu = \partial_\mu + A_\mu, \quad B_\mu = P_3\partial_\mu L^T.$$

Since \(L^T L = P_Q\) and \(P_Q P_3 = 0\),

$$\partial_\mu(L^T d) = L^T D_\mu d + B_\mu d,$$

and the two terms are orthogonal. The inherited kinetic term is therefore

$$\boxed{Z\,\partial_\mu\chi^T\partial^\mu\chi = Z(D_\mu d)^T D^\mu d + Z\, d^T B^T_\mu B^\mu d.}$$

The second term is the finite-dimensional Born–Huang correction. It is quadratic in frame gradients and is nonzero for every frame direction that mixes \(Q\) with \(V_3\). Its presence is forced by the earlier primitive-invariance obstruction: the full geometric generators cannot preserve the nine-dimensional complement. For spatial gradients it is a positive semidefinite induced scalar matrix; in the Lorentzian Hamiltonian the temporal and spatial contributions enter with the corresponding metric signs. The complete symmetric Hessian, including this term, is the one used in the exact transport theorem below. The mismatch contribution has lower gap \(3\kappa\), so leakage into the matched sector is perturbatively suppressed when frame gradients are small compared with that gap, but the term is not discarded in the exact statement.

For a relative rotation of one primitive spin-3/2 factor, let \(J^a_R = I \otimes J^a\), \(B_a = P_3 J^a_R P_Q\), and \(\omega^a_i\) denote its spatial frame gradients; the exact leakage sum gives the following isotropic contraction:

$$\sum_a B^\dagger_a B_a = \frac{21}{20}P_2, \quad \sum_i\omega^a_i\omega^b_i = \frac{\omega^2_{\text{fr}}}{3}\delta^{ab}, \quad \boxed{\sum_{i,a,b}\omega^a_i\omega^b_i B^\dagger_a B_b = \frac{7}{20}\omega^2_{\text{fr}}P_2.}$$

This correction acts on the internal \(V_2\) block, whose identification with a galactic response mode remains part of the open source–metric calculation.

Positive Lorentzian dynamics selects the initial polarization. At a fixed frame and fixed spatial momentum, define the positive frequency operator on the real marked carrier,

$$\boxed{\Omega = \left[k^2 I + \frac{M_c^2}{Z}I + \frac{\kappa}{Z}Q_f\big|_{Q_\mathbb{R}}\right]^{1/2}.}$$

Its eigenfrequencies are

$$\omega_J(k) = \sqrt{k^2 + M_c^2/Z + (\kappa/Z)q_J}, \quad (q_0, q_1, q_2) = (6, 5, 3).$$

The stability premise is \(\Omega^2 > 0\): the complement Hessian is positive in the selected phase. Its mismatch part is bounded below by \(3\kappa/Z\), so this premise holds on the retained branch used here.

Start only with the real canonical phase space

$$v = (q, p) \in W = Q_\mathbb{R} \oplus Q^*_\mathbb{R} \simeq \mathbb{R}^{18}, \quad \omega_{\text{can}} = \sum_{a=1}^9 dq_a \wedge dp_a.$$

Hamilton's equations have the real generator

$$K_0 = \begin{pmatrix}0 & Z^{-1}I\\-Z\Omega^2 & 0\end{pmatrix}.$$

The phase part of its polar decomposition is

$$\boxed{J_0 = -K_0(-K_0^2)^{-1/2} = \begin{pmatrix}0 & -(Z\Omega)^{-1}\\Z\Omega & 0\end{pmatrix}.}$$

It obeys \(J_0^2 = -I_{18}\), and

$$g_0(u, v) = \omega_{\text{can}}(u, J_0 v)$$

is positive because

$$g_0(v, v) = Zq^T\Omega q + \frac{1}{Z}p^T\Omega^{-1}p > 0 \quad (v \neq 0).$$

The corresponding complex amplitude is

$$\boxed{z_0 = \frac{1}{\sqrt{2}}\left[(Z\Omega)^{1/2}q + i(Z\Omega)^{-1/2}p\right].}$$

At a fixed frame the real equations give

$$\boxed{i\hbar\dot{z}_0 = \hbar\Omega z_0.}$$

Thus both \(i\) and the fixed-frame Schrödinger equation follow from the positive real Lorentzian dynamics rather than being inserted as kinematic postulates.

Exact moving-frame transport. A changing frame generally mixes the unequal-frequency blocks. Consequently the instantaneous polar structure \(J_{\text{inst}}(t) = -K_{\text{inst}}(t)[-K_{\text{inst}}(t)^2]^{-1/2}\) is not