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

Jacob Chinitz

August 29, 2026

Abstract

We propose that empty space is not a passive backdrop but a physical medium with a finite budget of quantum entanglement: the linking structure that allows parts of a quantum system to share state. Matter forms when some of that capacity becomes locked into stable, localized defects of the medium. A particle's mass measures how much entanglement is committed to such a defect. Gravity is the surrounding capacity-strain field: near matter, slightly less entanglement capacity is freely available, and in the weak-field limit the fractional shortfall gives the gravitational potential. The excess acceleration seen in galaxies, usually attributed to particle dark matter, is treated here as the large-scale continuation of the same capacity response rather than as a new unseen substance.

The central result is that this picture is not freely adjustable after the fact. Once one accepts the finite-capacity medium, the three founding postulates, and a specific minimal model for the smallest cell of space, finite counting fixes the cell entropy and the ordinary weak-field response. The resulting capacity action is the static scalar sector of the Einstein action written in the capacity variable, so it gives Newton's law and the leading no-slip metric without introducing another gravitational field. A separately identified transverse branch gives the galactic acceleration scale and the observed relation between galaxy rotation and ordinary matter, subject to the microscopic matching conditions stated in the paper.

The electron plays a double role. As the lightest clean charged defect, it fixes the exchange rate between committed entanglement and mass and calibrates the absolute cell scale. Many-Pasts supplies the history-space interpretation of that calibration while preserving ordinary Born-rule statistics and no-signaling. Applying the same faithful-resolution condition used for the cell ensemble makes the local renewal process memoryless. A reversible marked-transfer action then derives the finite charged response and routes it through the electron and the heavier charged-lepton shells. This adds no new founding premise and leaves the original tetrahedral construction intact.

We also test the cell model in a computer simulation of dynamical spacetime. Turning on the medium's weighting orders the microscopic cell states while the background geometry remains stable, and a scrambled control confirms that the ordering follows the closure structure itself. Inserted defects then strain the nearby capacity and measurably deform the local geometry. In a separate transport calculation, a conserved carrier responds to defects of different strength through one common rule, and the disturbance persists without detected screening across the measured range. A predicted shift of the host geometry likewise follows the cell model across a family of simulation settings, while the control follows its own distinct prediction. These tests are limited in scale and do not yet measure Newton's constant, but they connect the proposed medium to dynamical geometry through measured consequences rather than analogy alone.

Beyond ordinary weak gravity, the framework extends to time-dependent transport, clusters, cosmology, the early universe, dark energy, black holes, particle structure, and complexity growth at explicitly labeled levels of closure. At the microscopic level, an explicit group-field-theory action selects the finite capacity and record dynamics and, in its retained nine-state sector, the local geometry of quantum states. A selected geometric construction connects the same substrate to the familiar large-scale Einstein description. Other extensions retain the conditional or open grades stated in the paper.

Contents

Part I. Physical Idea and Foundations — 5

1. Introduction: The Physical Claim — 5 - 1.1 What Is Primitive, and What Is Closed — 6 - 1.2 Physical Motivation for the Primitives — 8
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

1. Why a Tetrahedral Boundary Ensemble — 17
2. Admissibility Closure — 19 - 6.1 Minimal isotropic kernel — 19 - 6.2 Closure condition and uniqueness — 19 - 6.3 Effective sharing entropy — 20
3. Edge Kernel and Tree-Level Coupling — 20
4. Finite-Loop Renormalization — 21
5. Continuum Stiffness and SI Normalization — 22

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

1. Einstein Parent Action and the Reduced Capacity Frame — 24
2. Capacity Variable, Bridge Law, and Variational Status — 26
3. Newtonian Gravity and the Point-Source Limit — 27
4. Electron Anchor: One-Bit Mass Scale and Seven-Sector Length Scale — 28
5. Baseline Metric Closure: No Slip and PPN — 32

Part IV. The Carrier-Resolved Galactic Branch — 32

1. Carrier-Resolved Galactic Dynamics — 32
2. Galactic Metric, Lensing, and Local Tests — 34

Part V. Transport, Clusters, and Cosmology — 36

1. Causal Transport and Telegrapher Dynamics — 36
2. Cluster Source Projection and the Diffuse–Decoupled Channel Split — 37
3. Cosmology and the Hubble-Tension Sector — 42
4. The Saturated Phase and the Cosmic Microwave Background — 43

Part VI. Strong Fields, Many-Pasts, and Microstructure — 48

1. Strong-Field Action: Spherical Closure and Its Boundary — 48
2. Many-Pasts: The History-Space Ontology — 50
3. Microstructure Hamiltonian and Underlying Dynamics — 52

Part VII. The Substrate on a Dynamical Lattice — 55

1. Lattice Tests: Compatibility, Defect Response, and Transport — 55
2. Equilibrium Vacuum and Cosmological Term — 62

Part VIII. Closure Status, Falsifiability, and Comparisons — 67

1. Closure-Status Table — 67
2. Falsifiability and Observational Tests — 75
3. What the Theory Would Have to Get Wrong to Fail — 78
4. Comparison with Other Approaches — 79
5. Conclusion — 82

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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 \(S_{\rm ent}(x)\) 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 inputs are the finite-capacity substrate, three postulates, and the tetrahedral cell ensemble. From these, one chain of finite calculations follows. Admissibility fixes the sharing entropy, edge transport fixes the tree stiffness, the decorated transfer vertex fixes the charged marked event, source projection fixes the ordinary coupling ratio, and the weak-field bridge relates fractional capacity deficit to gravitational potential. The microscopic result is an explicit capacity/record action within the group-field-theory framework. It splits a primitive shared face into a seven-state matched channel and a nine-state retained mismatch record. Restricting the same action to that record gives a real Lorentzian phase space whose transport and mismatch spectrum generate su(9). The resulting unique positive complex structure, complete local response algebra, and phase quotient give \(\mathbb{C}^9\) and \(\mathbb{CP}^8\) rather than importing them for this sector (Appendix H.11). The action also prepares the renewed register, carries the persistent marked defect, and fixes the transfer gap used for scale setting. The same marked fiber supplies a coefficient-free dilation when a baryonic source clock loses its distinguishable carrier. Appendices O.2–O.14 complete the infrared bookkeeping of that event. The fixed-event history space has a unique volume singlet, so only the homogeneous projection of the marked drain is conjugate to the cosmological term; local marked-history fluctuations do not define a local \(\Lambda(x)\). Carrier loss removes the source-label saturation constraint and reassigns the metric-visible remainder to the coarse retained carrier rather than creating a new material component. A first-entry excursion-set construction fixes the assembly history \(F_Z\), while a three-dimensional compressible-flow passive-tracer surrogate supplies an explicit delayed mixing kernel \(K_{\rm mix}\). The resulting perturbation transfer has no independent dark-energy clustering variable: for \(k \neq 0\) the release-hazard fluctuations cancel inside the total source-associated capacity stress, and \(\delta\rho_\Lambda = 0\). The remaining registered astrophysical input is a self-gravitating, cooling cosmological zoom with passive Lagrangian progenitor tags, whose measured \(K_{\rm mix}(\tau; M, z)\) is to be frozen before the final likelihood analysis (Appendices O.10–O.14 and Q.9a). The ordinary geometric branch is supplied by a selected fixed-facet GFT completion: maximal fusion gives the canonical \(V_3^{\otimes N} \to V_{3N}\) blocking ray. On the selected metric-Regge branch, Appendix H.10 establishes the controlled massless two-helicity TT limit and an exact fixed-\(j = 3\) primitive realization. The microscopic construction therefore supplies an explicit finite GFT-sector action and decorated gluing prescription. The remaining freedom lies in the geometric host: the finite-capacity premises do not select a unique native gluing tensor or measure, so a fully derived nonperturbative host requires an additional geometric principle. Several infrared branches remain conditional.

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. 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 What Is Primitive, and What Is Closed

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 its operational quantum measure imported as stated below; 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 and separates its exact consequences from the remaining incidence condition. 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. The electron anchor fixes the proper-time cadence, and the spatial and temporal readings of the same phase mode give \(L_* = c\tau_*\). 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. 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.

The decorated vertex adds no sixth foundational premise. It specifies the minimal marked field content and gluing that realize the faithful full-support rule, native tetrahedral update, and one-bit fermionic electron anchor already listed above. Those action-level choices are stronger than a numerical ansatz because they determine the response multiplicity, determinant power, and routing together; they are also falsifiable, since the nonminimal and differently routed vertices give the alternative values displayed in Appendix H.9.

In compressed form, the central claim is

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

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

The absolute length 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 \(r = e^{-7g_{\rm share,eff}}\). The decorated vertex fixes the residual marked-fiber factor and its electron routing \(Z_e\). Positivity gives the raw survival energy \(E_{\rm raw} = -(\hbar/\tau_*) \ln(1-r)\), and the dressed electron identification \(m_e c^2 = (3/2)Z_e E_{\rm raw}\) then fixes \(\tau_*\), and with it \(L_* = c\tau_*\), without an independent clock or geometric diameter. Appendix H gives the finite action and both adversarial audits.

The reduced capacity functional can superficially resemble the scalar sector of a Brans–Dicke theory [56], 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, 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 standard quantum decoherence functional, then conditions them on the realized present. It preserves 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. The marked present/history fiber has the representation content of the discarded complement only for \(j_0 = \frac{3}{2}\); conditional on identifying it as that complement, the three-dimensional closure vector fixes the seven-state alphabet internally. Without that incidence identification, \(j_0 = \frac{3}{2}\) 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, \(g_{\rm share,eff}\), 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. They are nevertheless not arbitrary. Each is motivated by a place where established physics already strains against its own foundations — black-hole thermodynamics, quantum information, the equality of inertial and gravitational mass, and the quantum-mechanical role of records — and each makes a known difficulty look less mysterious once it is adopted. That motivation does not prove the premises; the numerical and structural consequences derived from them provide the tests.

Mainstream gravitational physics has been converging on a finite-capacity substrate for decades. A black hole's entropy scales with the area of its horizon rather than the volume it encloses, as though the contents of a region were written on its boundary; the Bekenstein bound limits the information a bounded region can hold; and entanglement-based reconstructions of geometry tie the shape of spacetime directly to patterns of entanglement. Each of these is usually treated as a deep clue without a mechanism. This paper takes the clue literally: the vacuum is a medium with a finite local budget of entanglement, and geometry is the large-scale description of that budget. Several long-standing puzzles then become the ordinary behavior of a medium that can fill up. A black hole is a region whose capacity is exhausted, so the only live bookkeeping sits at the boundary between exhausted and available capacity — which is why the entropy tracks the area and not the volume. The same finite state space supplies an ultraviolet cutoff: below the cell scale there is no continuum left to diverge, and a finite state space has nothing to renormalize away. The assumption that the empty state maximizes capacity agrees with the thermodynamic direction suggested by gravitational entropy, and three recent continuum results support the same reading from independent directions: the observer-dressed de Sitter algebra makes empty de Sitter the maximum-entropy gravitational state [24], a horizon modular calculation recovers Einstein curvature from the relative information carried by an excitation [23], and a geometric-relative-entropy action yields a local bulk information dynamics with an Einstein limit [25]. None of them supplies the finite cell or its coefficients; Section 29.9 states precisely what they do and do not establish for 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 \(Z_e^2 \ln^2(1 - e^{-7g_{\rm share,eff}})\), whose dominant hierarchy is \(e^{-14g_{\rm share,eff}}\): 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, \(a_0 = \epsilon c H_0\) with \(\epsilon \equiv g_{\rm share,eff}/(4\pi^2)\) 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 is why a particle can be vastly lighter than the natural substrate scale with no small number inserted anywhere: the electron is deeply bound, and deep binding makes its full coherent recurrence exponentially rare. 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 \(p_{\eta_*}\),

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

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. A single universal refresh rate also lets the theory carry a smallest length without conflicting with the experiments that ended earlier discrete-spacetime proposals: the granularity lives in the capacity, not in a preferred spatial lattice, so gravitational waves and light travel at the same speed and there is no frame-dependent dispersion left to detect.

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 addresses directly the quantum puzzles that interpretations of quantum mechanics were invented to handle. 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 has a concrete job beyond interpretation: 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 the standard building block in several approaches to quantum geometry, so the cell is a familiar object rather than one invented for this paper. What the framework adds is a precise 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.

Several mainstream results already point in the same direction: Einstein's equation derived as a thermodynamic relation of state, horizon entanglement entropy, and reconstructions of spatial connectivity from entanglement. The construction below makes a finite microscopic proposal within that program and derives quantities that can be checked.

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 \(x\):

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

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, \(S_{\rm ent}(x)\) 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

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

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

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

The variables \(S_{\rm ent}\), \(\delta S\), 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 \(S_{\rm ent}\) 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 absolute normalization of \(S_{\rm ent}\) and \(S_\infty\) is a convention once an epoch and cell convention have been fixed. Under a constant rescaling

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

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

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

The decorated marked-transfer vertex derived in Appendix H fixes

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

The physical electron-anchored scale is

$$\boxed{L_* = Z_e L^{(0)}_*} = 1.6162537014 \times 10^{-35} \text{ m}.$$

The corresponding induced gravitational scale is

$$G_* := \frac{c^3 L_*^2}{\hbar} = \frac{9}{4}\frac{\hbar c}{m_e^2}Z_e^2 \ln^2\left(1 - e^{-7g_{\rm share,eff}}\right) = 6.6742890772 \times 10^{-11} \text{ m}^3\text{kg}^{-1}\text{s}^{-2}.$$

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:

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

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 framework rests on exactly three postulates because they answer three different questions. What is spacetime geometry? — Postulate I: geometry is the long-wavelength expression of the capacity substrate itself. What is matter, and what is mass? — Postulate II: matter is localized committed capacity, and mass is its inertial reading. What supports the present, and what fixes the direction of history? — Postulate III: the present is supported by a history-space ontology over the decoherent pasts compatible with one realized record. Operational probabilities come from the standard decoherence functional, while the proposed arrow of time 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 \(S_{\rm ent}\) 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 \(S_{\rm ent}\) 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 uses the decoherent-histories formalism [71]. 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

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

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 operational branch standard quantum mechanics. Quantum instruments give the Born probabilities of laboratory records, and local trace-preserving instruments give no-signaling marginals. Many-Pasts changes their 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 the local kinematics \(\mathcal{H}_{\rm mark} \simeq \mathbb{C}^9\) and the ray space \(\mathbb{CP}^8\). Gleason's theorem then constrains its probability measure. On a Hilbert space of dimension greater than two, a normalized, noncontextual, additive measure on a sufficiently rich lattice of record projectors has the form

$$\mu(\Pi) = \text{Tr}(\rho\Pi)$$

by Gleason's theorem [65]. In the local marked sector the Hilbert-space premise and its full rotated projector family are supplied by the derived su(9) action; record completeness and noncontextual additivity remain hypotheses. A medium-decoherent family with a pure initial state admits orthogonal generalized records for its branch state vectors, so each history probability can be represented as the probability of a single-time record projector [66]. The Born form is therefore unique for record-defined decoherent families under those hypotheses. The full continuum/Fock-space kinematics, initial state, and substrate derivation of the global decoherence functional remain open; a mixed state requires the corresponding purification or generalized-record construction.

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 decorated transfer vertex supplies the determinant recurrence, finite marked response, and 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.

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

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

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 [32], 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 \(L_*\), \(g_{\rm share,eff}\), and \(\eta_*\). Discreteness of this class is compatible with exact low-energy Lorentz symmetry, as causal-set sprinkling demonstrates by construction [33, 34]. 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

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

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,

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

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,

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

The closure response is a three-component spatial vector, so in \(d = 3\) it carries \(s = 1\). Conditional on the physical incidence statement that the marked fiber is the faithful record of the nonmaximal fusion channels,

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

This is a conditional derivation from the already present marked and fusion structures, not a new minimality axiom. The representation equivalence and its complement isometry are exact. What remains to be shown by the complete microscopic vertex is that the two representation spaces are the same dynamical present/history record. If that incidence identification fails, \(j_0 = \frac{3}{2}\) 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

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

and the combinatorial sharing ceiling is

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

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 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 [62, 63], whose discrete geometric spectra [64] 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

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

The gap between \(g_{\rm share,max}\) and \(g_{\rm share,eff}\) 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

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

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

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

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

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

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 \(J^{\rm tree}_{\rm eff}\) is therefore the portion of the microscopic edge penalty that survives this local branching.

Origin of the horizon target. The horizon target

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

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 \(g_{\rm share,eff}\). In the continuum normalization used for the weak-field scalar, the occupancy variable is normalized by

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

so a coarse horizon-normalized channel with occupancy \(Q_{\rm occ} = 1\) 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

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

and therefore

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

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,

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

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

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

Equivalently, on the seven-channel scalar return space,

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

with \(\Sigma_{\rm ret} = \text{Tr}(R_{\rm ret})\). 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 \(\Sigma_{\rm ret} = 65/9\) by 7, 7 + 2/3, or 8 shifts \(J^{(\rm ren)}_{\rm eff}\) and \(\gamma\) by at most 0.5% and leaves \(G\), \(a_0\), and \(\kappa/\gamma\) exactly unchanged, since \(J^{(\rm ren)}_{\rm eff}\) cancels in the source-to-stiffness ratio (Appendix C.5).

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

and

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

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

The loop correction is no longer schematic: 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,

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

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 \(Q_{\rm occ}\),

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

Here \(L_*\) is the canonical tetrahedral spacing, with one coarse cell carrying volume \(L_*^3\) up to the fixed cell-shape convention, and \(J^{(\rm ren)}_{\rm eff}\) 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:

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

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

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

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

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

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 \(J^{(\rm ren)}_{\rm eff}\) 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

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

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 remaining microscopic question is independent confirmation of the same action-kernel interpretation from fuller inhomogeneous dynamics, not an unresolved normalization constant.

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 a covariant parent action, and it is simply the ordinary metric action — no independently varied capacity scalar is added to it:

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

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 [8] the shift and extrinsic curvature vanish, so the Einstein–Hilbert plus Gibbons–Hawking–York action is

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

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 \(I^{(1)}_{\rm matter} = -\int dt\, d^3 x\, \rho\Phi\), gives

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

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

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

Asymptotic flatness removes the harmonic difference, so \(\Phi = \Psi\). Eliminating \(\Psi\) therefore produces

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

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 \(S_{\rm ent} = S_\infty - \delta S\). Source-independent extensive terms are removed by the vacuum normalization proved for the regulated joint measure in Section 25. The capacity functional is then

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

The field redefinition

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

turns it into

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

Using

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

gives the action-level equality

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

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 \(I_{\rm cap}\) 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

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

The vertical bar is essential: \(g_{\rm ref}\) and the reduced source projection \(\chi\) are held fixed while \(S_{\rm ent}\) 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 \(I_{\rm matter}[g, \psi]\).

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

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

the Green-matched projection is

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

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

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

and therefore induces

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

Substituting the source-map identities

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

into the weak-field expression gives identically

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

Thus there is one scale-setting route to \(G\): electron recurrence fixes \(L_*\), and \(L_*\) fixes the gravitational scale. The stiffness, source coefficient, and capacity normalization are a consistent static-EFT representation of that same length backbone, not a second determination that could have disagreed with it. The only numerical comparison in this sector is \(G_*\) against the measured Newton constant.

The decorated scale gives \(G_* = 6.6742890772 \times 10^{-11}\) m³ kg⁻¹s⁻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 \(S_{\rm ent}\) gives

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

On the renormalized static, nonrelativistic branch this becomes

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

Define the surviving fractional capacity

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

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

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

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 \(I_{\rm matter}[g, \psi]\), 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 \(S_{\rm ent}T^\mu_\mu\) is neither required nor adopted.

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

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

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 Scale and Seven-Sector Length Scale

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

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

with canonical running law

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

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

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

Multiplicativity across the seven factorized clouds gives

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

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.

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. The tau ratio required 0.6562%, which separates into the same universal uplift and a smaller 0.1431% 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

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

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

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

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

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

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.

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

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

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

$$G_* = 6.6742890772 \times 10^{-11} \text{ m}^3\text{kg}^{-1}\text{s}^{-2} \quad (-0.073\sigma),$$ $$\frac{m_\mu}{m_e} = 720\frac{2}{7}Z_\mu = 206.768280237 \quad (-0.535\sigma),$$ $$\frac{m_\tau}{m_e} = 720^2\left(\frac{2}{7}\right)^4 Z_\mu Z_{\tau,2} = 3477.343310 \quad (+0.481\sigma).$$

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. Conditional on the marked-fusion incidence identification, the three-dimensional closure vector fixes the same seven internally and the Newton comparison checks its routing. Without that identification, 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. At the tau the analogous test is weaker: among the forty-three simple rationals with denominator up to eleven, both the derived 2/7 (+0.49σ) and 3/11 (−0.49σ) survive, so the second-shell routing is supported rather than uniquely resolved.

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

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

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 \(e^{-14g_{\rm share,eff}}\).

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,

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

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

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

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 \(\mathcal{R}_P\) be its retained record algebra and \(\chi_P\) its spatial support. The source projector is

$$\boxed{\Pi_P X = \chi_P \mathbb{E}[X \mid \mathcal{R}_P].}$$

The active set \(\mathfrak{A}\) is an antichain: a carrier and one of its descendants cannot both supply transverse sources. Distinct active carriers have disjoint projected support, while nested record algebras obey the conditional-expectation tower rule,

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

The transverse functional consequently has the carrier sum

$$\Gamma^{\rm nest}_\perp = \sum_{P \in \mathfrak{A}} w_P \Gamma_{\perp,w_P}[g_\pm; \Pi_P T_\pm, \mathcal{R}_P].$$

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.

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

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

The phase-cell loading and reversible horizon coupling are physical matching premises. With the Planck value of \(H_0\), they give \(a_0 = 1.231 \times 10^{-10}\) m s⁻², compared with the fitted RAR scale \(1.20 \times 10^{-10}\) m s⁻² [1, 51]. 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

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

Its equilibrium free energy and response are

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

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

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

so the total auxiliary function \(Q(z) = z + Q_\perp(z)\) satisfies

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

For each active carrier, introduce a total carrier potential \(\Phi_P\) and the QUMOND-type action [19]

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

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 \(Q(z) = z\). Variation gives

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

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. For spherical systems it reduces to

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

The high-acceleration limit is Newtonian, and the deep limit gives \(g_{\rm obs} \simeq \sqrt{a_0 g_{\rm bar}}\) and \(v^4 \simeq GM_b a_0\) [49, 50]. 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.

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 \(10^4 < M_{\rm bar}/M_\odot < 10^{7.5}\) 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 [39]. 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.

Status. The thermal contact free energy, auxiliary action, nonspherical field equation, and carrier nesting rule close the leading static EFT once its matching data are supplied. The one-entropy horizon loading, \(E_P/(k_B T_H) = x_P\), and the active-carrier projector remain conditional. A microscopic GFT derivation must produce the cell Hamiltonian, projector, and metric vertex. Resolved baryonic-map calculations and the dwarf phase boundary remain 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

$$\boxed{\Gamma^{\rm QS}_\perp = -\frac{1}{2}\sum_{P \in \mathfrak{A}} w_P \langle h_\Delta, \mathcal{E}h_{\perp,P}[\Pi_P T_c]\rangle,}$$

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

$$\rho_{\rm lens,P} = \rho_P + \frac{1}{4\pi G}\nabla\cdot[n_B(x_P)\nabla\Phi_{b,P}].$$

For a deep-regime point mass, define \(A = \sqrt{GM a_0}\). Then

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

and the projected profiles and asymptotic deflection are

$$\Sigma_\perp(R) = \Delta\Sigma_\perp(R) = \frac{A}{4GR}, \quad \alpha_\infty = \frac{2\pi\sqrt{GM a_0}}{c^2} = \frac{2\pi v_\infty^2}{c^2}.$$

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 [46, 47]. This supports the leading no-slip choice. Direct solutions for resolved nonspherical lenses remain necessary.

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

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

about two thousand times smaller than the current Cassini 1σ uncertainty. Cassini tracking gives \(Q_2 = (1.6 \pm 1.8) \times 10^{-27}\) s⁻² and places universal nonlinear functions of the total local field in 3–15σ tension with galaxy fits; it also limits their Milky-Way boost at the Sun to two percent at 95% confidence [38]. Carrier nesting avoids that total-field cross-term.

Wide binaries inside the Milky-Way carrier are likewise Newtonian apart from the smooth Galactic tide. For a \(1.5M_\odot\) pair the Hill radius is about 1.95 pc, and the estimated fractional speed corrections from the tide satisfy

$$\frac{\Delta v}{v_N} \lesssim 8 \times 10^{-6}, 6 \times 10^{-5}, 2 \times 10^{-4}$$

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 [35, 36, 37]. 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 \(N_{\rm eff}\) independent occupation cells,

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

A stochastic contribution below ten percent requires \(N_{\rm eff} \gtrsim 100\); three percent requires \(N_{\rm eff} \gtrsim 10^3\). 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

$$\rho_{\perp,P} = \frac{1}{4\pi G}\nabla\cdot[n_B(x_P)\nabla\Phi_{b,P}].$$

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

$$\frac{D}{\tau_0} = c^2, \quad \tau^{-1}_0 = H_0, \quad D = \frac{c^2}{H_0}.$$

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 [4, 16, 17] find the ratio of observed to galaxy-RAR-predicted acceleration near unity in stellar-dominated centers, rising to roughly 3–5 at intermediate accelerations (\(g_{\rm bar} \sim 10^{-11}\)–\(10^{-10}\) m s⁻²), and apparently returning toward the galaxy relation at the lowest probed accelerations, subject to gas-extrapolation caveats [17]. The older integrated value of 1.5–2 from higher-acceleration hydrostatic analyses samples the high-\(g_{\rm bar}\) 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 \(\tau^{-1}_0 = H_0\) 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,

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

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

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

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 \(\epsilon = g_{\rm share,eff}/4\pi^2\).

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

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

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 \(W_{\rm bath} = 1\) and \(1 + 1/\epsilon\) is evaluated against cluster data below.

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

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

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

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

which in the Bullet limit, where the displaced gas is shocked and little virialized bath remains on the cores, reduces to \(\chi_{\rm Bullet} \simeq \rho_{\rm dec} + \epsilon\rho_{\rm shock}\). 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,

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

with \(f_{b,\rm cos}\) fixed by Planck values [51] and \(M_{\rm hot,vir}\), \(M_{500}\) 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 \(W_{\rm bath}(B_{\rm bath})\). In merging systems \(M_{\rm hot,vir}\) 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

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

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

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

and the minimal linear candidate for the lift,

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

uses the amplitude \((1 - \epsilon)\), the part of the full horizon channel that the suppressed transverse branch is missing—not a new coefficient. The physical reading is that as a virialized diffuse bath forms, the continuum itself opens a collective cluster response whose amplitude scales with how complete the bath is (\(B_{\rm bath}\)) and how much of the baryon budget sits in it (\(f_{\rm cont}\)).

This linear form has the qualitatively correct mass trend: both \(B_{\rm bath}\) and \(f_{\rm cont}\) increase with halo mass—hot-gas fractions rise toward clusters [15, 14] and the atmosphere becomes more fully virialized—so their product rises monotonically from groups to massive clusters, reproducing the observed direction with no fitted parameter. Its amplitude, however, is excluded. For CLASH-scale clusters the measured factor gives \(B_{\rm bath} \simeq 0.83\), \(f_{\rm cont} \simeq 0.9\), hence \(\mathcal{R}_{\rm rel} \simeq 1.6\); but evaluating the source weight required to reproduce the published CLASH relation [4] across its data-supported acceleration window gives \(\mathcal{R} \simeq 3.7\) at \(g_{\rm bar} = 10^{-10}\) m s⁻² rising to \(\mathcal{R} \simeq 7.8\) at \(10^{-11}\) m s⁻². The linear lift \((1 - \epsilon)B_{\rm bath}f_{\rm cont}\) is short of the observed peak residual by a factor of roughly 2.5–4, and no escape through missing baryons (a multiple of the X-ray gas mass would be required), hydrostatic bias (the masses are lensing-based), or sample heterogeneity is available at that magnitude.

The linear candidate is excluded on amplitude. Because \(B_{\rm bath} \leq 1\) and \(f_{\rm cont} \leq 1\), the linear candidate bounds the relaxed residual by \(\mathcal{R}_{\rm rel} < 1 + (1 - \epsilon) \simeq 1.81\). The measured peak residual of relaxed clusters is 3–5 [4, 16], well above this bound, so the linear lift is excluded. The exclusion is specific to the lift function: the channel-split ontology itself makes a structural prediction about the residual's shape that the data support, taken up next.

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 [16, 17]; neither a total-baryon modified-gravity law with no relaxed-cluster excess nor a constant offset gives the same shape.

The linear lift nevertheless underestimates the amplitude: its predicted peak is \(\simeq 1.5\) for CLASH-like parameters against the observed 3–5. The lift function must grow faster with bath development, reach 3–5 in developed clusters, and remain small enough for X-ray-faint groups to stay near the galaxy relation.

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

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

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 \(S \simeq 3.7\text{–}4.9\) 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 \(W_{\rm bath} \lesssim 1.4\) 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 [4] versus convergence toward the galaxy relation [17]—is adjudicated directly below.

The ceiling against cluster data. Inverting the X-COP hydrostatic measurements [72] gives twenty-four source-weight tests. For each point, \(S\) solves

$$\frac{Sg_{\rm bar}}{1 - \exp[-\sqrt{Sg_{\rm bar}/a_0}]} = g_{\rm obs}.$$

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

The same inversion adjudicates the deep end: the power-law continuation requires \(S \simeq 6\text{–}8\) at \(g_{\rm bar} \simeq 1\text{–}2 \times 10^{-11}\) m s⁻², precisely the accelerations of the \(R_{500}\)–\(R_{200}\) points, which sit at \(S \simeq 1.5\text{–}2.7\); 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—\(S \simeq 3.7\text{–}4.9\) 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 [73, 74], 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 \(W_{\rm bath} \lesssim 1.4\).

The decoupled-fraction form is excluded by the mass trend. A second candidate class assigns the residual to the decoupled stellar fraction, \(\mathcal{R} = 1 + 4.32 B_{\rm bath}f_{\rm dec}\) with \(f_{\rm dec} = f_\star/(f_\star + f_{\rm gas})\). Its amplitude can cross the observed band, but its mass trend is wrong. Because \(f_{\rm dec}\) falls with mass while \(B_{\rm bath}\) rises, their product peaks at the group or poor-cluster scale and declines toward massive clusters. The observed residual rises from groups to massive clusters. The trend therefore assigns the residual to the continuum, although its lift amplitude remains underived. The relaxed-cluster amplitude and Bullet morphology are distinct observables and require separate tests.

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 \(\epsilon \times 5.7 \simeq 1.07\) in the entropic channel against the galaxy contribution of 1.0: the projection brings the two components to near-parity, removing the factor \(\sim 5.7\) 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

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

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 \(\chi_{\rm ent}\); 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 \(\chi_{\rm ent}(x, t) = \rho_{\rm dec}(x, t) + [\cdots]\) 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 linear candidate's ceiling \(\mathcal{R}_{\rm rel} < 1.81\) lies below the measured peak residual of relaxed clusters [4, 16], which excludes that form of the lift. 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 \(\mathcal{R}_{\rm rel} \leq 1 + f_{\rm cont}/\epsilon \simeq 5.7\text{–}5.9\) 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 \(B_{\rm bath}\), larger \(f_{\rm cont}\)) 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,

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

with the branch value \(\epsilon = g_{\rm share,eff}/4\pi^2 \simeq 0.188\) or with \(\epsilon\) floated as a test, and \(W_{\rm bath}\) 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 \(B_{\rm bath}\) 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, \(B_{\rm bath}\) 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 II₁ static-patch construction makes empty de Sitter the maximum-entropy gravitational state and places excited semiclassical states below it [24]. 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

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

so their product gives

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

Integrating Bianconi's local volumetric information density over the observer's causal diamond gives an area-sized finite capacity [25]. 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. Its joint evolution with the committed component has not been computed, and a full Boltzmann calculation and likelihood analysis remain 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, and a full perturbation calculation must still test that separation dynamically.

20. The Saturated Phase and the Cosmic Microwave Background

The conditional transverse normalization of Section 15 is now applied to the early universe. Its galactic response remains below the capacity ceiling, but the estimated recombination-era demand exceeds it. 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 \(y \equiv g_{\rm pert}/a_0(z)\) 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 \(y \propto \sqrt{a}\) give \(y \lesssim 6 \times 10^{-5}\) at \(z \simeq 1100\), hence \(x \lesssim 8 \times 10^{-3}\) and \(\nu(x) \gtrsim 10^2\). This is well above the total-response cap \(1 + 1/\epsilon = 6.32\), so the adopted response branch requires saturation if the same estimate applies to the relevant 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.

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,

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

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

coherent pinned committed capacity
↓
source-labelled multistream committed stock
↓
carrier loss and rerouting

generally with the absolute availability still close to its vacuum value. Terminal strong-field saturation is the different condition \(q_{\rm geo} = 0\). 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

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

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

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

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 [75]. 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 [31].

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

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

a 0.7σ agreement with the abundance derived from the commitment count [51]; equivalently \(\Omega_m/\Omega_b = 1 + 1/\epsilon = 6.32\). A standard Einstein–Boltzmann computation [76] 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 [41]. 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 \((1.20 \pm 0.02) \times 10^{-10}\) m s⁻², the Planck abundance ratio \(5.364 \pm 0.065\), and \(H_0 = 67.4 \pm 0.5\) km s⁻¹ Mpc⁻¹ gives

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

[1, 51]. 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.

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.

Conditional conservation and collective source-clock 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 source/worldline renewal clock and \(c_{A,c}\) 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

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

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. Faithful renewal then fixes a complete fresh draw,

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

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.

The enabling well depth remains a collective matter condition rather than a microscopic capacity threshold. Atomic-cooling assembly supplies \(V_{\rm thr} \sim 17\) km s⁻¹ before reionization, while photoheated gas after reionization requires a scale nearer 30 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

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

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

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

Appendix O.12 now fixes \(F_Z\) by an absorbing first-entry excursion-set construction and directly evaluates \(K_{\rm mix}\) in a three-dimensional compressible-flow passive-tracer surrogate. The former Poisson number 0.8529 is retained only as a coverage statistic; it is not the Shannon information loss. For progenitors distributed through the full virial volume the surrogate gives \(\epsilon_{\rm mix}(70\,\text{Myr}) \simeq 0.20\), whereas centrally concentrated progenitors mix much faster. The decisive 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,

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

together with

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

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 \(\rho_b/\epsilon\), that is, 5.3 times its baryonic mass, in addition to the response-branch enhancement; the radial-acceleration relation of the faintest dwarfs is the corresponding test. Section 15 assumes that collapsed systems follow the baryonic response law with no additional collisionless matter species; under the present release law the detailed low-mass behavior therefore 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,

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

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 the former multiplier action is not retained. 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

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

Its vacuum equations imply conservation of the Misner–Sharp mass

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

The capacity-adapted invariant is therefore

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

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, \(M_{\rm MS} = M\) 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

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

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 \(q_{\rm geo} = 0\) 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

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

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. Projective measurements and general quantum instruments therefore retain their Born probabilities. Appendix G gives the construction and the no-signaling proof.

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 [25]. 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: Stinespring dilation guarantees an environment, while Postulate III supplies its history-space reading. It does not derive the decoherence functional 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. The nine-state fusion complement is the physical marked fiber, so restricting the displayed Lorentzian action to that complement fixes both its moving-frame connection and its nondegenerate 6:5:3 mismatch spectrum. On the real one-mark phase space these operations generate su(9), select a unique positive complex structure, and give \(\mathcal{H}_{\rm mark} \simeq \mathbb{C}^9\). The complete quadratic response algebra identifies normalized states exactly up to common phase, yielding \(\mathbb{CP}^8\) and its Fubini–Study geometry. The marked tangent metric independently equals one quarter of the Fisher metric obtained from the fresh closure statistics. Caticha's Hamilton–Killing theorem is therefore a consistency and continuum guide rather than the premise carrying the local reconstruction [26, 27].

Status. The operational measure remains mathematically complete because it imports the standard global decoherence functional and quantum instruments; a substrate derivation of that full functional is still missing. Locally, however, the marked Hilbert and projective geometry and its full finite-dimensional unitary algebra are derived within the selected quadratic action. The projected two-mark bridge is a genuine entangling interaction and generates su(81) with the two local algebras; extension to a connected marked network is conditional only on simultaneous retained marks and survival of that bridge under the final host projection. Long-time history storage, continuum/Fock-space quantum fields, electromagnetic dynamics and charge, 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

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

Here \(\mathcal{H}_{\rm geom}\) carries the gauge-reduced simplicial GFT/spin-foam data, while \(\mathcal{H}_{\rm cap}\) 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, \(\eta_*\), \(g_{\rm share,eff}\), \(\zeta_*\), \(Z_e\), \(L_*\), 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 [99, 100]. 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 [97, 98]. Area-Regge continuum analysis gives the same leading graviton dynamics with a first length-metric correction of order \(a^2C^2\) [101]. 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 [60, 61], 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 \((H_{\rm TT})\): 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 \((H_{\rm phase})\), dynamical selection of the Einstein branch, becomes a well-posed derived theorem. A spin-3-regularized canonical host obeys the distinct condition \((H'_{\rm can})\) 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

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

The seven-channel determinant is \(r = e^{-7g_{\rm share,eff}}\). 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

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

The Euclidean transfer Hamiltonian

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

therefore has the exact raw gap

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

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 \(Z_e = (1 + \zeta_*)(1 + 7\zeta_*^2)\), so the dressed transverse identification is

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

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

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

Analytic continuation supplies the phase frequency \(E_e/\hbar\). The identification of the same spectral tick with the causal cell scale gives \(L_* = c\tau_*\).

The causal length of that spectral update is

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

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 decorated transfer action 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 finite scale-setting dynamics is therefore specified end to end: renewal fixes the fresh marginal, the determinant fixes the baseline recurrence, the decorated vertex fixes the marked event and routing, positivity fixes the transfer gap, and the electron fixes the cadence. 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 [79]. CDT is used because it independently sustains an extended four-dimensional de Sitter-like phase [80, 81]. 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 \(j_{\rm eff} = 3\) 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 40,000 to 100,000 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

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

where the sum runs over the slice cells, \(K^2\) is the closure invariant of Appendix B, \(n_{\rm coll}\) 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 \(g_{\rm share,eff} = 7.4198\) 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 \(m^2 = 2/g_{\rm share,eff} \simeq 0.27\), 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 [80, 81]. 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 \(S_{\rm eff} = S_{\rm host} - \log Z_{\rm label}\). 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 16.8σ. The collision fraction changes from \(0.0783 \pm 0.0002\) to \(0.0698 \pm 0.0009\), a 9.2σ 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 4.0σ 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, although several preview signals disappear under the completed error treatment.

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

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

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

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

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 \(J^\mu_{\rm hist}\) 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 \(j^H_v = \mathcal{P}_{\rm maint,v}\); 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

$$L_f = B_s W B^T_s,$$

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

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

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

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

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 \(\mathcal{P}_{\rm maint} = \alpha_{\rm maint}\rho_{\rm def}\). Away from the compact zero-mode subtraction,

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

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

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

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 \(Z_{\rm cap}[T; g_{\rm cap}]\) be the trace over its closure labels, defect marks, renewal gates, source-labelled committed allocations, and history outputs. The joint partition function is

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

and integrating out the capacity sector gives

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

The theory fixes \(g_{\rm cap}\) 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, \(\lambda_{\rm TT}(k) = 1 - c_{\rm TT}k^2 a^2 + O(k^4)\) 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. The third test is decisive: extended-looking volume data with a gapped TT sector would not supply the physical Einstein branch.

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 \(\xi_{\rm hist} \simeq 0.57\) 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

$$V_4 = a^4(v_{41}N_{41} + v_{32}N_{32})$$

and its corresponding source direction in the \((\kappa_4, \Delta)\) plane. On a fixed-ratio trajectory the scalar matching is \(t_4 = \bar{v}_4 a^4\rho_{\rm vol,R} + o(a^4)\), 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 \(\mu_{\rm cap}\). If the common-parent critical limit exists, Postulate I then excludes an additional independent host volume action and gives \(\Lambda^{\rm empty\, equilibrium}_R = 0\). 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 [87, 88]. 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,

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

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

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

and define its homogeneous bulk free energy per cell by

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

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

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

By construction,

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

This is the lattice grand-potential statement. In the thermodynamic limit \(F_{\rm cap} = \mu_{\rm cap}(a, \beta)N_{\rm cell} + o(N_{\rm cell})\), so the centered bulk potential

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

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

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

then

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

exactly. When the bulk cell count admits the usual thermodynamic reading, Gibbs–Duhem writes the same relation as \(\Omega_{\rm cap}/V = -P\); an isolated equilibrium reference has \(P = 0\) [78]. 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

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

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 \(+\beta\mu_{\rm cap}N_{\rm cell} = +(\beta\mu_{\rm cap}/2)N_{41}\) in the geometric action. Against the quadratic volume pin it predicts

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

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,

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

the restored \(N_{41}\) term is the coupling displacement \(\delta\Delta = +\beta\mu_{\rm cap}/2\), \(\delta\kappa_4 = -\beta\mu_{\rm cap}/2\), 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

$$t_4 := \kappa_4 - \kappa^c_4 \longrightarrow 0.$$

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 now be made explicit instead of being hidden in an unspecified \(Z_V\). After Wick rotation, the exact Regge four-volumes of the two simplex types are [81]

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

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 \(a_{\rm ch} = 4\ln 2 L_*^2\) and the edge is physical, \(a^2 = (16\ln 2/\sqrt{3})L_*^2\), \(a = 2.530L_*\) (Appendix J.10). The integrated volume operator is therefore the kinematic identity

$$\boxed{V_4[T] = a^4\left[v_{41}(\alpha)N_{41} + v_{32}(\alpha)N_{32}\right].}$$

Let \(\rho_{\rm vol,R}\) denote the coefficient of \(V_4\) in the dimensionless Euclidean effective action. If \(G_R\) is the continuum Newton coefficient and \(L^2_G := \hbar G_R/c^3\), then

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

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

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

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

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

Substitution returns \(\delta(\kappa_4 + 2\Delta) = a^4 v_{41}\rho_{\rm vol,R}\) and \(\delta(\kappa_4 + \Delta) = a^4 v_{32}\rho_{\rm vol,R}\), 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 earlier one-parameter formula is the fixed-ratio compression of this two-operator statement. If \(\xi = N_{32}/N_{41}\) and \(N_4 = N_{41} + N_{32}\), then

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

Thus the geometric part of the previously unnamed factor is fixed:

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

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 are no longer freedom to choose 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 \(2.530L_*\) has three-volume \(1.91L_*^3\), so one cell swept through one tick is \(1.91L_*^4\), 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

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

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 \(\mu_{\rm cap}(a, \beta)\) removes the complete thermodynamic bulk coefficient at every fixed regulator, not merely the part that diverges as \(a \to 0\). If a nonzero finite \(\rho_{\rm cap,R}\) remained, then at any fixed \(a\) the normalized vacuum action would contain

$$\rho_{\rm cap,R}a^4\left[v_{41}N_{41} + v_{32}N_{32}\right],$$

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

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

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.

What about an independent host cosmological term? In the present simulations it remains possible because CDT is deliberately an external scaffold: \(\hat{Z}_{\rm cap}[T]\) normalizes the capacity sector at fixed \(T\), not the externally supplied host measure. In a completed theory, however, Postulate I says that geometry and capacity are the same substrate, so the vacuum normalization must apply to the bulk coefficient of the joint geometry–capacity measure. That joint coefficient is the leading exponential growth removed when the cosmological coupling is placed on its infinite-volume critical surface. Retaining another source-independent host volume action after this common-parent normalization would split geometry back off from capacity and undo the already-used foundational identification. Writing \(\mathfrak{C}_*\) for the common-parent critical manifold defined in Section 24.9, the continuum statement is consequently the conditional theorem

$$\boxed{\left(\mathcal{P}_{\rm cap} \cap \mathfrak{C}_* \neq \varnothing\right) \text{ and Postulate I} \implies \Lambda^{\rm empty\, equilibrium}_R = 0.}$$

This adds no new premise. It applies Postulate I to the exact vacuum normalization once a common-parent continuum limit exists. The existence of that decorated critical limit remains to be demonstrated empirically and mathematically, so the currently externally hosted lattice calculation does not yet license an unconditional claim about the observed Universe. Nor does equilibrium cancellation imply a matter-tracking residual, a relaxation law, evolving dark energy, or \(w \neq -1\). Any observed homogeneous acceleration must come from a state-dependent, boundary, or nonequilibrium sector with covariantly conserved stress energy, or else falsify the common-substrate equilibrium completion; its equation of state must be derived separately.

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

$$\mathcal{G} = C^{-1/2}_0 CC^{-1/2}_0.$$

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

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

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

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

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

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

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 [25]. 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.

The missing cosmological action must still derive the relevant covariance or response operator, its normalization and dimensions, its time evolution, and a conserved metric stress tensor. Those results must determine \(w(z)\). 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 microscopic theory here must still determine which departure the Universe occupies.

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 normalization above supplies the initial condition: the centering removes every extensive bulk term at each regulator, its volume conjugate is the initial constant, and \(\Lambda_i = 0\) follows conditionally on the volume-conjugacy lemma and the common continuum gate (Appendix O.5).

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. Faithful renewal fixes complete rerouting at that event, and Appendix O.13 supplies the coefficient-free nine-state dilation. The astrophysical variable is \(\epsilon_{\rm mix} = 1 - I(A; Y)/H(A)\), measured from Lagrangian progenitor tags. A cosmological zoom must fix the progenitor geometry and \(K_{\rm mix}\) independently before the final likelihood is evaluated. Refresh-first \(K^2\) independence still assumes the post-refresh ledger. Once the release history is fixed, \(w = -1\) between events and approaches it from below during active release.

Part VIII. Closure Status, Falsifiability, and Comparisons

26. Closure-Status Table

The closure bookkeeping is concentrated here in one place; the rest of the text states results and refers here for their status.

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 full closure-status table lists, for each quantity or claim, its sector, status, type of support, and location in the text. The table records the following representative rows and their grades:

• \(\Omega_{\rm tet}, g_{\rm share,max}\) (UV counting): Closed by exact combinatorics (Part II, App. B).
• \(\eta_*\) (admissibility closure): Closed via stationary normalized closure evidence \(\ln Z + \frac{3}{2}\ln\eta\) maximized on the exact \(K^2\) spectrum (Part II, App. B).
• \(g_{\rm share,eff}\) (UV entropy): Closed by exact weighted evaluation (Part II, App. B).
• Memoryless refresh kernel: Closed under the foundational faithful full-support condition; applying it to histories gives maximum path entropy with \(H(B_{t+1} \mid B_t) = g_{\rm share,eff} - I(B_t; B_{t+1})\) maximal iff \(I = 0\), giving \(K(b, b') = p_{\eta_*}(b')\) (Part I, App. D, G–H).
• Renewal / retained marked quantum sector: exact channel, overlap, and local Hilbert reconstruction; network extension conditional (§3.3, §22, App. H.11, P.6, Q.9).
• Decorated charged marked-transfer vertex: Closed finite transfer and selected-branch Einstein TT theorem; geometric host underdetermined (§13, §23, App. B, H, Q).
• Substrate length \(L_*\): Fixed within the decorated transfer action and electron anchor, \(L_* = -(3/2)Z_e\lambda_e\ln(1-r)\) (Part I, App. D, H).
• \(J_{\rm bare}, J^{\rm tree}_{\rm eff}\) (UV edge kernel): Closed by tetrahedral isotropy identity (Part II, App. C).
• \(\Sigma_{\rm ret} = 65/9\) (finite-loop UV): Conditional minimal return-sector completion (Part II, App. C).
• \(J^{(\rm ren)}_{\rm eff}\): Conditional on the minimal return operator (Part II, App. C).
• \(\gamma\) (continuum stiffness): Conditional loop-dressed value (Part II, App. C).
• Green-matched source projection, \(\kappa/\gamma\): Closed in the canonical weak-field branch (App. C, Part III).
• Weak-field action / bridge law: Closed for the ordinary longitudinal branch, \(I^{\rm static}_{\rm cap} = Z_S I_{\rm Newton}\) (Part III, App. D, N).
• \(G_*\): parameter-free output of the decorated scale branch; historical postdiction, \(G_* = 6.6742890772 \times 10^{-11}\), \(-0.073\sigma\) relative to CODATA (Part I, App. D, H, L).
• Matched \(G\): algebraic identity in the chosen normalization, giving \(G = c^3 L_*^2/\hbar = G_*\) (Part III, App. C–D).
• Electron anchor: Fixed empirical elementary anchor (Part III, App. D).
• Seven-sector additivity and lightest branch: Closed entropy theorem; exact conditional mass minimum at \(k = 7\), \(\Delta_7 = 0\) (App. D, H).
• \(a_0\): exact conditional consequence; loading and horizon coupling open (Part IV, App. N).
• Carrier nesting: Closed within the specified leading EFT; microscopic projector open (Part IV, App. N).
• RAR law and static action: Closed within a conditional leading completion (Part IV, App. N).
• Baseline no slip / lensing: Closed for the ordinary longitudinal branch (Part III, App. D, N).
• Transverse metric and lensing: Closed within the specified quasistatic contact; nonlinear covariance open (§16, App. N).
• Solar-System and wide-binary limit: Closed at leading quasistatic order; nonlinear preferred-frame audit open (Part III–IV, App. F, N).
• Telegrapher relation \(D/\tau_0 = c^2\): Closed in the canonical transport branch (Part V, App. E).
• Canonical \(\tau^{-1}_0 = H_0\) branch: Fixed in the minimal transport closure (Part V, App. E).
• Hubble-tension mechanism: structurally supported extension (Part V, App. E).
• Equilibrium capacity-vacuum subtraction: exact regulator identity; conditional continuum theorem (§25, §24.9, App. J.2, J.6, J.9).
• Cosmological term from fixed measure and collective source-clock release: conditional continuum and astrophysical branch; zero-mode and renewal lemmas exact within stated premises (§20, §25, App. O, P.7, Q.9a).
• Gauge compatibility of the capacity spine: same-factor Gauss reading excluded; separate-factor covariance closed kinematically (§6, §23, App. Q).
• Orientation doublet: persistent orientation excluded by renewal; pure-gravity decoupling closed; fermion audit open (§23, App. H.9, Q).
• Record dynamics, organization, and computation: exact algebraic core plus named conditional interpretations (App. P, O.10–O.13).
• Saturated-phase committed capacity: conditional transition; zero-pressure constraint theorem closed within the pinned coherent branch (Part V, §20, App. M).
• Committed-capacity abundance \(\Omega_c/\Omega_b = 1/\epsilon\): conditional on transverse normalization, pinned reading, and per-defect bookkeeping (Part V, §20).
• \(a_0(\Omega_c/\Omega_b) = cH_0\): exact conditional identity independent of the cell value, current values give \(0.983 \pm 0.022\) (Part IV, Part V, §20).
• Post-fixation tests of \(\epsilon\): supported across heterogeneous tests; no combined significance assigned (§15, §18, §20, §27.3).
• Collective source-clock carrier loss / conditional conservation: exact no-stream, renewal, and volume-projection lemmas within stated premises; 3-D tracer surrogate completed; cosmological kernel open (Part V, §20, App. O.7–O.14).
• Diffuse source projection \(\epsilon = g_{\rm share,eff}/4\pi^2\): conditional, with no additional coefficient (Part V, §18).
• Hot-atmosphere factor \(B_{\rm bath}\): operationally closed; microscopic origin open (Part V, §18).
• Cluster residual \(\mathcal{R}_{\rm rel} = 1 + (W_{\rm bath} - 1)f_{\rm cont}\): hook morphology and trend direction supported; linear lift candidate excluded; lift function open (Part V, §18).
• Relaxed vs. merger source expressions: conditional — two regimes, one coefficient (Part V, §18).
• Channel-selection rule (suppression + collective lift): conditional — suppression open; linear lift excluded (Part V, §18).
• Bullet gas/lensing inversion: structurally supported; consistent but untested (Part V, §18).
• Resolved cluster lensing-map test: open — principal empirical task (Part V, §18).
• Bounded capacity \(q\), \(N^2 = q\): fixed static constitutive rule; covariant generalization open (Part VI, App. F, N).
• Former ADM multiplier action: excluded as a parent completion (Part VI, App. F, N).
• Spherical parent reduction: Closed in the metric-only branch (Part VI, App. F, N).
• Capacity-exhaustion horizon \(q = 0\): geometric zero closed in spherical symmetry; domain termination conditional (Part VI, App. F, N).
• Hawking temperature and exterior ringdown: GR-matching branch closed; capacity boundary condition open (App. F).
• Bekenstein–Hawking / channel-area bridge: channel-area normalization closed within the factorized \(j = 3\) embedding, \(\gamma_{\rm BI} = \ln 2/(6\pi)\), \(a_{\rm ch} = 4\ln 2 L_*^2\) (App. B, C, F, Q).
• Horizon formation and boundary microphysics: geometric marginal-surface formation closed; saturation dynamics open (Part VI, App. F).
• Rotating / charged stationary exteriors: baseline metric solutions closed; capacity map open (App. F).
• Many-Pasts Born compatibility: local marked kinematics derived; global measure conditional on stated record hypotheses (§3.3, §22, App. G, H.11).
• No-signaling in operational branch: exact within the operational branch (Part VI, App. G).
• Record-retention reading of Many-Pasts: closed as an ontological consequence of Postulate III (§3.3, §22, App. G.6).
• Arrow-of-time account: open conditional extension (Part VI, App. G).
• Microstructure Hamiltonian: finite marked-transfer action and capacity rigging lift closed; selected-branch Einstein theorem closed; host selection underdetermined (Part VI, §23, App. B, H, Q).
• Charged-lepton spectrum: fixed within the decorated marked-transfer and shell action; no fitted lepton coefficient, giving \(m_\mu/m_e = 206.768280237\) and \(m_\tau/m_e = 3477.343310\), both within 1σ (§13, App. H–I).
• Abelian / weak gauge hosting: coherent extension; Standard Model identification external (App. I.2).
• Persistent open-route color algebra: conditional derivation from one open-route premise, giving \(PU(3)\) with su(3) (App. I.2).
• Primitive color transfer \(t_8 = 13/14\): fixed within the minimal identity-kernel completion (App. H.9, I.2).
• Diamond causal color regulator: conditional on the minimal product causal stack and equal primitive heat times (App. I.2).
• Physical QCD matter sector: open and outside the present premises (App. I.2).
• Vacuum stiffness from the substrate: excluded, five branches (§24.3, App. J.4).
• Vacuum geometry in the lattice realization: open; externally hosted (§24.4, App. G, J.4).
• Capacity-decorated CDT phase / continuum tests: open, with explicit tests; finite-volume host behavior measured (§24.9, App. J.1, J.9, J.10).
• Ensemble fracture under local dynamics: verified explicit construction, 48 components of 35 states (§24.3, App. D.4, H.6, J.4).
• Substrate contribution to the host volume term: measured as a scaling family, demonstration volume, single point \(-62.2\) predicted, \(-62.8\) measured (§24.4, App. J.6).
• Admissibility weighting on dynamical geometry: finite-lattice compatibility measured, collision fraction from 0.654 to 0.077 with shuffled control at 0.736 (§24.5, App. J.5).
• Local response of geometry to persistent closure failure: demonstrated in two independent matched comparisons, first-shell separations 16.8σ and 9.2σ, third-shell coordination 4.0σ (§24.6, App. J.7).
• Long-range propagation of the strain field: conditional Ward theorem; finite-lattice source-to-field behavior measured; continuum junction open (§24.7–24.9, App. J.8–J.9).
• Numerical consistency checks: supportive audit layer (App. K, R).
• EFT consistency checklist: supportive audit layer (App. D).

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. The dynamics that drive \(\sigma_A \to 1\), 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 five direct checks. Rotation curves test the exponential transition and baryonic Tully–Fisher limit [1, 45]; resolved disks test the geometry corrections in the nonspherical Poisson equation; galaxy–galaxy lensing tests the no-slip metric; Cassini tests the carrier nesting rule; and dispersion-supported dwarfs test the boundary of the equilibrium carrier phase.

Stacked weak lensing first established a low-acceleration continuation and reported an early/late-type split [46]. 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 [47]. 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 \(Q_2 = (1.6 \pm 1.8) \times 10^{-27}\) s⁻²; current universal total-field implementations are in 3–15σ tension with that result [38]. Wide-binary analyses remain statistically divided [35, 36, 37]. A confirmed order-unity binary boost would reject carrier nesting as stated, while a significant Solar quadrupole correlated with the Galactic field would reject its local projection rule.

At lower baryonic mass, the GravSphere dwarf sample of Júlio et al. [39] 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.

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 [4, 16, 17]—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 [2] 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 [51], distance-ladder determinations ranging from \(\simeq 70\) [53] to 73 [52], and a tension whose proposed resolutions are reviewed in Di Valentino et al. [54].

The relation \(a_0(z) = cH(z)g_{\rm share,eff}/(4\pi^2)\) predicts measurable redshift evolution. For a Planck-like background, \(H(2.3)/H_0 \simeq 3.47\). A disk with \(g_{\rm bar} \simeq 2 \times 10^{-10}\) m s⁻² then has \(g_{\rm obs}/g_{\rm bar} \simeq 2.02\), compared with 1.39 for an epoch-independent scale. Observed outer rotation curves at these redshifts indicate strong baryon dominance [40, 44], 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 has separate continuum requirements, now with no free volume normalization. The capacity-decorated trajectory must approach a common-parent critical limit, show finite limiting \(\alpha\) and simplex ratio \(\xi\), set \(a\) by a target-blind observable, and recover the exact two-simplex volume source after mixed-eigenoperator projection. A nonzero source-independent capacity bulk coefficient after vacuum normalization would directly falsify the regulator identity. A continuum completion that still requires an independent host cosmological action would instead falsify the Postulate I common-substrate identification. Only if both conditions hold does the conditional theorem \(\Lambda^{\rm empty\, equilibrium}_R = 0\) apply. The value or constancy of \(w\) tests whatever state-dependent or nonequilibrium cosmological dynamics is eventually supplied; it does not test the exact equilibrium partition identity by itself.

A MUSE sample of 79 star-forming galaxies at \(0.33 < z < 1.44\) finds a radial-acceleration scale that rises with redshift [42]; a resolved low-redshift H i sample tentatively reports the same direction [43]. An epoch-independent \(a_0\) predicts no such evolution. The comparison is already quantitative. The reported scale at \(z \sim 1\), \(a_0 = 2.38^{+0.12}_{-0.10} \times 10^{-10}\) 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 decisive 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 \(I_{\rm rel}\) in the narrow background-compatible region 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: an independent cosmological zoom fails the collective trigger if it finds substantial minihalo-era loss, a slow full-halo-like kernel, or excessive early mixing that overshoots the background-compatible release integral. A background fit of the frozen hydro history 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 imports ordinary quantum instruments, so it predicts no laboratory departure from the Born rule or no-signaling. Its discriminating burdens are theoretical and now separable. The substrate must reproduce the decoherence functional rather than assume it; the retained \((P, \Phi)\) sector must derive the symplectic and Fisher-compatible metric structures needed for the Caticha Hamilton–Killing bridge; and a conditional-typicality calculation must suppress high-entropy-past and Boltzmann-fluctuation histories. Failure of the first two would remove the proposed microscopic quantum completion, while failure of the third would remove the proposed arrow; none would constitute a new experimental violation of standard quantum mechanics.

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, 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 \(q_{\rm cap}\) does not track \(q_{\rm geo}\) 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 sevenfold 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 [18, 59, 21] 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 [22, 58, 29, 30], 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 [20] 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 [31], 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 [56]. 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 is the decoherent-histories formalism: 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. Born statistics and no-signaling are inherited from the quantum instruments; a substrate derivation of the decoherence functional remains 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 [62, 63, 64]. 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

$$\mathcal{H}_{\rm cell} = \mathcal{H}_{\rm geom} \otimes \mathcal{H}_{\rm cap} \otimes \mathbb{C}^2_{\rm orient}.$$

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 [99, 100]. 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 [97, 98]. The area-Regge continuum calculation gives the same leading graviton dynamics and an \(O(a^2 C^2)\) correction [101]. Appendix H.10 turns that selected-branch symbol into the strong-resolvent theorem \((H_{\rm TT})\) 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 \((H_{\rm phase})\) 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 [80, 81]. 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 [24, 23, 25]. 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

$$S_\infty = \frac{A_{\rm dS}}{4L_*^2} = \frac{c^3 A_{\rm dS}}{4\hbar G_*}.$$

The observer-dressed de Sitter construction uses this area coefficient and gives \(S_{\rm max} - S_{\rm gen}(E) = \beta_{\rm dS}E + O(E^2)\) for a small central excitation. The horizon modular calculation uses \(S_{\rm rel} = c^3\delta A/(4\hbar G_*)\) 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 now fixes the microscopic channel-to-area normalization within the factorized coherent \(j = 3\) embedding, so the former area-normalization rider is removed; 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 \((H_{\rm TT})\). 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 \((H_{\rm phase})\). The distinct canonical completion retains \((H'_{\rm can})\).

The cosmological scaling also agrees at the level of form. With \(R_A = c/H\),

$$S_\infty = \frac{\pi c^2}{H^2 L_*^2}, \quad S_{\rm GfE} \simeq \frac{\bar\omega_{[1]}c^2}{\ell_P^4 H^2}, \quad \frac{S_{\rm GfE}}{S_\infty} \simeq \frac{\bar\omega_{[1]}L_*^2}{\pi\ell_P^4}.$$

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 theory derives general relativity as the medium's low-energy geometry. 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; 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 now 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. 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 \(+0.481\sigma\). 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 \(q_{\rm geo}\) 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 16.8σ and 9.2σ and a third-shell coordination separation of 4.0σ 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 joint measure also closes a sharper equilibrium-vacuum statement. Normalizing by the complete homogeneous capacity free energy at each regulator makes the theory invariant under arbitrary extensive shifts of microscopic energy origin, and the uncentered control verifies the predicted \(N_{41}\) coupling displacement. Conditional on the decorated common-parent continuum limit, the fixed primitive event measure removes an independent host volume action and gives zero cosmological term for the empty equilibrium branch. Appendix O now supplies a more restrictive subsequent dynamics. The fixed-event history space has a unique volume-conjugate singlet, so only the homogeneous projection of the universal marked drain changes \(\Lambda\); a local \(\Lambda(x)\) branch would require additional cell-by-cell volume couplings absent from the construction. Collective baryonic loss of progenitor identity triggers complete renewal, but it does not create a new material or scalar component. The metric-visible remainder stays in source-associated capacity of the retained coarse carrier, while the orthogonal marked-history output has no independent leading local stress. The absorbing first-entry excursion-set construction fixes the assembly history and the three-dimensional passive-tracer surrogate supplies an explicit delayed mutual-information kernel. For nonzero spatial modes, release-hazard fluctuations only redistribute the internal source labels and cancel from the leading total metric source, leaving \(\delta\rho_\Lambda = 0\) and no new short-scale gradient force. The remaining astrophysical input is therefore sharply isolated: a self-gravitating cosmological progenitor-tagged zoom must determine \(K_{\rm mix}(\tau; M, z)\) independently, after which the background and perturbation likelihood contains no fitted release amplitude or bias parameter.

Three independent continuum constructions support different parts of the proposed ontology. The type II₁ 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 [24, 23, 25]. 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 \((H_{\rm TT})\). 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 \((H_{\rm phase})\) 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: no-signaling follows from standard quantum instruments, and the Born form is the unique normalized noncontextual additive measure on a sufficiently rich record-projector lattice. The local marked kinematics are no longer wholly imported. The retained \(Q\) complement inherits the displayed Lorentzian action; its transport and 6:5:3 phase splitting generate su(9) on a real canonical phase space, select a unique positive complex structure, and yield \(\mathbb{C}^9/\mathbb{CP}^8\). The fresh-state response metric agrees exactly with one quarter of the Fisher metric, and a projected neighboring bridge supplies the finite entangling algebra before the final host projection. The full continuum/Fock space, global decoherence functional, initial state, and record completeness remain premises. 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 complementary-channel calculation shows how renewal can coexist with interference: the marked position fiber remains in the coherent system, and the discarded free-renewal record carries no path label. 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 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.

The remaining work is specific. 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 \(L_{\rm block} \sim 3.3\text{–}5.3L_*\) bracket now has a computed representative, \(L_{\rm block} \simeq 3.4L_*\) 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 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, the remaining theory work is a microscopic derivation of 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. Resolved nonspherical SPARC and 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.

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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, \(\delta S = S_\infty - S_{\rm ent}\).
• 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 \(\kappa/\gamma = 3L_*/(4G_{\rm tet}(0)\kappa_m(L_*))\) are simultaneously exact (Appendix C.4–C.5). In the decorated electron-anchored marked-transfer branch,

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

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

$$S_{\rm ent}(x),$$

with vacuum baseline \(S_\infty\) and deficit

$$\delta S(x) = S_\infty - S_{\rm ent}(x).$$

For nonlinear work the bounded occupancy fraction is

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

The absolute entropy unit is fixed only after choosing a cell or horizon normalization. Under a constant rescaling of \(S_{\rm ent}\), 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

$$\gamma : \text{entanglement-field stiffness}, \tag{10}$$ $$\kappa : \text{continuum defect–entropy coupling}, \tag{11}$$ $$\kappa_m(\ell) : \text{mass-per-entropy map at scale } \ell, \tag{12}$$ $$\zeta_* \equiv 9e^{-g_{\rm share,eff}}\left(1 - \frac{8\eta_*}{49}\right)^{21/2}, \tag{13}$$ $$Z_e \equiv (1 + \zeta_*)(1 + 7\zeta_*^2), \tag{14}$$ $$L_* \equiv -\frac{3}{2}Z_e\lambda_e \ln\left(1 - e^{-7g_{\rm share,eff}}\right) \text{ in the decorated marked-transfer branch}, \tag{15}$$ $$G_* = \frac{c^3 L_*^2}{\hbar} = \frac{9}{4}\frac{\hbar c}{m_e^2}Z_e^2 \ln^2\left(1 - e^{-7g_{\rm share,eff}}\right), \tag{16}$$ $$G_{\rm tet}(0) : \text{tetrahedral on-site Green constant}, \tag{17}$$ $$g_{\rm share,max} = \ln(1680), \tag{18}$$ $$g_{\rm share,eff} : \text{admissibility-weighted sharing entropy}, \tag{19}$$ $$J_{\rm bare}, J^{\rm tree}_{\rm eff}, J^{(\rm ren)}_{\rm eff} : \text{UV edge couplings}, \tag{20}$$ $$a_0 = \frac{cH_0 g_{\rm share,eff}}{4\pi^2}. \tag{21}$$

The canonical weak-field bridge and Newton closure are

$$\frac{\Phi}{c^2} = -\frac{\delta S}{2S_\infty}, \quad \frac{\kappa}{\gamma} = \frac{3L_*}{4G_{\rm tet}(0)\kappa_m(L_*)}, \quad G = \frac{c^2\kappa}{8\pi\gamma S_\infty}.$$

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 \(g_{\rm share,eff}\), the scalar variable is always the vacuum-relative field \(S_{\rm ent}\) or its deficit \(\delta S\), and the weak-field bridge is used in the single form stated above. The principal extension-sector symbols are:

$$\epsilon \equiv \frac{g_{\rm share,eff}}{4\pi^2}, \quad \nu(x) \equiv \frac{1}{1 - e^{-x}}, \quad x \equiv \frac{E_\perp}{k_B T_H},$$

with \(W_{\rm bath}\) the cluster bath source weight, \(B_{\rm bath}\) the measured bath-development fraction, \(D\) the capacity diffusivity, \(\tau_0\) the transport relaxation time, \(D/\tau_0 = c^2\), \(\sigma_* \equiv \pi/g_{\rm share,eff}\), and \(a_{\rm UV} \equiv 1/\text{Var}_{\eta_*}(K^2)\). 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 \(Q_\perp(z_P)\). 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.

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[The remaining technical appendices B through R and the reference list follow the same structure of derivations, tables, and closure grades detailed above. The full reference list is reproduced below.]

References

1. S. S. McGaugh, F. Lelli, and J. M. Schombert, "Radial acceleration relation in rotationally supported galaxies," Physical Review Letters 117, 201101 (2016).
2. D. Clowe et al., "A direct empirical proof of the existence of dark matter," The Astrophysical Journal Letters 648, L109–L113 (2006).
3. B. Bertotti, L. Iess, and P. Tortora, "A test of general relativity using radio links with the Cassini spacecraft," Nature 425, 374–376 (2003).
4. Y. Tian et al., "The radial acceleration relation in CLASH galaxy clusters," arXiv:2001.08340 (2020).
5. A. J. Guttmann, "Lattice Green functions in all dimensions," Journal of Physics A: Mathematical and Theoretical 43, 305205 (2010).
6. K. G. Wilson, "Confinement of quarks," Physical Review D 10, 2445–2459 (1974).
7. G. S. Joyce, "On the cubic lattice Green functions," Proceedings of the Royal Society of London A 445, 463–477 (1994).
8. R. Arnowitt, S. Deser, and C. W. Misner, "The dynamics of general relativity," in Gravitation: An Introduction to Current Research, ed. L. Witten (Wiley, New York, 1962).
9. J. D. Bekenstein, "Black holes and entropy," Physical Review D 7, 2333–2346 (1973).
10. S. W. Hawking, "Particle creation by black holes," Communications in Mathematical Physics 43, 199–220 (1975).
11. G. W. Gibbons and S. W. Hawking, "Action integrals and partition functions in quantum gravity," Physical Review D 15, 2752–2756 (1977).
12. T. Regge and J. A. Wheeler, "Stability of a Schwarzschild singularity," Physical Review 108, 1063–1069 (1957).
13. F. J. Zerilli, "Effective potential for even-parity Regge–Wheeler gravitational perturbation equations," Physical Review Letters 24, 737–738 (1970).
14. A. J. R. Sanderson, T. J. Ponman, A. Finoguenov, E. J. Lloyd-Davies, and M. Markevitch, "The Birmingham–CfA cluster scaling project — I. Gas fraction and the M–T_X relation," Monthly Notices of the Royal Astronomical Society 340, 989–1010 (2003).
15. A. Vikhlinin, A. Kravtsov, W. Forman, C. Jones, M. Markevitch, S. S. Murray, and L. Van Speybroeck, "Chandra sample of nearby relaxed galaxy clusters: mass, gas fraction, and mass–temperature relation," The Astrophysical Journal 640, 691–709 (2006).
16. D. Eckert, S. Ettori, E. Pointecouteau, R. F. J. van der Burg, and S. I. Loubser, "The gravitational field of X-COP galaxy clusters," Astronomy & Astrophysics 662, A123 (2022).
17. P. Li, Y. Tian, M. P. Júlio, M. S. Pawlowski, F. Lelli, S. S. McGaugh, J. M. Schombert, J. I. Read, P.-C. Yu, and C.-M. Ko, "Measuring galaxy cluster mass profiles into the low-acceleration regime with galaxy kinematics," Astronomy & Astrophysics 677, A24 (2023).
18. M. Milgrom, "A modification of the Newtonian dynamics as a possible alternative to the hidden mass hypothesis," Astrophysical Journal 270, 365 (1983).
19. M. Milgrom, "Quasi-linear formulation of MOND," Monthly Notices of the Royal Astronomical Society 403, 886 (2010), doi:10.1111/j.1365-2966.2009.16184.x.
20. J. D. Bekenstein, "Relativistic gravitation theory for the modified Newtonian dynamics paradigm," Physical Review D 70, 083509 (2004).
21. B. Famaey and S. S. McGaugh, "Modified Newtonian dynamics (MOND): observational phenomenology and relativistic extensions," Living Reviews in Relativity 15, 10 (2012).
22. T. Jacobson, "Thermodynamics of spacetime: the Einstein equation of state," Physical Review Letters 75, 1260 (1995).
23. P. Dorau and A. Much, "From quantum relative entropy to the semiclassical Einstein equations," Physical Review Letters 136, 091602 (2026).
24. V. Chandrasekaran, R. Longo, G. Penington, and E. Witten, "An algebra of observables for de Sitter space," Journal of High Energy Physics 2023(2), 082 (2023), doi:10.1007/JHEP02(2023)082.
25. G. Bianconi, "Thermodynamics of the gravity from entropy theory," Physical Review D 114, 024042 (2026), doi:10.1103/26kn-thgp.
26. A. Caticha, "Quantum mechanics as Hamilton–Killing flows on a statistical manifold," Physical Sciences Forum 3, 12 (2021), doi:10.3390/psf2021003012.
27. A. Caticha, "Entropic dynamics approach to quantum electrodynamics," Entropy 27, 1247 (2025), doi:10.3390/e27121247.
28. A. Tonomura, J. Endo, T. Matsuda, T. Kawasaki, and H. Ezawa, "Demonstration of single-electron buildup of an interference pattern," American Journal of Physics 57, 117–120 (1989), doi:10.1119/1.16104.
29. E. Verlinde, "On the origin of gravity and the laws of Newton," Journal of High Energy Physics 2011(4), 29 (2011).
30. E. Verlinde, "Emergent gravity and the dark universe," SciPost Physics 2, 016 (2017).
31. C. Skordis and T. Złośnik, "New relativistic theory for modified Newtonian dynamics," Physical Review Letters 127, 161302 (2021).
32. J. Collins, A. Perez, D. Sudarsky, L. Urrutia, and H. Vucetich, "Lorentz invariance and quantum gravity: an additional fine-tuning problem?," Physical Review Letters 93, 191301 (2004).
33. L. Bombelli, J. Lee, D. Meyer, and R. D. Sorkin, "Space-time as a causal set," Physical Review Letters 59, 521 (1987).
34. F. Dowker, J. Henson, and R. D. Sorkin, "Quantum gravity phenomenology, Lorentz invariance and discreteness," Modern Physics Letters A 19, 1829 (2004).
35. C. Pittordis, W. Sutherland, and P. Shepherd, "Wide binaries from Gaia DR3: testing GR versus MOND with realistic triple modelling," The Open Journal of Astrophysics 8 (2025), doi:10.33232/001c.142887.
36. K.-H. Chae, "Low-acceleration gravitational anomaly from Bayesian 3D modeling of wide binary orbits: methodology and results with Gaia Data Release 3," Astrophysical Journal 985, 210 (2025), doi:10.3847/1538-4357/adce09.
37. K.-H. Chae and Y. Yoon, "Revisiting data quality control and multiple-star modeling in wide binary gravity tests: confirmation of MOND-type gravitational anomaly at low acceleration," arXiv:2607.14450 (2026).
38. R. S. Park, A. Hees, B. Famaey, H. Desmond, and A. Durakovic, "Improved constraints on modified Newtonian gravity from Cassini radio tracking data," Physical Review D 114, 024066 (2026), doi:10.1103/r7n8-kw38.
39. M. P. Júlio, J. I. Read, M. S. Pawlowski, P. Li, D. Vaz, J. Brinchmann, M. P. Rey, O. Agertz, and T. Holmes, "The radial acceleration relation at the EDGE of galaxy formation: testing its universality in low-mass dwarf galaxies," Astronomy & Astrophysics 704, A330 (2025), doi:10.1051/0004-6361/202557106.
40. R. Genzel et al., "Strongly baryon-dominated disk galaxies at the peak of galaxy formation ten billion years ago," Nature 543, 397 (2017).
41. H. Prince and J. Dunkley, "Data compression in cosmology: A compressed likelihood for Planck data," Physical Review D 100, 083502 (2019).
42. B. I. Ciocan, N. F. Bouché, J. Fensch, D. Krajnović, J. Freundlich, H. Desmond, B. Famaey, and R. Techi, "MUSE-DARK III: The evolution of the radial acceleration relation at intermediate redshifts," Astronomy & Astrophysics 709, L16 (2026).
43. A. A. Vărăşteanu, M. J. Jarvis, A. A. Ponomareva, H. Desmond, I. Heywood, T. Yasin, N. Maddox, M. Glowacki, M. Maksymowicz-Maciata, P. E. M. Mancera Piña, and H. Pan, "MIGHTEE-HI: The radial acceleration relation with resolved stellar mass measurements," Monthly Notices of the Royal Astronomical Society 541, 2366 (2025).
44. P. Lang et al., "Falling rotation curves of star-forming galaxies at 0.6 < z < 2.6," Astrophysical Journal 840, 92 (2017).
45. F. Lelli, S. S. McGaugh, and J. M. Schombert, "SPARC: mass models for 175 disk galaxies with Spitzer photometry and accurate rotation curves," Astronomical Journal 152, 157 (2016).
46. M. M. Brouwer et al., "The weak lensing radial acceleration relation: measuring the dark matter law with KiDS-1000," Astronomy & Astrophysics 650, A113 (2021).
47. T. Mistele, S. S. McGaugh, F. Lelli, J. M. Schombert, and P. Li, "Radial acceleration relation of galaxies with joint kinematic and weak-lensing data," Journal of Cosmology and Astroparticle Physics 2024(04), 020 (2024), doi:10.1088/1475-7516/2024/04/020.
48. W. G. Unruh, "Notes on black-hole evaporation," Physical Review D 14, 870 (1976).
49. R. B. Tully and J. R. Fisher, "A new method of determining distances to galaxies," Astronomy & Astrophysics 54, 661 (1977).
50. S. S. McGaugh, J. M. Schombert, G. D. Bothun, and W. J. G. de Blok, "The baryonic Tully–Fisher relation," Astrophysical Journal Letters 533, L99 (2000).
51. Planck Collaboration, "Planck 2018 results. VI. Cosmological parameters," Astronomy & Astrophysics 641, A6 (2020).
52. A. G. Riess et al., "A comprehensive measurement of the local value of the Hubble constant with 1 km s⁻¹ Mpc⁻¹ uncertainty from the Hubble Space Telescope and the SH0ES team," Astrophysical Journal Letters 934, L7 (2022).
53. W. L. Freedman, B. F. Madore, I. S. Jang, et al., "Status report on the Chicago–Carnegie Hubble Program (CCHP): measurement of the Hubble constant using the Hubble and James Webb Space Telescopes," The Astrophysical Journal (2025), doi:10.3847/1538-4357/adce78.
54. E. Di Valentino et al., "In the realm of the Hubble tension — a review of solutions," Classical and Quantum Gravity 38, 153001 (2021).
55. C. M. Will, "The confrontation between general relativity and experiment," Living Reviews in Relativity 17, 4 (2014).
56. C. Brans and R. H. Dicke, "Mach's principle and a relativistic theory of gravitation," Physical Review 124, 925 (1961).
57. E. G. Adelberger, B. R. Heckel, and A. E. Nelson, "Tests of the gravitational inverse-square law," Annual Review of Nuclear and Particle Science 53, 77 (2003).
58. T. Padmanabhan, "Thermodynamical aspects of gravity: new insights," Reports on Progress in Physics 73, 046901 (2010).
59. R. H. Sanders and S. S. McGaugh, "Modified Newtonian dynamics as an alternative to dark matter," Annual Review of Astronomy and Astrophysics 40, 263 (2002).
60. D. Oriti, "Group field theory as the second quantization of loop quantum gravity," Classical and Quantum Gravity 33, 085005 (2016).
61. S. Gielen, D. Oriti, and L. Sindoni, "Cosmology from group field theory formalism for quantum gravity," Physical Review Letters 111, 031301 (2013).
62. A. Barbieri, "Quantum tetrahedra and simplicial spin networks," Nuclear Physics B 518, 714 (1998).
63. J. C. Baez and J. W. Barrett, "The quantum tetrahedron in 3 and 4 dimensions," Advances in Theoretical and Mathematical Physics 3, 815 (1999).
64. C. Rovelli and L. Smolin, "Discreteness of area and volume in quantum gravity," Nuclear Physics B 442, 593 (1995).
65. A. M. Gleason, "Measures on the closed subspaces of a Hilbert space," Journal of Mathematics and Mechanics 6, 885–893 (1957).
66. J. B. Hartle, "Decoherent histories quantum mechanics starting with records of what happens," arXiv:1608.04145 (2016).
67. E. T. Jaynes, "The minimum entropy production principle," Annual Review of Physical Chemistry 31, 579–601 (1980), doi:10.1146/annurev.pc.31.100180.003051.
68. W. F. Stinespring, "Positive functions on C-algebras," Proceedings of the American Mathematical Society 6*, 211–216 (1955), doi:10.1090/S0002-9939-1955-0069403-4.
69. A. D. Wyner and J. Ziv, "Some asymptotic properties of the entropy of a stationary ergodic data source with applications to data compression," IEEE Transactions on Information Theory 35, 1250–1258 (1989).
70. D. S. Ornstein and B. Weiss, "Entropy and data compression schemes," IEEE Transactions on Information Theory 39, 78–83 (1993).
71. J. J. Halliwell, "A review of the decoherent histories approach to quantum mechanics," arXiv:gr-qc/9407040 (1994).
72. D. Eckert et al., "Non-thermal pressure support in X-COP galaxy clusters," Astronomy & Astrophysics 621, A40 (2019).
73. S. Dupourqué, N. Clerc, E. Pointecouteau, D. Eckert, S. Ettori, and F. Vazza, "Investigating the turbulent hot gas in X-COP galaxy clusters," Astronomy & Astrophysics 673, A91 (2023).
74. XRISM Collaboration, "XRISM reveals low nonthermal pressure in the core of the hot, relaxed galaxy cluster Abell 2029," Astrophysical Journal Letters 982, L5 (2025).
75. A. H. Chamseddine and V. Mukhanov, "Mimetic dark matter," Journal of High Energy Physics 1311, 135 (2013).
76. A. Lewis, A. Challinor, and A. Lasenby, "Efficient computation of cosmic microwave background anisotropies in closed Friedmann–Robertson–Walker models," Astrophysical Journal 538, 473 (2000).
77. S. More, B. Diemer, and A. Kravtsov, "The splashback radius as a physical halo boundary and the growth of halo mass," Astrophysical Journal 810, 36 (2015).
78. G. E. Volovik, "Cosmological constant and vacuum energy," Annalen der Physik 14, 165–176 (2005).
79. T. Regge, "General relativity without coordinates," Il Nuovo Cimento 19, 558–571 (1961).
80. J. Ambjørn, J. Jurkiewicz, and R. Loll, "Emergence of a 4D world from causal quantum gravity," Physical Review Letters 93, 131301 (2004).
81. J. Ambjørn, A. Görlich, J. Jurkiewicz, and R. Loll, "Nonperturbative quantum gravity," Physics Reports 519, 127–210 (2012).
82. J. Ambjørn, J. Jurkiewicz, and R. Loll, "Dynamically triangulating Lorentzian quantum gravity," Nuclear Physics B 610, 347–382 (2001).
83. J. Ambjørn, J. Gizbert-Studnicki, A. Görlich, J. Jurkiewicz, N. Klitgaard, and R. Loll, "Characteristics of the new phase in CDT," European Physical Journal C 77, 152 (2017).
84. J. Ambjørn, A. Görlich, J. Jurkiewicz, and R. Loll, "The nonperturbative quantum de Sitter universe," Physical Review D 78, 063544 (2008).
85. A. A. Thoul and D. H. Weinberg, "Hydrodynamic simulations of galaxy formation. II. Photoionization and the formation of low-mass galaxies," Astrophysical Journal 465, 608 (1996).
86. T. Okamoto, L. Gao, and T. Theuns, "Mass loss of galaxies due to an ultraviolet background," Monthly Notices of the Royal Astronomical Society 390, 920–928 (2008).
87. J. Ambjørn, S. Jordan, J. Jurkiewicz, and R. Loll, "A second-order phase transition in CDT," Physical Review Letters 107, 211303 (2011).
88. J. Ambjørn, D. Coumbe, J. Gizbert-Studnicki, A. Görlich, and J. Jurkiewicz, "Critical phenomena in causal dynamical triangulations," Classical and Quantum Gravity 36, 224001 (2019).
89. A. Perez, "The spin-foam approach to quantum gravity," Living Reviews in Relativity 16, 3 (2013).
90. M. Han, "Semiclassical behavior of spinfoam amplitude with small spins and entanglement entropy," Physical Review D 100, 084049 (2019).
91. M. Han, "Summation and triangulation independence of Lorentzian spinfoam amplitudes for all LQG," Physical Review D 113, 084034 (2026), doi:10.1103/n76f-31gf.
92. T. Thiemann and A. Zipfel, "Linking covariant and canonical LQG II: spin foam projector," Classical and Quantum Gravity 31, 125008 (2014).
93. A. Calcinari and S. Gielen, "Relational dynamics and Page–Wootters formalism in group field theory," Quantum 9, 1610 (2025).
94. M. Han and Y. Ma, "Master constraint operators in loop quantum gravity," Physics Letters B 635, 225–231 (2006).
95. M. Han and T. Thiemann, "On the relation between rigging inner product and master constraint direct integral decomposition," arXiv:0911.3431 (2010).
96. W. J. Fairbairn and C. Meusburger, "Quantum deformation of two four-dimensional spin foam models," Journal of Mathematical Physics 53, 022501 (2012).
97. M. Han, "Einstein equation from covariant loop quantum gravity in semiclassical continuum limit," Physical Review D 96, 024047 (2017).
98. M. Han, Z. Huang, and A. Zipfel, "Emergent four-dimensional linearized gravity from a spin foam model," Physical Review D 100, 024060 (2019).
99. J. Engle, "Proposed proper Engle–Pereira–Rovelli–Livine vertex amplitude," Physical Review D 87, 084048 (2013).
100. J. Engle, I. Vilensky, and A. Zipfel, "The Lorentzian proper vertex amplitude: Asymptotics," Physical Review D 94, 064025 (2016).
101. B. Dittrich and A. Kogios, "From spin foams to area metric dynamics to gravitons," Classical and Quantum Gravity 40, 095011 (2023).
102. P. A. Höhn, "Canonical linearized Regge calculus: Counting lattice gravitons with Pachner moves," Physical Review D 91, 124034 (2015).
103. M. Roček and R. M. Williams, "Quantum Regge calculus," Physics Letters B 104, 31–37 (1981).
104. B. Dittrich, W. Kamiński, and S. Steinhaus, "Discretization independence implies nonlocality in 4D discrete quantum gravity," Classical and Quantum Gravity 31, 245009 (2014).
105. D. Lovelock, "The Einstein tensor and its generalizations," Journal of Mathematical Physics 12, 498–501 (1971).
106. S. Deser, "Self-interaction and gauge invariance," General Relativity and Gravitation 1, 9–18 (1970).
107. J. Borissova, B. Dittrich, A. Eichhorn, and M. Schiffer, "Renormalization group flows in area-metric gravity," arXiv:2507.02034 [gr-qc] (2025).
108. M. Bruno, E. Colafranceschi, F. M. Mele, and C. Rovelli, "The structure of the continuum limit of spin foams," arXiv:2603.16999 [gr-qc] (2026).
109. J. Engle, E. Livine, R. Pereira, and C. Rovelli, "LQG vertex with finite Immirzi parameter," Nuclear Physics B 799, 136–149 (2008).
110. F. Gozzini, "A high-performance code for EPRL spin foam amplitudes," Classical and Quantum Gravity 38, 225010 (2021).
111. W. G. Unruh, "Unimodular theory of canonical quantum gravity," Physical Review D 40, 1048 (1989).
112. M. Henneaux and C. Teitelboim, "The cosmological constant and general covariance," Physics Letters B 222, 195 (1989).
113. S. Weinberg, "The cosmological constant problem," Reviews of Modern Physics 61, 1 (1989).
114. T. Josset, A. Perez, and D. Sudarsky, "Dark energy from violation of energy conservation," Physical Review Letters 118, 021102 (2017).
115. A. Perez and D. Sudarsky, "Dark energy from quantum gravity discreteness," Physical Review Letters 122, 221302 (2019).
116. M. Ahmed, S. Dodelson, P. B. Greene, and R. D. Sorkin, "Everpresent Λ," Physical Review D 69, 103523 (2004).
117. S. Das, A. Nasiri, and Y. K. Yazdi, "Everpresent Λ. Part II. Structural stability," Journal of Cosmology and Astroparticle Physics 10, 047 (2023).
118. J. D. Barrow, "A strong constraint on ever-present lambda," Physical Review D 75, 067301 (2007).
119. S. D. H. Hsu, "Entropy bounds and dark energy," Physics Letters B 594, 13 (2004).
120. M. Li, "A model of holographic dark energy," Physics Letters B 603, 1 (2004).
121. DESI Collaboration, "DESI DR2 results II: Measurements of baryon acoustic oscillations and cosmological constraints," arXiv:2503.14738 (2025).
122. D. Brout et al., "The Pantheon+ analysis: Cosmological constraints," The Astrophysical Journal 938, 110 (2022).
123. W. H. Press and P. Schechter, "Formation of galaxies and clusters of galaxies by self-similar gravitational condensation," The Astrophysical Journal 187, 425 (1974).
124. A. G. Doroshkevich, "Spatial structure of perturbations and origin of galactic rotation in fluctuation theory," Astrophysics 6, 320 (1970).
125. D. J. Eisenstein and W. Hu, "Baryonic features in the matter transfer function," The Astrophysical Journal 496, 605 (1998).
126. A. N. Ormondroyd, W. J. Handley, M. P. Hobson, A. N. Lasenby, and D. Yallup, "Dynamic or systematic? Bayesian model selection between dark energy and supernova biases," Monthly Notices of the Royal Astronomical Society 548, stag615 (2026).
127. DES Collaboration, "Dark Energy Survey supernova program: A reanalysis of cosmology results and evidence for evolving dark energy with an updated Type Ia supernova calibration," Monthly Notices of the Royal Astronomical Society 548, stag632 (2026).
128. G. F. Lesci, C. Giocoli, F. Marulli, M. Romanello, L. Moscardini, et al., "AMICO galaxy clusters in KiDS-1000: Splashback radius from weak lensing and cluster–galaxy correlation function," Astronomy & Astrophysics 710, A269 (2026).
129. C. Hidalgo, Why Information Grows: The Evolution of Order, from Atoms to Economies (Basic Books, New York, 2015).
130. R. Landauer, "Irreversibility and heat generation in the computing process," IBM Journal of Research and Development 5, 183–191 (1961).
131. G. L. Bryan and M. L. Norman, "Statistical properties of X-ray clusters: Analytic and numerical comparisons," The Astrophysical Journal 495, 80 (1998).
132. T. Shin et al., "Measurement of the splashback feature around SZ-selected galaxy clusters with DES, SPT and ACT," Monthly Notices of the Royal Astronomical Society 487, 2900 (2019).
133. D. Rana et al., "The eROSITA Final Equatorial-Depth Survey (eFEDS): Splashback radius of X-ray galaxy clusters using galaxies from HSC survey," Monthly Notices of the Royal Astronomical Society 522, 4181 (2023).
134. D. Rubin et al., "Union through UNITY: Cosmology with 2,000 SNe using a unified Bayesian framework," arXiv:2311.12098 (2023).
135. T. H. Greif, J. L. Johnson, R. S. Klessen, and V. Bromm, "The first galaxies: assembly, cooling and the onset of turbulence," Monthly Notices of the Royal Astronomical Society 387, 1021–1036 (2008).
136. R. Barkana and A. Loeb, "The photoevaporation of dwarf galaxies during reionization," The Astrophysical Journal 523, 54–65 (1999).
137. A. P. Ji, A. Frebel, and V. Bromm, "Preserving chemical signatures of primordial star formation in the first low-mass stars," Monthly Notices of the Royal Astronomical Society 454, 659–674 (2015).
138. B. F. Griffen, G. A. Dooley, A. P. Ji, B. W. O'Shea, F. A. Gomez, and A. Frebel, "Tracing the first stars and galaxies of the Milky Way," Monthly Notices of the Royal Astronomical Society 474, 443–459 (2018).

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