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

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

Jacob Chinitz

May 16, 2026

Abstract

We propose that the vacuum is not an inert stage but a finite-capacity entanglement substrate, and that matter is the localized depletion of that capacity. On this view, inertial mass is the entanglement content of a defect, gravity is the long-wavelength relaxation of the surrounding substrate, and the galactic excess usually attributed to dark matter is the extended reach of that relaxation rather than a separate particulate component. The main result is a closed conditional derivation of the static weak-field sector. A minimal tetrahedral boundary ensemble, together with a one-bit fermionic defect anchor, turns the ultraviolet problem into a finite counting problem. Admissibility weighting, edge transport, finite-loop dressing, continuum matching, and source projection then determine the coefficients of a scalar entanglement EFT. In the resulting weak-field limit, Newtonian gravity, the galactic acceleration scale, the radial-acceleration relation, and leading no-slip lensing structure arise from one coefficient chain, without per-galaxy interpolation functions or phenomenological tuning. The same construction also gives a non-gravitational route to the substrate length scale by treating the electron as the lightest elementary defect. The induced gravitational scale agrees with the observed Newton constant to about one percent, providing a quantitative cross-check on the matched weak-field normalization. The proposal is therefore not simply a scalar addition to general relativity: the metric field and the entanglement scalar are two continuum expressions of the same finite-capacity medium. Time-dependent transport, cosmology, strong-field black-hole physics, and the Many- Pasts interpretation are developed as further consequences of this ontology. These sectors are not presented with equal closure. The static weak-field derivation is the central completed result; the cosmological, strong-field, and microscopic boundary sectors remain conditional or frontier completions where explicitly stated.

1. Introduction: The Physical Claim 4 1.1 What Is Primitive, and What Is Closed . . . . . . . . . . . . . . . . . . . . . . . . 5

2. Canonical Field Content and Definitions 6

3. The Three Postulates 8 3.1 Information–Geometry Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.2 Mass–Entropy Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.3 Many-Pasts Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

4. Relativistic Continuum Structure 9 4.1 Capacity budget and continuum symmetry . . . . . . . . . . . . . . . . . . . . . . 9 4.2 Dependency Map of the Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

5. Why a Tetrahedral Boundary Ensemble 10

6. Admissibility Closure 11 6.1 Minimal isotropic kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 6.2 Closure condition and uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 6.3 Effective sharing entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

7. Edge Kernel and Tree-Level Coupling 12

8. Finite-Loop Renormalization 13

9. Continuum Stiffness and SI Normalization 14

10. Covariant Action 15

11. Field Equations and Bridge Law 17

12. Newtonian Gravity and the Point-Source Limit 18

13. Electron Anchor and the Mass–Entropy Relation 19

14. Galactic Dynamics 20

15. Lensing, PPN, and Weak-Field Consistency 21

16. Why Dynamics Requires Extension Beyond the Static Branch 22

17. Causal Transport and Telegrapher Dynamics 22

18. Cosmology and the Hubble-Tension Sector 23

19. Why These Sectors Belong 24

20. Strong-Field Branch and Bounded Occupancy 24

21. Many-Pasts: Operational Reduction and Arrow of Time 25

22. Candidate Microstructure Hamiltonian and Underlying Dynamics 26

23. Closure-Status Table 26

24. Falsifiability and Observational Tests 28 24.1 Static weak-field falsifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 24.2 Dynamical falsifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 24.3 Cosmological falsifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 24.4 Correlated-constant falsifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 24.5 Many-Pasts status . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

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

26. Comparison with Other Approaches 29 26.1 Relative to ΛCDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 26.2 Relative to MOND-like interpolation programs . . . . . . . . . . . . . . . . . . . . 29 26.3 Relative to Verlinde-style emergent gravity . . . . . . . . . . . . . . . . . . . . . . 29 26.4 Relative to TeVeS and other multi-field modified gravities . . . . . . . . . . . . . . 29

27. Conclusion 30

Part I. Physical Idea and Canonical Definitions

1. Introduction: The Physical Claim

The central hypothesis is that the vacuum’s local entanglement capacity is a real dynamical re- source and that the localized objects we call matter are defects of that same resource rather than independent agents acting on it from outside. In this framework inertial mass is the entangle- ment content of a localized defect read through κm, gravity is the long-wavelength field carried by the surrounding restructuring of vacuum capacity around that defect, and what is usually modeled by particulate dark matter is the long-range reach of that same restructuring. The weak-field manifestation of that medium is a scalar EFT written in terms of a vacuum-relative entanglement field Sent(x) and its deficit relative to the background capacity. At continuum scale the defect sector is written in ordinary stress-energy variables, but its ontology is unchanged: it is still the coarse description of localized entanglement defects rather than a separate substance.

This is meant as a genuine replacement proposal for part of the usual dark-sector story, not simply as a new vocabulary laid on top of it. In the standard picture, one keeps visible matter and Einstein gravity, then introduces additional dark components to account for the missing gravitational response. Here the alternative hypothesis is that the vacuum already carries a finite entanglement-capacity structure, and that what we call matter is a localized defect of that structure. The same medium is then asked to explain ordinary weak-field gravity, the galactic excess usually attributed to dark matter, and the homogeneous mode relevant in cosmology.

The proposal is also meant to reach deeper than an ordinary scalar extension of Einstein gravity. If the underlying substrate has one finite update budget, an exact isotropic local struc- ture, and no external background manifold, then the continuum description should already be Lorentzian and covariant before any further phenomenology is added. In that reading, GR is not an external geometric stage to which the entanglement field is later appended. It is the low-energy capacity geometry of the same substrate. The scalar sector then keeps track of how that geometry is depleted and redistributed by localized defects.

The derivation proceeds from microstructure to observables. After introducing the field con- tent, postulates, and normalization conventions, the static weak-field coefficient chain is derived from a minimal tetrahedral boundary ensemble with admissibility closure, edge coupling, and finite loop dressing. The resulting EFT then recovers Newtonian gravity, the galactic accel- eration scale, the radial acceleration relation, lensing consistency, and the leading weak-field post-Newtonian structure without per-system tuning. Time-dependent transport, cosmology, strong-field completion, and the Many-Pasts sector are taken up afterward as extensions of the same framework, though not all of those sectors are developed to the same degree of closure.

The central claim is sharper than a heuristic analogy: one scalar capacity mode can be followed from microscopic counting all the way to weak-field observables. In compressed form, the chain is microstructure −→coefficient chain −→continuum EFT −→observables.

That chain is the primary claim: the same mode is counted in the UV, stiffened by edge transport, sourced by localized defects, normalized by the weak-field bridge, and observed as Newtonian gravity plus galactic excess acceleration.

The logical order is simple. Part I says what the theory is about, fixes the variables, and explains why the continuum description should already be relativistic and covariant if the sub- strate picture is correct. Part II asks whether a minimal UV boundary structure can actually determine the coefficients that later appear in that continuum theory. Part III asks whether those coefficients reproduce ordinary gravity and the galactic weak-field phenomenology. Only

after that chain is visible do the later parts discuss time dependence, cosmology, strong-field completion, and quantum-foundational interpretation.

The physical hypothesis is global, but the most complete derivational closure is the static weak- field UV-to-EFT chain. Other sectors are developed as controlled consequences or structured frontier extensions.

The framework is unusual in five ways at once: general relativity is derived rather than posited, Newton’s G is predicted from the electron Compton scale and a single UV combinatorial number to about one percent, the galactic acceleration scale and radial-acceleration law are derived outputs with no per-system tuning, the horizon area coefficient 1/4 closes from microscopic graph counting using the same Joyce diamond-lattice constant that fixes the weak-field source projection, and a closure-status table separates what is derivationally tight from what remains frontier. The point of the manuscript is that these features are not independent design choices but consequences of one finite UV counting problem.

1.1 What Is Primitive, and What Is Closed

The word “closure” is used here in a specific sense. The paper does not attempt to derive the existence of a finite entanglement substrate from no assumptions, nor does it derive the tetrahedral boundary ensemble from a still deeper microscopic Hamiltonian in the main text. Those are theory-defining ultraviolet inputs. The closure claim begins after those inputs are specified.

The primitive inputs are finite local entanglement capacity, matter as localized defects of that capacity, mass–entropy equivalence, the minimal tetrahedral boundary ensemble, and the ground-state faithful sector-resolution principle that fixes the absolute UV cell scale from the electron Compton length and the already-derived sharing entropy. Given those inputs, the static weak-field sector is claimed to close: the finite boundary counting problem fixes the admissibility weighting, effective sharing entropy, substrate cell length, edge kernel, loop dressing, continuum stiffness, source projection, weak-field bridge, Newtonian limit, galactic acceleration scale, radial- acceleration relation, and leading no-slip metric structure without per-system tuning.

In compressed form, the claim is

primitive UV capacity hypothesis →finite counting + admissibility →L∗

→γ, κ/γ →δS ↔Φ →G, a0, RAR, no slip .

The microstructure is therefore not itself the result being proven. It is the finite ultraviolet counting problem from which the weak-field sector is derived. The absolute length calibration is supplied by the sector-resolution principle

ln  λe

 = 7gshare,eff −ln 3

 ,

L∗

2

which identifies the electron with the maximally-extended (seven-pass) ground state of the history-space sector-resolution operator, supported over its reduced Compton scale. The seven- fold additive form 7gshare,eff is product-measure entropy additivity in the history space, and the ln(3/2) correction is the transverse export factor derived in Appendix C.5; the microscopic pos- tulate proper is the electron identification alone. The central technical claim is conditional but nontrivial: given the postulates, the minimal tetrahedral ensemble, and this sector-resolution principle, the static weak-field branch has no remaining phenomenological knobs.

This distinction matters because the continuum action can superficially resemble an ordinary scalar-tensor theory. In a conventional scalar-tensor model, one usually asks whether variation

of a covariant action containing a scalar field can produce the desired metric phenomenology. Here the scalar is the continuum order parameter of an underlying capacity substrate, and the weak-field bridge is the emergent-geometry dictionary relating fractional capacity depletion to the metric lapse. The closure question is therefore not whether the tetrahedral microstructure has been derived from nothing, but whether the specified microscopic capacity ensemble fixes the weak-field gravitational response once it is adopted.

The remaining external tasks are correspondingly clear: derive the same microscopic ensemble from a deeper substrate Hamiltonian, derive the faithful sector-resolution principle from that Hamiltonian rather than taking it as the elementary-defect support rule, independently audit the graph-level return operator, confront the resulting radial-acceleration law and no-slip pre- diction with data, derive the horizon boundary action from the graph ensemble, and extend the constrained-capacity black-hole branch into a fully dynamical collapse theory. These are recon- struction and validation tasks outside the already-closed static weak-field branch, not hidden fit parameters inside it.

2. Canonical Field Content and Definitions

We define the fundamental continuum variable as the vacuum-relative coarse-grained entangle- ment assigned to a UV probe cell of size L∗centered at x:

Sent(x) ∈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, Sent(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∞, and the deficit field is

δS(x) ≡S∞−Sent(x).

Positive δS denotes reduced available vacuum entanglement capacity in the neighborhood of a localized defect or defect distribution. It is the extended restructuring 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) ≡Sent(x)

S∞ = 1 −δS

S∞ ∈[0, 1].

The variables Sent, δS, and q therefore describe the same local physics in three closely related ways: available capacity, missing capacity relative to vacuum, and surviving-capacity fraction. The weak-field theory is most transparent in δS because it talks directly to the Newtonian potential. The nonlinear completion is most transparent in q because boundedness is built in from the start. The operational meanings are:

• q = 1: vacuum capacity fully available in the absence of local defect-induced restructuring;

• 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 Sent and S∞is a convention once an epoch and cell convention have been fixed. Under a constant rescaling

Sent 7→KSent, S∞7→KS∞, δS 7→KδS,

the observable bridge Φ c2 = −δS

2S∞ 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 κ/(γS∞) rather than on S∞alone. A cell-normalized descrip- tion and a horizon-normalized description can therefore assign different numerical values to S∞ without changing Φ, 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. It is fixed by the ground-state faithful sector-resolution principle

ln  λe

 , λe = ℏ mec.

 = 7gshare,eff −ln 3

L∗

2

Equivalently,

L∗= 3

2 λee−7gshare,eff = 1.60771947 × 10−35 m.

The corresponding induced gravitational scale is

4 ℏc m2e e−14gshare,eff = 6.60399128 × 10−11 m3 kg−1 s−2.

G∗:= c3L2 ∗ ℏ = 9

The principle says that the lightest one-bit fermionic defect coherently resolves the seven face sectors exactly once and exports only the transverse 2/3 share of that dressing block. The conventional Planck length LP = p

ℏG/c3 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:

γ : entanglement-field stiffness, (1)

κ : defect–entropy coupling, (2)

κm(ℓ) : mass-per-entropy map at scale ℓ, (3)

L∗: substrate cell length fixed by faithful sector resolution, (4)

G∗: gravitational scale induced by L∗, (5)

gshare,max = ln(1680), (6)

gshare,eff : admissibility-weighted effective sharing entropy, (7)

Jbare, Jtree eff , J(ren) eff : UV edge-kernel couplings, (8)

a0 = cH0gshare,eff

4π2 . (9)

The gravitational potentials are denoted Φ and Ψ, and the canonical weak-field bridge will be written as Φ c2 = −δS

2S∞ .

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

These definitions are fixed canonically and used without further redefinition below.

3. The Three Postulates

3.1 Information–Geometry Equivalence

The first postulate states that vacuum-relative entanglement structure contributes to spacetime curvature on equal footing with ordinary stress-energy. In the EFT this means that the scalar field Sent(x) enters a covariant action, contributes its own stress-energy, and couples to a trace- equivalent defect source. At continuum scale that source is written in the usual stress-energy variables, but ontologically it is the coarse description of the localized defect sector. In weak field, metric response is governed not by absolute entropy but by the deficit relative to the vacuum-capacity baseline.

The role of Postulate I is to say what gravity is sensitive to. Einstein gravity already tells us that geometry responds to physical content. The present extension says that local entanglement- capacity structure is part of that content. Once that is accepted, gradients and deficits of the entanglement field are no longer metaphorical; they belong in the gravitational bookkeeping alongside the usual stress-energy variables.

3.2 Mass–Entropy Equivalence

The second postulate identifies inertial mass with the entanglement content of a localized defect. At scale ℓ, m(ℓ) = κm(ℓ) ∆S.

For elementary fermionic sectors the canonical defect increment is

∆Sf = ln 2,

because a spin-1/2 fermionic face exclusion creates a binary occupied/unoccupied defect of the local network and therefore carries exactly one bit of missing entanglement. This provides the cleanest anchor for the mass–entropy map. Mass and entanglement are therefore not two separate substances linked by an empirical proportionality; they are two descriptions of the same localized defect sector at different levels of coarse-graining. For composite sectors, the relevant quantity is the fully dressed bound-state entanglement budget rather than a bare constituent count.

The purpose of this postulate is to remove the temptation to think of matter as external to the medium. In the present ontology, a particle is already a localized defect of the entanglement substrate. Writing m = κm∆S therefore does not assert an analogy between two independent things. It asserts that the inertial content of the defect is the entanglement content of the defect, read in mass units.

3.3 Many-Pasts Hypothesis

The third postulate is part of the full framework, but not every weak-field derivation depends on it directly. In canonical closed form the operational history weight is

P(H|P) ∝e−D(H,P),

equivalently the branch α = 1, β = 0 of the generalized family. This closed operational branch is fixed because exact Born recovery forces α = 1 and forbidding any extra signaling-sensitive operational bias channel forces β = 0. Its consequences are developed later as part of the theory’s interpretive and cosmological completion sector.

Postulates I and II define the gravitational ontology directly. Postulate III belongs to the broader theory because the same entanglement substrate is also asked to support an account

of branch realization and temporal asymmetry. It is part of the total theory but enters the derivational order later than the weak-field gravity chain.

The three postulates define the ontology of the theory. The main text treats them as theory- defining inputs, not as derived outputs.

4. Relativistic Continuum Structure

4.1 Capacity budget and continuum symmetry

In the present framework the continuum description is expected to be covariant not because a geometric axiom is added at the outset, but because the substrate itself is finite-capacity, isotropic, and relational.

The first ingredient is a finite maximal update rate, denoted by the same constant c that later appears in the transport relation D/τ0 = c2. 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 restructuring of the surrounding network. Because the substrate is isotropic, the cost of motion can depend only on the rotational scalar v2 at leading order, and the boundary conditions are fixed: the temporal rate is maximal at v = 0 and vanishes when the full budget is exhausted at v = c. The surviving temporal fraction is therefore

r

1 −v2

dτ

dt =

c2 .

In this sense special-relativistic time dilation is read here as a capacity-budget relation rather than as an independent postulate about flat spacetime. Once a finite invariant speed, vacuum homogeneity, and exact isotropy are in place, the Lorentz transformation law follows as the corresponding inertial symmetry rather than the Galilean one.

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

q = Sent

S∞ ,

so smaller q means that less local update capacity remains available. Motion reduces the tempo- ral 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. It is not embedded in a prior physical manifold whose coordinate labels carry independent meaning. The physical con- tent is the pattern of local capacities, defects, and neighborhood relations within the network itself. Continuum coordinates are therefore descriptive labels imposed on that relational struc- ture, not additional physical data. Smooth changes of coordinates relabel the same underlying configuration rather than altering the physics. In continuum language this is precisely why the low-energy description should be written in generally covariant form.

The upshot is that the metric sector of the EFT is not being 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. The entanglement scalar then tracks how that same capacity geometry is redistributed by localized defects. The resulting low-energy theory can therefore be written in the usual covariant lan- guage, but the intended logic runs from substrate properties to geometry, not the other way

around. As with any discrete substrate, this is a continuum statement: exact Lorentz and dif- feomorphism symmetry belong to the coarse theory, while lattice-scale corrections may survive near the UV cutoff.

4.2 Dependency Map of the Theory

The logical flow of the theory can be summarized compactly as

Postulates →UV boundary ensemble →admissibility closure

→edge kernel →finite renormalization →continuum matching →weak-field EFT

→static observables →transport / cosmology /

strong field / Many-Pasts.

This is a dependency graph, not an epistemic-equality graph. The static weak-field sector, the UV coefficient chain that feeds it, and the operational Born-recovery branch are more tightly closed than the cosmological or strong-field sectors. Part VI makes that difference explicit in a closure-status table.

The remainder of the argument follows this order so that each later result can build on the same coefficient choices and the same field dictionary.

Part II. UV Coefficient Chain

5. Why a Tetrahedral Boundary Ensemble

The microstructural problem is to identify a minimal discrete boundary-cell architecture capable of supporting finite channel entropy, isotropic closure data, and a continuum scalar response. The canonical choice adopted here is a tetrahedral cell with four structural ingredients:

• a tetrahedral volumetric cell;

• half-integer fermionic face data on each face;

• injective face assignment;

• binary orientation/parity.

This package is not presented as the only imaginable UV completion of emergent gravity. It is the minimal architecture used here to support the needed closure properties. The tetrahedron is the minimal volumetric simplex in d = 3, injectivity preserves independent boundary information across the four faces, and parity doubling captures the two orientations of the cell. The face- state multiplicity is then not chosen from a menu. Postulate II identifies elementary defects as fermionic, so each face carries half-integer base spin

2, 3

2, 5

j0 = 1

2, . . .

Two cells sharing a face therefore generate the effective boundary sector

j0 ⊗j0 = 0 ⊕1 ⊕· · · ⊕2j0.

Postulate I selects the maximum-capacity boundary channel, so the effective face label is the top channel jeff = 2j0,

with |M| = 2jeff + 1 = 4j0 + 1

distinguishable face states. Injectivity across four tetrahedral faces requires at least four distinct labels, so |M| ≥4 =⇒ 4j0 + 1 ≥4.

The only half-integer option below j0 = 3/2 is j0 = 1/2, which gives jeff = 1 and |M| = 3, so it fails the injectivity condition. The first fermionic choice that works is therefore

j0 = 3

2, jeff = 3, |M| = 7.

In that sense the seven-state face sector is derived from fermionic face data, maximum-capacity channel selection, tetrahedral injectivity, and minimality. Its appearance is fixed before any contact with the weak-field observables it later feeds. The same face-level structure is also where the elementary matter sector enters: fermionic face exclusion creates the binary one-bit defect increment ∆Sf = ln 2 used later in the electron anchor.

The resulting combinatorial state count is

Ωtet = 2 × P(7, 4) = 2 × 840 = 1680,

so the combinatorial sharing ceiling is

gshare,max = ln(1680) ≈7.427.

The exact K2 spectrum and multiplicities are carried in the appendices. The essential physical point is that the UV theory begins with a finite microscopic counting problem rather than a free continuum ansatz.

This fixes the minimal structural package used in the ultraviolet construction; the supporting derivations are collected in the appendices.

6. Admissibility Closure

6.1 Minimal isotropic kernel

The UV boundary ensemble is not used with a flat weighting. The admissibility family is

pη(b) ∝e−ηK2(b),

where K2 is the unique leading quadratic closure-defect scalar compatible with tetrahedral symmetry. This choice is not made because it “works” phenomenologically. It is the minimal isotropic maximum-entropy kernel under normalization and fixed quadratic closure moment. Higher invariants such as K4 would correspond to additional UV information and therefore to subleading refinements rather than competing leading kernels.

The reason for introducing this weighting is that the raw combinatorial ensemble is too per- missive to be the whole UV story. Some boundary configurations are closer to the regular closure pattern expected of a smooth medium, while others are more distorted. The scalar K2 is the minimal rotationally invariant way to measure that distortion. The admissibility kernel there- fore says, in the mildest possible form, that more badly closed configurations should contribute less to the effective coarse ensemble.

6.2 Closure condition and uniqueness

The canonical closure condition is ⟨K2⟩η = 3

This is the self-consistency condition for the admissibility kernel itself. The parameter η sets how strongly the weighting suppresses badly closed configurations, so the ensemble generated by that weighting must in turn exhibit the fluctuation scale that η presumes. In that sense the equation is the entropic analogue of a mean-field fixed-point condition: η is not chosen externally, but fixed by requiring the admissibility kernel to be consistent with its own induced closure fluctuations.

On the exact discrete spectrum this equation has a unique solution,

η∗= 0.0298668443935.

The closed branch is locally stiff: small fractional changes in η produce only small fractional changes in the downstream effective sharing entropy.

6.3 Effective sharing entropy

The admissibility-weighted effective sharing entropy is

gshare,eff = 7.41980002357.

The gap between gshare,max and gshare,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.

This distinction is one of the conceptual pivots of the theory. 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.

At this stage the effective sharing entropy is no longer a free choice. The exact spectrum, multiplicities, uniqueness proof, and stiffness numerics are preserved in the appendices for ref- erence.

7. Edge Kernel and Tree-Level Coupling

The same UV closure data fix the tree-level edge kernel. The geometric bridge is the tetrahedral identity 4 X

i=1 ˆniˆnT i = 4

3I3,

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

Jbare = 2

3η∗.

The interpretation of Jbare is straightforward: it is the cost assigned to mismatch between the occupancy variables of two neighboring coarse cells. 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 tetrahedral channels into the isotropic continuum limit.

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

Jtree eff = Jbare

3 = 2η∗

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 the forward transport

into the tree. Thus Jtree eff is not simply the microscopic edge penalty itself, but the part of that penalty that survives as net long-range transport after the local branching structure is taken into account.

The same chain also fixes the horizon-normalization target

σ∗= π gshare,eff ,

and the rooted shell observable converges to that target rapidly enough that the nonlocal cor- rection is already strongly constrained by small shell depth.

The story is no longer just one of state counting. The edge kernel measures how costly it is for neighboring coarse cells to disagree in local occupancy. The tetrahedral identity is what makes this bridge controlled: it is the statement that the four discrete channel directions average to the correct isotropic tensor structure in the continuum limit.

Tree-level edge transport is fixed by the same microscopic data that fixed the admissibility closure. The shell hierarchy and phase-selection checks are preserved in Appendix C.

8. Finite-Loop Renormalization

Tree level is not the whole UV story. 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(ren) eff = Jtree eff 1 + Jtree eff Σret .

The need for this step is physically straightforward. 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 structural decomposition is the key result. There are seven sector-diagonal local returns, together with one permutation-symmetric shared closure-singlet. The singlet is weighted by the same transverse projection and branch dilution factors that define the tree edge map, 2

 1

 = 2

9,

3

3

so the leading local self-energy is

Σret = 7 + 2

9 = 65

9 .

Equivalently, on the seven-channel scalar return space,

Rret = I7 + 2

9Psing, Psing = |u⟩⟨u|, u = 1 √

7(1, . . . , 1),

and Σret = Tr(Rret). The orthogonal six-dimensional sum-zero sector carries no net scalar charge in the coarse branch and therefore does not add a separate scalar return. Information sent along an edge need not simply move outward once and for all. Some of it can circulate through short local loops before contributing to coarse transport. The seven sector-diagonal returns are the seven face-label channels that return independently without mixing. In addition there is one collective mode, symmetric across channels, that returns as a shared closure-singlet rather than

as a channel-specific loop. The self-energy is therefore not a generic loop number but a sum of seven independent return channels plus one shared mode weighted by the same projection and branching factors already present in the tree map. Hence

c(ren) loop ≡J(ren) eff Jtree eff = 1 1 + Jtree eff Σret ≈0.95426,

and J(ren) eff ≈0.00633348.

This reproduces the shell-target crossing near Jbare,cross ∼0.019 at the 0.05% level.

What matters here is that the loop correction is no longer schematic. The finite renormal- ization is written as an explicit local self-energy. The remaining audit task is independent graph-level confirmation of the same scalar-return operator, not the introduction of any new loop parameter.

9. Continuum Stiffness and SI Normalization

The last UV step is not a thermodynamic one. The lattice quadratic form is interpreted as a Euclidean action weight, IE

ℏ= J(ren) eff

X

⟨ab⟩ (Qa −Qb)2,

2

over a microscopic four-cell of size

∆V4 = L4 ∗ c .

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 is what 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 Qocc,

γQ = 4ℏc

3L2∗ J(ren) eff .

Here L∗is the canonical tetrahedral spacing, with one coarse cell carrying volume L3 ∗up to the fixed cell-shape convention, and J(ren) eff is the loop-dressed edge coupling. The numerical factor 4/3 is the isotropic projection X

i ˆniˆnT i = 4

3I3

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

The field normalization is fixed by horizon capacity:

S = πQocc.

2(∂S)2 convention is

Therefore the canonical EFT coefficient in the γ

γ = 4ℏc 3π2L2∗ J(ren) eff .

The faithful sector-resolution principle fixes L∗without using G. It is nevertheless useful to define the gravitational scale induced by this length,

G∗:= c3L2 ∗ ℏ .

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

γ = 4J(ren) eff 3π2 c4

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 convention already assumed by the EFT, the SI-normalized weak-field stiffness coefficient is fixed rather than left schematic.

At a high level, Part II has now completed the micro-to-continuum coefficient story. The tetrahedral ensemble determines the effective sharing entropy, the edge kernel and loop dressing turn that entropy data into a discrete stiffness, and the Euclidean matching turns the discrete stiffness into the continuum coefficient γ that appears in the weak-field EFT.

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

{Ωtet, K2, η∗, gshare,eff, L∗, Jbare, Jtree eff , Σret, J(ren) eff , γ} −→{κ, G, a0, gobs(gbar)}.

The first bracket is the micro-to-continuum closure chain; the second bracket collects the weak- field observables it feeds. The rest of the manuscript uses this chain rather than introducing independent weak-field coefficients.

With that normalization in place, the stiffness chain runs from the discrete ensemble to the weak-field EFT without an extra coefficient choice. The remaining microscopic question is independent confirmation of the same action-kernel interpretation from fuller inhomogeneous dynamics, not an unresolved normalization constant.

Part III. Weak-Field EFT and Static Phenomenology

10. Covariant Action

With that continuum symmetry structure in place, the canonical weak-field EFT takes the covariant form

I = Z d4x √−g  c4

 ,

16πGR −γ

2gµν∂µSent∂νSent −λSent −κχSent

with χ(x) ≡−T µµ

c2 .

The action should be read as the simplest weak-field continuum realization of the ontology already stated in Part I and the coefficient chain already derived in Part II. The metric sector remains the familiar Einstein one at low energy, but it is now interpreted as the continuum capacity geometry of the substrate rather than as an independent starting theory. It is coupled to a scalar field that tracks available entanglement capacity and to a source channel written in ordinary stress-energy notation while still being interpreted microscopically as the defect sector of the same medium.

At the EFT level χ is written in ordinary stress-energy language, but ontologically it is the coarse trace channel of the localized defect sector. Here γ is the continuum stiffness fixed by the UV chain, while κ is the defect–entropy coupling fixed by the canonical source map,

κ = Ξρ L2∗κm(L∗),

and λ controls the background branch. The source map is determined by matching the scalar mode on the lattice and in the continuum. Appendix C fixes the rigid defect amplitude through the isotropic defect benchmark ∆Sdef = ln(7/6) and the exact tetrahedral on-site Green constant Gtet(0) = 0.448220394 . . .. With defect-entropy density

σdef = ρ κm(L∗),

the Green-matched continuum projection is

∇2δS = − 3L∗ 4Gtet(0) σdef,

and comparison with ∇2δS = −(κ/γ)ρ gives

κ γ = 3L∗ 4Gtet(0)κm(L∗).

This is the source-side counterpart of the stiffness derivation: the same tetrahedral scalar mode that carries the edge stiffness is the mode sourced by localized mass defects. The fixed-epoch normalization against S∞still enters the final gravity dictionary, but it is not an additional source freedom; S∞and κ rescale together under a change of entropy units, while κ/(γS∞) is invariant. Local weak-field dynamics are studied in the renormalized branch

λren ≡λ + γ□Sbg = 0,

so that local perturbations are sourced only by the defect sector, written at continuum level in ordinary matter variables.

The Einstein–Hilbert coefficient is written here in the already-matched Einstein normalization of the weak-field EFT. This is the metric normalization in which weak-field gravity is observed, not a separate microscopic parameter added on top of the entanglement chain.

Three roles of G in the manuscript. There are three distinct uses of G in what follows, and it is useful to keep them separate. First, the weak-field metric normalization gives the identity

G = c2κ 8πγS∞ .

This is the gravitational scale implied by the scalar source coupling, scalar stiffness, and capacity- to-lapse bridge. The second use is the non-gravitational scale-setting route. The faithful sector- resolution principle fixes

2 λee−7gshare,eff, λe = ℏ mec,

L∗= 3

and therefore induces

4 ℏc m2e e−14gshare,eff.

G∗= c3L2 ∗ ℏ = 9

The electron anchor enters this route through the reduced Compton length and through the identification of the electron as the lightest one-bit fermionic defect, while the mass–entropy an- chor still fixes κm(λe) = me/ ln 2 for the source map. Third, formulas involving the conventional Planck length,

ℏG

r

LP =

c3 ,

are matched representations after the gravitational scale has been identified. They are useful for compactness, comparison, and black-hole thermodynamic notation, but the scale-setting argument runs through L∗, not through a Planck-cell rewriting.

The claim, then, is that the weak-field gravitational normalization is overconstrained: the same substrate that fixes the scalar stiffness also fixes a non-gravitational cell scale, while the scalar bridge and Green-matched source map give the observed weak-field normalization through the invariant ratio κ/(γS∞). Faithful sector resolution derives G∗as the gravitational scale induced by the substrate length. The matched weak-field bridge reads

G = c2

8π κ γS∞ ,

so the numerical comparison between G and G∗is a genuine cross-check of the shared substrate normalization, not a separate opportunity to tune the source strength. The further microscopic task is to derive the faithful sector-resolution principle from a deeper Hamiltonian. This is the simplest covariant realization of the closure chain: one metric, one scalar entanglement field, one trace-equivalent defect-source channel, and one renormalized background branch.

The logical order matters. Once this weak-field covariant form is accepted as the correct low-energy language of the substrate, the earlier UV closure chain fixes the entanglement-side coefficients that appear in it; the terms in the action are then read off from that chain, not chosen term by term to fit phenomenology.

The weak-field action used below is now fully written.

11. Field Equations and Bridge Law

Varying the action with respect to Sent gives the sourced scalar equation

γ□Sent = λ + κχ.

Varying with respect to the metric yields

Gµν = 8πG

 T (matter) µν + T (ent) µν  ,

c4

with the canonical scalar stress-energy induced by the entanglement sector.

These two equations separate the two jobs played by the scalar. The scalar equation tells us how the local entanglement-capacity variable responds to defect sources. The metric equation tells us how that scalar response then contributes back to spacetime curvature. The bridge law below is what turns those two statements into an ordinary weak-field gravitational potential.

The weak-field bridge law is not inserted as an arbitrary interpolation. It is the normalization map between fractional capacity depletion and the weak-field lapse. Let

ϵ(x) ≡δS(x)

Vacuum normalization requires the lapse to approach its asymptotic value when ϵ = 0. Locality requires the leading metric response to depend on the local fractional deficit. Independent small deficits must add at leading order, while independent redshift factors multiply. These requirements force the lapse to take exponential form,

N(ϵ) = exp[−Cϵ],

with a single constant C. The standard weak-field metric normalization,

N = 1 + Φ

c2 + O(Φ2/c4),

fixes C = 1/2. Hence

N = exp  −δS

 = 1 −δS

2S∞ + O(δS2/S2 ∞),

2S∞

and therefore Φ c2 = −δS

2S∞ .

In the static weak-field branch the emergent Newton constant is therefore

G = c2κ 8πγS∞ .

This bridge law is where the derivation stops speaking only in the language of entropy variables and starts speaking directly in the language of observable gravity. Without it the theory would remain a scalar model with an entanglement interpretation. With it, the deficit field acquires a unique weak-field normalization in terms of the ordinary gravitational potential.

With this bridge in place, the weak-field continuum dictionary is complete.

12. Newtonian Gravity and the Point-Source Limit

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

∇2δS = −κ

γ ρ.

This is the point where the micro-to-macro chain becomes operationally familiar. Once the background is renormalized away and the source is nonrelativistic, the scalar sector obeys an ordinary Poisson equation for the deficit field. The unusual quantity is δS, but the mathematical structure is the same one that underlies standard weak-field gravity.

For a point source M,

δS(r) = κM

4πγr.

Using the bridge law, Φ c2 = −δS

2S∞ ,

the gravitational acceleration becomes

g(r) = c2κ 8πγS∞

M

r2 = GM

r2 .

Thus Newtonian gravity is recovered as the weak-field response of the entanglement-capacity medium.

Nothing qualitatively exotic has to be inserted at the last step to recover ordinary gravity. The same sourced scalar equation and the same bridge law already imply the familiar point-mass force law. In that sense Newtonian gravity appears here not as a starting axiom but as the first infrared limit of the entanglement medium.

Interpretation. Ordinary gravity is the small-deficit, weak-curvature limit of the extended entanglement restructuring around localized defects. The Newtonian 1/r2 law is therefore emer- gent, not fundamental.

Once the bridge law, Green-matched source projection, and UV stiffness are fixed, the New- tonian limit is closed.

13. Electron Anchor and the Mass–Entropy Relation

The mass–entropy relation requires a clean elementary anchor because, in this framework, the elementary matter sector is the localized defect sector itself. For a single fermionic face-exclusion defect the canonical increment is ∆Sf = ln 2.

The physical content is that a single excluded face is a binary occupied/unoccupied topological defect and therefore carries exactly one bit of missing entanglement. At the electron Compton scale ℓ= λe the mass–entropy map gives

κm(λe) = me

ln 2.

This is the first step of the anchor logic. If the elementary fermionic defect carries ∆Sf = ln 2, then dividing the electron mass by that fixed entropy increment gives the mass-per-entropy conversion at the electron’s own scale. The electron is the cleanest place to do this because it is the lightest simple fermionic defect and is not obscured by hadronic compositeness.

The UV normalization is κm,UV = ℏ cL∗

1 ln 2,

and the canonical running law is

1+αcl , αcl = 0

L∗

κm(ℓ) = κm,UV

ℓ

in the closed branch. The electron sector supplies a sharp anchor for the mass–entropy map and a fixed-normalization input to the source projection. The next step is then to run that conversion back to the UV scale. The quantity κm,UV is the fundamental mass-per-entropy conversion attached to the cutoff cell itself, and the running law tells us how that conversion appears at longer physical scales. So the logic of the section is: one bit fixes the electron-scale conversion, the running law connects the electron scale back to the UV scale, and the same conversion then feeds the weak-field normalization chain.

The same electron also supplies the non-gravitational length anchor used in faithful sector resolution. Its reduced Compton wavelength λe fixes the absolute cell length L∗through the seven-sector history-space dressing relation in Appendix D, while its mass me fixes the mass- per-entropy map through κm(λe) = me/ ln 2. This does not mean that an electron is a single tetrahedral cell carrying seven simultaneous labels. The local tetrahedral ensemble supplies the seven-sector alphabet; the electron is the lightest coherent one-bit defect whose ground-state dressing resolves those sectors once and exports the transverse share of the resulting support. These are distinct uses of the same elementary defect: the Compton scale calibrates the sub- strate length hierarchy, and the one-bit mass anchor calibrates the source map. Together they overconstrain the role of the electron in the weak-field gravitational normalization rather than duplicating the same input twice.

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

mhadron = κm(ℓH)Sdressed ent,H ,

with the dressed entropy budget generated by confinement, gluonic structure, trace-anomaly dynamics, and chiral vacuum reorganization. A finished lattice derivation of that dressed entropy is not yet available. What is claimed here is structural compatibility between the mass–entropy map and the standard QCD mass budget.

The elementary-fermion anchor is settled in the simple sectors, while the hadronic sector remains structurally compatible but not yet fully coefficient-complete.

14. Galactic Dynamics

The galactic sector is one of the main payoffs of the coefficient chain. The characteristic accel- eration scale is a0 = cH0gshare,eff

4π2 .

This formula already shows why the galactic phenomenology is not independent of the UV story. The same effective sharing entropy that appears in the closure chain now reappears in the acceleration scale governing departure from the Newtonian branch on galactic outskirts.

The corresponding 1 + 2 channel decomposition separates one longitudinal slot aligned with the baryonic acceleration gradient and two transverse slots carrying the cosmic background scale. This fixes the galactic dimensionless variable as

x = rgbar

a0 .

This is the right variable because it measures baryonic forcing relative to the intrinsic acceleration scale set by the same closure chain. Baryonic matter sources the single longitudinal channel, while the two transverse modes form the quadratic capacity bath, so the occupancy depends on the amplitude ratio p

gbar/a0 rather than directly on gbar/a0. When gbar ≫a0, the system should reduce to the ordinary Newtonian branch; when gbar ≪a0, the response should cross into the low-acceleration completion. The square-root form is the one selected by the canonical 1+2 channel structure and is exactly what reproduces the deep-MOND scaling later in the section.

For the massless bosonic entanglement mode, the minimal stationary completion is therefore the Bose–Einstein occupancy branch

1 + nB(x) = 1 1 −e−x ,

This bosonic language is not an extra fit ingredient added after the fact. The collective excitation of a scalar entanglement field is itself bosonic, so once the weak-field response is organized as occupancy of a massless scalar mode, Bose–Einstein statistics are the minimal stationary completion. The role of a0 is then to provide the effective scale against which that occupancy is measured, so the galactic law becomes an occupancy statement rather than an empirical interpolation formula chosen by hand.

The resulting radial-acceleration law is

gobs = gbar 1 + nB(x)
= gbar

gbar/a0 .

1 −exp  − p

This has the correct asymptotic limits:

gbar ≫a0 =⇒gobs ≈gbar, (10)

gbar ≪a0 =⇒gobs ≈√a0gbar. (11)

The deep-MOND branch therefore gives the baryonic Tully–Fisher law

v4 ≈a0GMb.

Once the channel geometry is fixed, the weak-field medium has a minimal bosonic occupancy completion. The interpolation law is the way the entanglement response fills the available modes when the baryonic source is weak compared with the intrinsic acceleration scale, and the scale itself is set by the same microstructural chain.

The UV channel geometry that fixes the microscopic coefficient chain also feeds the galactic EFT, so the radial-acceleration law is read off rather than interpolated per galaxy.

The galactic branch is fixed up to the same channel-identification structure already used elsewhere in the weak-field EFT.

15. Lensing, PPN, and Weak-Field Consistency

Because the entanglement sector is scalar, it does not generate anisotropic stress at leading weak-field order. The scalar anisotropic stress is quadratic in gradients, schematically ∂iS ∂jS = O(Φ2/c4), so it enters beyond the linear branch. Hence

Φ = Ψ

to the order treated in the present EFT. This means that light bending and dynamical mass estimates are sourced by the same leading metric response. In effective-halo language, the entanglement response can be rewritten as

ρhalo(r) = 1 4πGr2 d dr

h r2(gobs −gbar) i ,

which yields the familiar 1/r2 outer-halo profile in the asymptotic branch.

This matters because a theory can match galactic rotation curves and still fail lensing if the two metric potentials slip apart. The weak-field branch here avoids that problem at leading order. The same response that governs the dynamics also governs light deflection, so the theory is not buying galactic support at the price of a leading weak-field inconsistency. Solar-system constraints such as Cassini require any PPN slip to remain at the ∼10−5 level, consistent with the no-slip structure of the linear branch.

The same weak-field structure also returns the GR post-Newtonian values at the order treated:

γPPN = βPPN = 1 + O(Φ2/c4).

Thus the leading weak-field EFT does not purchase galactic phenomenology by introducing gravitational slip or obvious solar-system-scale pathologies.

At leading weak-field order this sector is closed. Higher-order precision confrontation remains an audit task rather than an architectural gap.

Part IV. Time-Dependent, Transport, and Cosmological Sectors

16. Why Dynamics Requires Extension Beyond the Static Branch

The static weak-field branch is not the whole theory. If the entanglement-capacity medium is physical, it must admit relaxation, propagation, and causal response to changing sources. The time-dependent sector should therefore not be read as an optional add-on. It is the natural dynamical extension of the same medium that produces the static weak-field EFT.

This section is only a bridge into the dynamical sectors; no independent closure claim is being made here.

17. Causal Transport and Telegrapher Dynamics

The canonical time-dependent completion is most cleanly written relative to the substrate four- velocity uµ:

τ0(uµ∇µ)2δS + uµ∇µδS = Dhµν∇µ∇νδS + Aχ, hµν = gµν + uµuν.

The vector uµ is the local rest frame of the entanglement-capacity medium, not an additional ad hoc force carrier. In that frame, uµ = (1, 0, 0, 0), the equation reduces to the familiar telegrapher form τ0∂2 t δS + ∂tδS = D∇2δS + Aχ(x, t),

with static-matching condition A D = κ

γ .

This equation is introduced because a physical medium should not respond instantaneously to changing sources. The static Poisson equation is appropriate when the source has already settled, but once sources evolve in time one needs both propagation and relaxation. The telegrapher form is the minimal causal extension that still reduces to the static branch when time dependence becomes negligible.

Causality requires D τ0 = c2,

so the transport sector propagates disturbances at finite speed. In the canonical no-new-IR-scale branch,

τ −1 0 = H0, D = c2

H0 .

This transport equation separates two roles that must remain distinct. Ordinary galactic support belongs to the near-stationary static branch. The telegrapher sector governs how the same medium propagates, relaxes, and develops lag when sources evolve in time. It is therefore not used to generate ordinary static galactic support; it governs transport, lag, relaxation, and merger phenomenology around the near-stationary weak-field branch.

For galactic modes the Appendix E analysis shows that the long relaxation time does not destroy the static limit. Galactic modes lie deep in the underdamped regime, so the static Poisson branch is recovered as the exact time average relevant to ordinary galactic dynamics. The assumption here is that the source is quasi-static on galactic timescales and supported on wavelengths far shorter than the critical scale λc ∼4πc/H0; under those conditions the oscillatory transient averages out instead of competing with the static branch.

The static branch still handles the ordinary galactic law. The transport equation describes what happens when the source history is no longer quasi-static: propagation delay, relaxation, and merger-era lag.

The transport branch is closed at the level of D/τ0 = c2 and the preferred choice τ −1 0 = H0. Detailed merger phenomenology remains frontier.

18. Cosmology and the Hubble-Tension Sector

The cosmological sector should be read as the homogeneous continuation of the same scalar medium, not as an unrelated dark-energy add-on bolted onto the weak-field theory. What changes here is not the ontology but the kinematic regime: the background mode becomes dynamically relevant on horizon scales while the local weak-field branch remains encoded in the inhomogeneous fluctuations.

The cosmological sector uses the same field split,

S(x, t) = S(t) + s(x, t),

where 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∞(t) = πRA(t)2

L2∗ .

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 δS/S∞ and κ/(γS∞) rather than on an absolute entropy unit.

This decomposition is essential. The homogeneous mode and the local weak-field fluctuations are not two unrelated scalar fields. They are two kinematic sectors of the same field. The split lets the background mode change cosmological evolution without automatically rewriting the local weak-field equations that already fixed the galactic phenomenology.

Because the entanglement field couples to the trace of the stress-energy tensor, the homoge- neous mode is largely dormant during radiation domination but becomes active near matter– radiation equality. This gives a transient early-energy component of the same general type used in early-dark-energy resolutions of the Hubble tension. In the closed cosmological branch treated here, the effect reduces the sound horizon and shifts the CMB-inferred Hubble constant upward from the high-67 range toward the high-68 to low-69 range.

What matters here is not just the direction of the shift but the timing. A successful Hubble- tension mechanism must turn on near the right epoch, alter the sound horizon in the right direction, and then decouple cleanly enough from the local weak-field sector that the galactic branch is not spoiled. The entanglement medium has exactly that qualitative structure.

The local weak-field predictions are protected by the separation between 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 claim is therefore a mechanism with the right direction, timing, and qualitative separation of scales, not a finished precision cosmology package. What is shown is that the trace-coupled homogeneous mode turns on in the relevant epoch and pushes the sound horizon in the re- quired direction; what remains open is the full perturbation propagation and likelihood-level confrontation. The homogeneous mode modifies the background history; the inhomogeneous

branch continues to govern the local weak-field observables already fixed earlier in the deriva- tion. That separation is what allows the cosmological extension to remain part of the same scalar medium rather than a re-tuning of the galactic sector.

This is a structurally supported and directionally successful extension, but it is not yet Boltzmann-closed.

Part V. Nonlinear, Interpretive, and Completion Sectors

19. Why These Sectors Belong

The most directly constrained micro-to-weak-field chain is now in hand. The next sectors de- velop three further pieces required for overall framework completeness: nonlinear completion, operational quantum reduction, and candidate underlying dynamics. They belong to the same ontology, but they should not be read as resting on identical evidence.

20. Strong-Field Branch and Bounded Occupancy

The strong-field branch begins with the same variable that already appeared in the weak-field bridge, but it uses the form in which boundedness is explicit:

q(x) = Sent(x)

S∞ ∈[0, 1].

The physical meaning is direct: q = 1 is undepleted vacuum capacity, 0 < q < 1 is a partially depleted region, and q = 0 is complete local exhaustion of continuum-defining capacity. The strong-field question is therefore not how to continue the weak-field scalar after it becomes large, but how to write the continuum theory on the domain where the capacity variable remains physically allowed.

The local lapse associated with the static asymptotic time must be a function of surviving capacity. If N2 = f(q),

then vacuum normalization gives f(1) = 1, horizon normalization gives f(0) = 0, the weak-field bridge gives f′(1) = 1, and serial composition of independent capacity-reduction layers gives

f(q1q2) = f(q1)f(q2).

The continuous solutions are f(q) = qα, and weak-field matching fixes α = 1. Thus

N2 = q

is the unique continuous multiplicative completion compatible with the weak-field branch. This is a stronger statement than a useful ansatz: once the capacity variable is adopted, the static lapse rule is fixed.

The constrained-capacity theory is then posed on the physical domain

Mq = {x | q(x) > 0},

with a Lagrange multiplier enforcing N 2 = q. In bulk vacuum, away from the q = 0 boundary and with no explicit matter source, the multiplier vanishes and the equations reduce to the ordinary vacuum Einstein equations on Mq. The difference from GR is not in the exterior

vacuum equations; it is in the physical domain on which those equations are allowed to live and in the boundary condition at capacity exhaustion.

Consequently the static spherical asymptotically flat vacuum branch is the Schwarzschild exterior,

q(r) = N2(r) = 1 −2GM

c2r , r > rh = 2GM

c2 .

The continuum branch terminates at q = 0. A real continuation of this static branch to r < rh would require q < 0, which is outside the state space of the capacity variable. The classi- cal Schwarzschild interior is therefore not interpreted as a physical continuation of the same entanglement-capacity EFT. It is a formal continuation of the GR manifold beyond the point where the substrate has no remaining local continuum channels.

This preserves the tested exterior physics. The near-horizon Euclidean regularity argument gives the usual Hawking temperature, and exterior perturbation theory gives the usual absorp- tion and ringdown provided the q = 0 boundary is effectively absorbing. Echoes are not a generic prediction; they require finite microscopic reflectivity or a stretched layer q = ϵ > 0 outside the capacity-exhaustion surface. The entropy story also separates cleanly: the continuum exterior saddle gives the usual Bekenstein–Hawking area law, while the coefficient 1/4 is recovered by the boundary channel-counting rule built from the exact local graph response. What remains open is the microscopic Hamiltonian derivation of that boundary rule and the dynamical formation of the saturated surface.

The strong-field branch is therefore sharper than a mere kinematic horizon criterion but still not a finished collapse theory. The static spherical exterior closes within the constrained- capacity EFT, and the static area coefficient closes under the boundary channel-counting rule. The open problems are the graph-derived horizon boundary Hamiltonian, the boundary reflectiv- ity/relaxation spectrum, rotating and charged generalizations, and the bounded causal transport law that forms the q = 0 surface dynamically during collapse.

21. Many-Pasts: Operational Reduction and Arrow of Time

Many-Pasts belongs in the full framework because the theory is not only a gravity mechanism. It is also a proposal about branch realization and temporal asymmetry on the same entropic substrate. Operationally, however, it is deliberately conservative.

With P(H|P) ∝e−D(H,P),

the Born rule is recovered exactly because

e−D(H,P) = Tr(ΠP ρH→now)

in the projective laboratory limit. Exact Born recovery forces α = 1, and forbidding any extra signaling-sensitive operational bias channel forces β = 0. No-signaling is preserved exactly in this operational branch.

The remaining content is interpretive and cosmological. The arrow of time is recovered through conditional typicality: among histories consistent with present macroscopic records, overwhelm- ingly many exhibit entropy growth toward the future direction defined by those records. This adds no new laboratory probability law; it offers a global consistency account of branch realiza- tion and temporal asymmetry.

Operational closure is exact in the laboratory sector; the extra content added here is inter- pretive and cosmological.

22. Candidate Microstructure Hamiltonian and Underlying Dy- namics

The UV closure chain should have a microscopic dynamical realization. The candidate developed in the appendices is a GFT/condensate picture in which spacetime emerges from a condensate of discrete tetrahedral building blocks, while what is macroscopically read as matter appears as fermionic defects of that same substrate. In Madelung form,

n(x)eiθ(x),

σ(x) = p

the condensate hydrodynamics generically generate a positive scalar stiffness for the logarithmic- density variable, providing the condensate-side origin of the EFT kinetic term. This is a struc- tural compatibility check, not a replacement for the explicit coefficient closure in Appendix C; it shows that the EFT kinetic term has a natural microscopic origin.

This provides a coherent microscopic realization supporting the UV closure chain, but not yet a first-principles inhomogeneous derivation of every continuum term.

Part VI. Closure Status, Falsifiability, and Research Program

23. Closure-Status Table

The closure bookkeeping is concentrated here in one place so the rest of the text can simply derive, state, and move on.

Quantity / Claim Sector Status Type of Support Where Established Ωtet, gshare,max UV counting Closed exact combinatorics Part II, App. B η∗ admissibility closure

Closed exact K2 spectrum and multiplicity closure

Part II, App. B

gshare,eff UV entropy Closed exact weighted evaluation Part II, App. B Substrate length L∗ scale setting Conditional derivation

faithful seven-sector support relation from λe, gshare,eff, and Ccl = 3/2

Part I, App. D

Jbare, Jtree eff UV edge kernel Closed tetrahedral isotropy identity Part II, App. C Σret = 65/9 finite-loop UV Fixed in the minimal UV return sector

explicit seven-channel plus singlet return count

Part II, App. C

J(ren) eff finite-loop UV Fixed by the UV return resummation

derived from Σret Part II, App. C

γ continuum stiffness

Closed in canonical EFT convention

Euclidean-action normalization Part II, App. C

Green-matched source projection

UV source map Closed in the canonical weak-field branch

exact defect counting + tetrahedral on-site Green function

App. C

κ/γ source-to- stiffness ratio

σdef = ρ/κm(L∗) plus Gtet(0) and tetrahedral 4/3 projection

Closed in the canonical weak-field branch

Part III, App. C–D

Weak-field bridge law EFT / gravity Closed in canonical weak-field branch

uniqueness from multiplicative composition

Part III, App. D G∗ gravitational scale

G∗= (9/4)(ℏc/m2 e)e−14gshare,eff Part I, App. D

Conditional derivation at percent level

Matched G weak-field gravity

bridge law + Green-matched κ/γ + fixed-epoch normalization of κ/(γS∞); comparison with G∗is an overconstraint check

Closed in the matched weak-field branch

Part III, App. C–D

one-bit fermionic defect supplies λe for length setting and me/ ln 2 for mass–entropy map

Electron anchor mass / length sector

Fixed empirical elementary anchor

Part III, App. D

Quantity / Claim Sector Status Type of Support Where Established Faithful sector-resolution principle

λe/L∗= dimeff(De,⊥) from the seven-sector history-space operator

UV scale theorem

Primitive support rule with a Hamiltonian audit target

App. D

a0 galactic EFT Fixed in the closed weak-field realization

UV entropy + cosmic scale Part III, App. C

1 + 2 channel geometry + bosonic occupancy

RAR law galactic EFT Fixed in the closed weak-field realization

Part III, App. C

No slip / lensing consistency

weak-field metric

Closed at leading weak-field order

scalar-stress structure Part III, App. D PPN leading values weak-field metric

Structurally supported

weak-field expansion Part III, App. F Telegrapher relation D/τ0 = c2

transport Closed in canonical transport branch

causal closure Part IV, App. E Canonical τ −1 0 = H0 branch

transport Fixed in the minimal transport closure

no-new-IR-scale choice Part IV, App. E

Hubble-tension mechanism

cosmology Structurally supported extension

homogeneous trace-coupled mode Part IV, App. E

Bounded capacity q, N2 = q

strong field Closed as constrained- capacity rule

uniqueness from multiplicative composition and weak-field matching

Part V, App. F

Constrained action on Mq = {q > 0}

strong field Closed within the EFT

ADM action plus lapse-capacity constraint

Part V, App. F

Static spherical exterior

strong field Closed within the constrained- capacity branch

bulk vacuum reduces to Einstein vacuum on Mq

Part V, App. F

Capacity-exhaustion horizon q = 0

strong field Closed as static-domain boundary

Schwarzschild exterior terminates where q leaves the physical state space

Part V, App. F

Hawking temperature and exterior ringdown

strong field Conditional exterior result

unchanged Schwarzschild exterior with absorbing q = 0 boundary

App. F

Bekenstein–Hawking area-law bridge

strong field Conditionally closed under boundary channel-counting rule; boundary Hamiltonian open

exact graph response + G⊥= (2/3)Gtet(0) + nhorScell ∞ = 1/4

App. C, F

Dynamical collapse and boundary microphysics

strong field Frontier free-boundary transport, reflectivity, and horizon action still to derive

Part V, App. F

Rotating / charged strong-field branches

strong field Frontier requires generalized stationary capacity variable

App. F

α = 1 theorem Part V, App. G

Many-Pasts Born recovery

quantum foundations

Closed operationally

β = 0 theorem Part V, App. G

No-signaling in operational branch

quantum foundations

Closed operationally

Arrow-of-time account

quantum foundations

Coherent extension conditional typicality / counting Part V, App. G

Candidate microstructure Hamiltonian

UV realization Coherent extension condensate / GFT realization sketch Part V, App. H

Lepton-shell mass extension

particle-sector extension

Coherent extension constrained shell ladder with finite generation count

App. I

Gauge-redundancy extension

gauge sector Coherent extension baseline-redundancy construction with Maxwell/Yang–Mills form

App. I

Numerical robustness checks

validation layer Supportive audit layer

cross-sector consistency tests App. J

EFT consistency checklist

field-theory audit

Supportive audit layer

no-ghost / no-tachyon / causal-propagation checklist with explicit vacuum dispersion stability

App. D

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

24. Falsifiability and Observational Tests

24.1 Static weak-field falsifiers

The static weak-field sector stands or falls on a small number of concrete checks. The most direct are the shape and tightness of the galaxy RAR transition [1], the baryonic Tully–Fisher scaling in systems where the EFT should apply, and the weak-field lensing sector. A persistent need for gravitational slip where the scalar stress predicts none would be especially damaging, because it would break the same no-slip structure used to keep dynamics and lensing aligned. Solar-system bounds, especially Cassini-class PPN tests [3], also require the leading no-slip branch to survive at high precision.

24.2 Dynamical falsifiers

The dynamical extension is more vulnerable, and its failure modes are correspondingly sharper. Time-dependent halo lag, cluster-scale acceleration relations, merger offsets, or relaxation signa- tures that cannot be reconciled with the telegrapher relation D/τ0 = c2 would indicate that the causal completion has the wrong propagation structure even if the static branch survives. Clus- ter data and systems such as the Bullet Cluster [2, 4] should therefore be treated as dynamical and transport tests, not merely as larger versions of the static galaxy problem.

24.3 Cosmological falsifiers

Cosmology presents a different kind of test. The question there is not whether the mechanism points in the right direction, but 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.

24.4 Correlated-constant falsifiers

One of the more distinctive signatures of the framework is that the same microstructural chain feeds the substrate scale, the weak-field gravitational normalization, and the galactic acceleration scale. A precision program that could test the inferred L∗and G∗scale against matched G, a0, and the RAR normalization would probe the theory more sharply than isolated single-observable fits, because it would confront the shared coefficient origin directly.

24.5 Many-Pasts status

The Many-Pasts sector is not likely to be challenged first by ordinary laboratory deviations from quantum mechanics, because it is built to reproduce the usual operational structure there. Its more immediate points of failure are internal ones: failure of exact Born recovery, failure of no-signaling, or incompatibility with the thermodynamic arrow structure it is supposed to illuminate.

These points define the canonical falsifiability map.

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

Placed together, the main failure modes have a simple shape:

• If future weak-field observations require persistent gravitational slip in the relevant galactic or cluster regimes, the canonical weak-field branch fails.

• If the RAR transition shape systematically departs from the derived bosonic occupancy law in systems well described by the static branch, the canonical galactic EFT fails.

• If the faithful sector-resolution principle cannot be recovered from a deeper microscopic Hamiltonian, the percent-level G∗scale prediction remains a conditional support principle rather than a microscopic theorem.

• If the weak-field UV coefficient chain cannot be reconciled with an independently validated microscopic derivation, the canonical UV closure loses support.

• If the cosmological trace-coupled homogeneous mode cannot survive full Boltzmann likeli- hood confrontation, the cosmology sector fails even if the static weak-field branch survives.

• If the constrained-capacity branch cannot retain the tested Schwarzschild exterior, stan- dard absorbing-boundary ringdown, or the required horizon thermodynamics once its boundary action is derived, the strong-field completion fails while the weak-field EFT may still remain viable.

These are failure modes rather than a separate derivational sector.

26. Comparison with Other Approaches

Because the framework aims to replace dark matter and partially reorganize the usual dark- energy story, it is useful to state briefly how its logic differs from nearby alternatives.

26.1 Relative to ΛCDM

The contrast with ΛCDM begins at the level of ontology. Standard cosmology explains the relevant phenomenology by adding dark matter and an independent cosmological constant or dark-energy sector to otherwise standard gravity. Here the visible matter sector is retained, but it is interpreted as the macroscopic description of localized defects in a vacuum-capacity medium whose weak-field response supplies the effective extra gravitating component. The same closure chain is then asked to feed G, a0, the RAR law, weak-field lensing consistency, and the homogeneous cosmological mode.

26.2 Relative to MOND-like interpolation programs

MOND-like programs usually begin from an acceleration law or interpolation function and ask how much galaxy phenomenology it can explain. The present logic runs the other way. The interpolation law is not taken as primary; it is downstream of the UV entropy, the 1 + 2 channel geometry, and the bosonic occupancy branch. The galactic law is thus treated as an output of the same micro-to-IR closure chain rather than as the phenomenological starting point.

26.3 Relative to Verlinde-style emergent gravity

Verlinde-style emergent-gravity programs share the broad intuition that gravity may be entropic, but they are usually formulated at the level of thermodynamic reasoning or horizon-inspired force laws. The present framework is trying to do something narrower and more explicit: fi- nite tetrahedral boundary counting, admissibility closure, edge coupling, finite renormalization, Euclidean-action normalization, and only then a continuum scalar EFT. Whether that chain is ultimately correct is an empirical matter, but it is a different kind of proposal from a purely macroscopic entropic argument.

26.4 Relative to TeVeS and other multi-field modified gravities

Multi-field relativistic MOND completions such as TeVeS typically introduce additional vector or tensor sectors in order to repair lensing or cosmological problems. The present weak-field

construction instead keeps a single scalar entanglement field within a low-energy Einstein con- tinuum sector that is itself interpreted as emergent from the substrate, and relies on the no-slip structure Φ = Ψ at leading order to keep lensing and dynamics aligned. That economy is at- tractive if the branch survives confrontation with data, and immediately vulnerable if future observations demand persistent slip or extra weak-field structure.

This comparison is meant as context rather than as a derivational sector.

27. Conclusion

The central achievement of the manuscript is a weak-field closure result. A finite entanglement- capacity microstructure is carried through admissibility closure, substrate length setting, edge transport, finite renormalization, continuum matching, Green-function source matching, and a covariant scalar EFT to produce Newtonian gravity, the galactic acceleration scale, the RAR law, and weak-field lensing consistency without per-system tuning. More broadly, the manuscript argues that the continuum metric sector itself should be understood as the low-energy capacity geometry of the same substrate, so Einstein gravity appears here as a continuum limit of the theory rather than as an independent foundation underneath it.

In compact form, the static chain is now: tetrahedral counting fixes gshare,eff, faithful sector resolution fixes L∗= (3/2)λee−7gshare,eff and hence G∗= (9/4)(ℏc/m2 e)e−14gshare,eff, finite returns give Σret = 65/9, continuum matching fixes γ, Green matching fixes κ/γ, and the weak-field bridge gives the Newtonian branch, a0, the RAR law, and Φ = Ψ at leading order. Transport also closes at the level of finite propagation, D/τ0 = c2.

That does not finish the whole theory, but it does change the shape of the open problems. What remains is no longer the invention of a missing theory; it is the hard completion work of an existing one: independent graph-level confirmation of the finite-loop self-energy, Hamiltonian derivation of the faithful sector-resolution principle, precision comparison of the matched weak- field G with the induced scale G∗, full Boltzmann cosmology, the microscopic horizon boundary Hamiltonian behind the channel-counting rule, dynamical collapse simulations, and rotating or charged strong-field generalizations. Cosmology, constrained-capacity strong field, and Many- Pasts all remain part of the same ontology, though they do not yet stand at the same level of derivational support as the static weak-field chain.

Appendix A: Symbol Dictionary and Canonical Conventions

Appendix A gathers the conventions used throughout the technical material that follows. Its purpose is simply to keep the later appendices readable by fixing the units, field definitions, and couplings in one place before the denser calculations begin.

A.1 Units, signature, and entropy normalization

All dimensional quantities are expressed in SI units unless noted otherwise. The metric signature is (−, +, +, +). 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∗= L3 ∗. In the canonical branch L∗is fixed by faithful sector resolution,

2 λee−7gshare,eff, λe = ℏ mec.

L∗= 3

ℏG/c3 is used only as a comparison scale or in standard gravitational thermodynamic expressions after the gravitational scale has been identified.

The conventional Planck length LP = p

These conventions matter because the argument repeatedly moves between a dimensionless UV counting problem and a dimensionful continuum EFT. The units and signature are what make those two descriptions comparable rather than merely suggestive.

A.2 Core scalar variables

The canonical continuum variable is the vacuum-relative coarse-grained entanglement field

Sent(x),

with vacuum baseline S∞and deficit

δS(x) = S∞−Sent(x).

For nonlinear work the bounded occupancy fraction is

q(x) = Sent(x)

S∞ = 1 −δS

S∞ ∈[0, 1].

The absolute entropy unit is fixed only after choosing a cell or horizon normalization. Under a constant rescaling of Sent, the quantities S∞and κ/γ rescale together, leaving δS/S∞and κ/(γS∞) invariant. The source channel is

χ(x) = −T µµ

c2 ,

which is the continuum trace channel of the localized defect sector and reduces to the ordinary mass density ρ in the nonrelativistic static limit.

A.3 Couplings and derived observables

The main-text conventions are

γ : entanglement-field stiffness, (12)

κ : continuum defect–entropy coupling, (13)

κm(ℓ) : mass-per-entropy map at scale ℓ, (14)

L∗= 3

2 λee−7gshare,eff, (15)

4 ℏc m2e e−14gshare,eff, (16)

G∗= c3L2 ∗ ℏ = 9

Gtet(0) : tetrahedral on-site Green constant, (17)

gshare,max = ln(1680), (18)

gshare,eff : admissibility-weighted sharing entropy, (19)

Jbare, Jtree eff , J(ren) eff : UV edge couplings, (20)

a0 = cH0gshare,eff

4π2 . (21)

The canonical weak-field bridge and Newton closure are

Φ c2 = −δS

2S∞ , κ γ = 3L∗ 4Gtet(0)κm(L∗),

G = c2κ 8πγS∞ .

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∞normalization are applied.

A.4 Notation map

One notation set is used throughout. The effective sharing entropy is denoted gshare,eff, the scalar variable is always the vacuum-relative field Sent or its deficit δS, and the weak-field bridge is used in the single form stated above.

Appendix A serves as the reference layer for those conventions.

Appendix B: UV Boundary Ensemble and Admissibility Closure

Appendix B records the finite ultraviolet counting problem in its explicit form. It shows how the theory begins from a discrete boundary ensemble and ends with a unique admissibility-closed entropy rather than with an unconstrained continuum ansatz.

B.1 Minimal tetrahedral package

The canonical UV cell is a tetrahedron with four structural ingredients:

• a tetrahedral volumetric cell;

• half-integer fermionic face data on each face;

• injective face assignment across the four faces;

• binary orientation/parity.

Postulate II identifies the elementary defect sector as fermionic, so each face carries half-integer base spin j0. For a shared face the effective boundary sector is

j0 ⊗j0 = 0 ⊕1 ⊕· · · ⊕2j0.

Postulate I selects the maximum-capacity channel, hence jeff = 2j0 with |M| = 2jeff +1 = 4j0+1 distinguishable face states. Injectivity across four faces requires |M| ≥4. The j0 = 1/2 option fails because it gives jeff = 1 and |M| = 3. The first half-integer choice that works is therefore j0 = 3/2, giving jeff = 3 and the canonical seven-state face sector. The resulting state count is

Ωtet = 2 × P(7, 4) = 1680, gshare,max = ln(1680) = 7.42654907240.

This is the minimal discrete package used in the framework to obtain a finite, isotropic, auditable boundary-channel structure.

The important feature is not just that the counting closes, but that it closes for structural reasons. Fermionic face data, injectivity, and maximum-capacity channel selection together force the seven-state face sector instead of leaving it as a tunable menu choice.

The minimality statement can also be written as a short proof. A volumetric cell in d = 3 needs at least four faces, so a tetrahedron is the first admissible simplex. The closure surrogate is three-component, so the face sector must be rich enough to support a nontrivial quadratic spectrum in d = 3 rather than a degenerate one-dimensional label count. Postulate II makes the face data fermionic, hence half-integer. Maximum-capacity channel selection then gives

jeff = 2j0, |M| = 2jeff + 1 = 4j0 + 1.

Injectivity across four faces requires |M| ≥4. The only half-integer option below j0 = 3/2 is j0 = 1/2, which gives jeff = 1 and |M| = 3, so it fails. The first admissible fermionic choice is therefore j0 = 3/2, giving jeff = 3 and the canonical seven-state face sector. In that precise sense, the (4-face, 7-state) tetrahedral package is the minimal architecture compatible with a three-component isotropic closure mode, injective boundary information, and finite volumetric counting.

B.2 Closure invariant, kernel, and unique fixed point

The canonical scalar closure invariant is

4 X

4 X

K2(b) = 48 −1

3 S2 −Σ2
, S =

i=1 mi, Σ2 =

i=1 m2 i .

The admissibility family is

pη(b) = 1 Z(η)e−ηK2(b), Z(η) = X

b∈B e−ηK2(b).

The closure condition ⟨K2⟩η = 3

2η has the unique solution η∗= 0.0298668443935.

Because the parity-symmetric ensemble is finite, the root-finding problem can be written directly from the exact discrete spectrum itself. The distinct closure-defect values and their degeneracies are K2 122

3 mult 96 96 96 288 192 144

3 mult 384 192 48 96 48

with total multiplicity 1680 as required. In particular,

a nae−ηK2 a, ⟨K2⟩η = P a naK2 ae−ηK2 a P a nae−ηK2a ,

Z(η) = X

where (K2 a, na) run over the table above. The closed-branch value η∗is therefore the unique root of an exact finite-spectrum equation, not an unseen numerical fit. The corresponding effective sharing entropy is gshare,eff = − X

b∈B pη∗(b) ln pη∗(b) = 7.41980002357.

The closed-branch moments used in the UV stiffness discussion are

⟨K2⟩η∗= 50.2229154254, Varη∗(K2) = 15.6889750078, aUV = 0.0637390269.

These values quantify the local stiffness of the canonical closure point rather than a tunable phenomenological uncertainty. The closure-saturation product is

2, C−1 cl = 2

Ccl := η∗⟨K2⟩η∗= 3

3.

This same reciprocal 2/3 reappears as the tetrahedral transverse export fraction in Appendix C and as the global correction in the faithful sector-resolution scale setting.

This is where the admissibility parameter stops being free. The kernel introduces η, and the closure condition removes its arbitrariness again by demanding that the fluctuation scale produced by the weighting agree with the weighting itself.

B.3 Rooted reduction and local benchmarks

Rooting on the shared face reduces the exact parity-symmetric ensemble to 140 rooted mi- crostates and 69 rooted closure classes. The rooted classes can be labeled by α = (m•, K2), so the same reduced state space already supports the local evaluation, the cavity benchmark, and the later shell propagation. The local information observable

σ(r) ind = H(X | Yr)

H(X)

has the principal pre-nonlocal benchmarks

σtoy ind = 0.44997, (22)

σloc ind = 0.44708, (23)

σBethe ind (J = 0) = 0.44749. (24)

Here the Bethe value is the homogeneous cavity evaluation on the 69×69 rooted-class interaction graph at zero transport coupling,

z−1





X

, X

µα ∝wα

β Uαβ(0)µβ

α µα = 1,



with z = 4 and Uαβ(0) the rooted shared-face compatibility matrix before shell transport is turned on. In other words, σBethe ind (J = 0) is the cavity-theory benchmark of the same explicit

rooted ensemble, not a disconnected numerical insert. The horizon target implied by the effective sharing entropy is σ∗= π gshare,eff = 0.42340665.

The gap between the local benchmarks and σ∗is therefore a genuinely shell / loop problem rather than a failure of the local admissibility closure.

That separation matters for the later UV story. It means the remaining work is not to repair the local closure ensemble, but to propagate it more accurately through transport and return structure.

B.4 What is fixed at this stage

By the end of the admissibility calculation, the framework has already fixed the microscopic counting ceiling, the unique closure point, the effective sharing entropy, and the local stiffness moments. What remains for the next appendix is not another entropy choice, but the propaga- tion of those quantities into edge transport, finite renormalization, and continuum normalization.

Appendix B completes the UV counting problem and records the unique admissibility closure together with the local benchmarks needed by the coefficient chain.

Appendix C: Edge Kernel, Finite Renormalization, and Contin- uum Matching

Appendix C carries the middle part of the UV-to-IR derivation. Appendix B fixed what the local boundary ensemble is. Appendix C asks how that local data propagate into edge transport, loop dressing, and finally the continuum stiffness coefficient of the weak-field EFT.

C.1 Channel-averaged isotropy identity and tree coupling

Let ˆni be the four face normals of a regular tetrahedron. The exact identity

4 X

i=1 ˆniˆnT i = 4

3I3

implies a channel-averaged transverse fraction of 2/3. The bare edge stiffness is therefore

Jbare = 2

3η∗= 0.0199112296.

For a rooted z = 4 coarse adjacency graph, the tree-to-lattice map gives

Jtree eff = Jbare

z −1 = 2η∗

9 = 0.0066370765.

This is the first place where local closure data become a transport law. The tetrahedral identity fixes the isotropic projection, and the rooted branching structure determines how much of the microscopic edge penalty survives as net outward propagation on the coarse graph.

C.2 Horizon target and shell convergence

The horizon-capacity target is

σ∗= π gshare,eff = 0.42340665.

At the derived coupling the explicit shell values are

σ(2) ind = 0.42143, σ(3) ind = 0.42166, ∆2→3 = 0.00023.

The residual shift from the target is already small and stable by shell depth r = 2, isolating the remaining correction to the loopy local-return sector rather than a broad nonlocal ambiguity.

So the shell calculation narrows the open problem substantially. The tree branch already lands very near the target, and the residual discrepancy can be assigned specifically to local returns rather than to an uncontrolled long-range correction.

C.3 Finite-loop self-energy closure

The leading loopy correction is organized as a local Dyson dressing:

J(ren) eff = Jtree eff 1 + Jtree eff Σret .

The structural decomposition is

Σret = 7 + 2

9 = 65

9 ,

and each term has a concrete return-channel origin. A short return motif leaves a shared face, explores a local closed loop, and re-enters the same coarse edge before contributing to net long- range transport. In the canonical label basis m = −3, −2, . . . , 3, there are exactly seven ways to do this without changing sector. These are the seven sector-diagonal returns, one for each face-label channel, and together they contribute

Tr(I7) = 7.

In addition to these label-preserving loops, permutation symmetry allows one collective mode shared across all channels. Writing

Psing = |u⟩⟨u|, u = 1 √

7(1, 1, . . . , 1),

this shared return is rank one. Any additional off-diagonal return sector would break the per- mutation symmetry of the canonical local ensemble, so there is no second independent collective channel to count. Only the transverse scalar branch feeds back into the coarse transport law, so the singlet first acquires the same 2/3 projection factor that appeared in the tree coupling. It is then diluted by the rooted branching factor 1/(z −1) = 1/3 on the z = 4 graph, because only one of the three outward branches returns to the original edge. The collective contribution is therefore

 = 2

Tr 2

3 1 3Psing

9,

since Tr(Psing) = 1. Equivalently,

Rret = I7 + 2

9Psing, Σret = Tr(Rret) = 7 + 2

9.

This is the sense in which the finite-loop coefficient is counted rather than guessed: seven inde- pendent label-preserving returns plus one shared singlet return with exactly the same projection and branching weights already fixed in the tree map. Hence

c(ren) loop ≡J(ren) eff Jtree eff = 1 1 + Jtree eff Σret ≈0.95426,

and J(ren) eff ≈0.00633348.

This reproduces the shell-target crossing near Jbare,cross ∼0.019 at the stated level of agreement.

The local Dyson dressing is therefore doing one precise job: it corrects the tree branch by accounting for the short motifs that recycle amplitude before it contributes to true coarse trans- port. The renormalized coupling is not a new parameter, but the tree coupling after local returns have been summed.

C.4 Euclidean-action normalization and continuum stiffness

The lattice quadratic form is interpreted canonically as a Euclidean action weight,

ℏ= J(ren) eff

IE

X

⟨ab⟩ (Qa −Qb)2,

2

over the microscopic four-cell

∆V4 = L4 ∗ c .

The same tetrahedral identity then yields

γQ = 4ℏc

3L2∗ J(ren) eff

for the occupancy field Qocc. With the horizon-capacity normalization

S = πQocc,

2(∂S)2 gives

the canonical convention γ

γ = 4ℏc 3π2L2∗ J(ren) eff = 4ℏc 3π2L2∗

2η∗/9 1 + (2η∗/9)(65/9).

Using the substrate-induced scale

G∗:= c3L2 ∗ ℏ ,

this is

γ = 4J(ren) eff 3π2 c4

G∗ ≈8.556 × 10−4 c4

G∗ .

This is the decisive stiffness-side matching step. Up to here the derivation has produced a dimensionless lattice weighting; after Euclidean normalization and faithful sector-resolution scale setting, that same weighting becomes the dimensionful continuum stiffness that appears in the weak-field action.

C.5 Local defect insertion and the source-side lattice constant

The stiffness-side matching is not the only UV quantity that can be closed locally. For the canonical rigid defect insertion, excluding one of the seven admissible face labels from one face removes exactly one-seventh of the isotropically averaged local partition weight. Therefore the logarithm of the isotropically averaged partition ratio is exactly

∆Sdef := −ln DZdef

This is the exact isotropic source benchmark in the canonical seven-label ensemble. The isotrop- ically averaged defect free-energy cost differs from it only at O(10−5) because the admissibility weighting breaks label symmetry only weakly.

The local source benchmark ln(7/6) should not be confused with the elementary fermionic one-bit anchor ln 2. The former is the isotropically averaged partition-ratio shift produced by removing one admissible label from the seven-label boundary ensemble. The latter is the intrinsic binary entropy of an elementary occupied/unoccupied fermionic face-exclusion defect. The source theorem uses ln(7/6) to normalize the local scalar insertion into the lattice response, while the electron anchor uses ln 2 to fix the mass–entropy unit of the elementary fermionic defect.

To propagate that local defect into the lattice field equation one needs the on-site Green function of the tetrahedral/diamond nearest-neighbor Laplacian. The dual graph of face-sharing tetrahedral cells is the four-valent diamond graph. Eliminating the two-sublattice structure gives the standard Brillouin-zone representation for the diamond lattice Green function [5]

[−π,π]3 4 d3k

Gtet(0) = 1 (2π)3

Z

16 −|1 + eik1 + eik2 + eik3|2 .

This integral is the reproducible source-side lattice constant: it is the self-energy of a unit point insertion for the same scalar mode whose long-wavelength stiffness was matched in Appendix C.4. Joyce’s exact evaluation of the diamond-lattice Green function, in this normalization, gives [6, 5]

Gtet(0) = 3Γ(1/3)6

214/3π4 = 0.4482203943883814 . . . .

Thus the source-side graph constant is an exact lattice invariant rather than a fitted numerical coefficient. Direct quadrature with endpoint extrapolation reproduces the same value.

Green-tensor transverse response. The same graph response also fixes the transverse ex- port weight w⊥, the single graph quantity that propagates downstream into the horizon channel count of Appendix F.5 and into the global ln(3/2) closure-saturation factor of the scale-setting relation. We compute it directly from the four nearest-neighbor bond frame, with no fitting freedom.

Let di, i = 1, . . . , 4, be the four diamond nearest-neighbor bond directions,

3 , d2 = (1, −1, −1) √

3 , d3 = (−1, 1, −1) √

3 , d4 = (−1, −1, 1) √

d1 = (1, 1, 1) √

3 .

They obey 4 X

i=1 da i db i = 4

3δab.

Distributing the scalar on-site Green response over the tetrahedral bond frame gives

Gab loc = Gtet(0)1

i da i db i = Gtet(0)

X

3 δab.

4

For a local horizon normal ˆr, P ab ⊥= δab −ˆraˆrb,

so G⊥= P ab ⊥Gab loc = 2

Thus the transverse export weight w⊥= 2/3 is the transverse part of the exact local graph response.

Using the field normalization S = πQocc, the rigid local defect shift is

δQdef = ∆Sdef

π = ln(7/6)

π .

The corresponding local source amplitude in lattice units is therefore

sdef = J(ren) eff δQdef Gtet(0),

so that sdef J(ren) eff = ln(7/6)

πGtet(0) = 0.109472228 . . .

is a pure number fixed by the same UV lattice geometry.

The Green-function constant turns the local insertion into a continuum source theorem. Defin- ing the defect-entropy density by σdef = ρ κm(L∗),

the tetrahedral projection used in the stiffness mapping gives

∇2δS = − 3L∗ 4Gtet(0) σdef.

Equating this with the weak-field source equation

∇2δS = −κ

γ ρ

closes the canonical source-to-stiffness ratio:

κ γ = 3L∗ 4Gtet(0)κm(L∗) .

This is the source-side counterpart of the stiffness derivation. The edge-kernel calculation fixes how the scalar capacity mode resists gradients; the Green-matched defect calculation fixes how localized matter defects source that same mode.

Equivalently, the same source closure fixes the continuum quantity Ξρ appearing in

κ = Ξρ L2∗κm(L∗).

The absolute scale of S∞remains a fixed-epoch normalization convention in the bridge law, not a residual freedom in the source projection. In the cell-normalized gauge natural to the local source theorem, one may write

Scell ∞= 3 ln 2 32πGtet(0) = 0.0461482516 . . . .

In a horizon-normalized gauge, S∞instead carries the much larger apparent-horizon capacity. These are not two physical constants. A constant rescaling of the entropy field rescales S∞ and κ/γ together, leaving κ/(γS∞) and hence G unchanged. Thus the weak-field source map is closed in the canonical branch once the mass–entropy map κm(L∗), the tetrahedral Green constant, and the standard cell convention are specified.

C.6 Local susceptibility cross-check

The exact local moments of the admissibility-closed ensemble supply an independent scale check on the source-side result. From the variance in Appendix B,

aUV := 1 Varη∗(K2) = 0.0637390269,

which is the local zero-mode inverse susceptibility of the closure scalar. Combining this with the same nearest-neighbor gradient template used in the stiffness matching, together with the fixed normalization S = πQocc, gives the susceptibility estimate

κ γ ≈1.487 × 103 Ξρ c(ren) loop L4∗κm(L∗)

.

This estimate is valuable as an internal audit: it uses the local branch curvature aUV, the derived gradient template, the loop-renormalization factor, and the same source map to check the scale of the source response. The actual closure of κ/γ in the canonical branch is the Green-matched theorem in Appendix C.5; the susceptibility estimate is retained as a consistency check rather than as the primary source derivation.

C.7 UV-to-IR payoff

At this stage the weak-field UV coefficient chain is explicit:

Ωtet →gshare,eff →L∗→Jbare →Jtree eff →Σret →J(ren) eff →γ.

The same chain feeds a0 = cH0gshare,eff

4π2 ,

and faithful sector resolution gives the independent scale prediction

4 ℏc m2e e−14gshare,eff.

G∗= c3L2 ∗ ℏ = 9

and the Green-matched source theorem fixes

κ γ = 3L∗ 4Gtet(0)κm(L∗).

The weak-field bridge then converts this source-to-stiffness ratio into the observed Newtonian normalization through the invariant combination κ/(γS∞). The comparison with the same G∗ fixed by faithful sector resolution is an overconstraint check, not an additional fit.

Appendix C completes the UV-to-continuum chain through the SI-normalized weak-field stiff- ness coefficient and the Green-matched source-to-stiffness ratio. The remaining uses of S∞ belong to the fixed normalization of the weak-field bridge, not to the source sector itself.

Appendix D: Weak-Field Technical Derivations, Electron Anchor, and EFT Consistency

Appendix D gathers the weak-field derivations that are conceptually central but too dense to repeat in full in the main line. It is best read as a technical support layer for the bridge law, Newtonian recovery, the electron anchor, and the basic consistency checks of the EFT.

D.1 Bridge-law uniqueness

The weak-field bridge is derived once and then used throughout. Let the lapse be written as

N = e−F(δS/S∞).

Additivity of independent deficits requires F(x+y) = F(x)+F(y), so continuity implies F(x) = cx. Standard weak-field metric normalization fixes c = 1/2, giving

N = e−δS/(2S∞)

and therefore Φ c2 = −δS

2S∞ to leading order. This is the unique weak-field bridge compatible with locality, additive inde- pendent deficits, and multiplicative redshift composition.

Writing the argument this explicitly removes one of the most common ambiguities in modified- gravity proposals. The bridge from entropy deficit to gravitational potential is not being chosen phenomenologically after the fact; it is fixed by the structural requirements of the weak-field limit itself.

D.2 Point source, Newton limit, and lensing

In the renormalized static branch, ∇2δS = −κ

γ ρ.

For a point source M,

4πγr, g(r) = c2κ 8πγS∞

δS(r) = κM

M

r2 = GM

r2 .

Because the leading entanglement stress carries no anisotropic stress,

Φ = Ψ

at the order treated. The effective-halo rewrite is

ρhalo(r) = 1 4πGr2 d dr

h r2(gobs −gbar) i .

Thus the same deficit field controls both orbital dynamics and light bending in the leading weak-field regime.

That shared control is the key weak-field consistency test. A viable branch must not reproduce galactic support only by sacrificing lensing, and the scalar deficit sector avoids that failure at the order treated.

D.3 Electron anchor and composite matter

The canonical fermionic entropy increment is

∆Sf = ln 2.

The UV mass normalization is κm,UV = ℏ cL∗

and the running law in the closed branch is

1+αcl , αcl = 0.

L∗

κm(ℓ) = κm,UV

ℓ

At the electron Compton scale ℓ= λe this gives

κm(λe) = me

ln 2,

which is the clean elementary anchor used here. Composite hadrons are not reduced to a bare constituent count. Their mass budget is assigned to a dressed bound-state entropy

mhadron = κm(ℓH)Sdressed ent,H ,

whose microscopic decomposition must include confinement, gluonic structure, trace-anomaly contributions, and chiral vacuum reorganization.

The contrast between the two sectors is deliberate. The electron is a clean one-bit defect anchor; hadrons are not. Their inertial content must therefore be assigned to a dressed entropy budget rather than to a naive constituent count.

D.4 Faithful sector resolution and induced G∗

The electron anchor enters the gravitational normalization through the absolute substrate length. The reduced electron Compton wavelength is

λe = ℏ mec,

and the admissibility-closed sharing entropy is

gshare,eff = 7.41980002357.

The closure fixed point also gives

⟨K2⟩η∗= 3 2η∗ , η∗= 0.0298668443935,

so the closure-saturation factor is

2, C−1 cl = 2

Ccl := η∗⟨K2⟩η∗= 3

3.

The faithful sector-resolution principle is

ln  λe

 = 7gshare,eff −ln Ccl = 7gshare,eff −ln 3



.

L∗

2

Equivalently,

L∗= 3

2 λee−7gshare,eff .

This is the named sector-resolution principle of the canonical branch. Once it is adopted, the cell length is fixed from non-gravitational input: the electron Compton scale and the already-derived substrate combinatorics. It gives

L∗= 1.60771947 × 10−35 m,

which is about 0.528% below the conventional CODATA Planck length.

The induced gravitational scale follows from the same algebra used in the stiffness matching:

G∗:= c3L2 ∗ ℏ .

Substituting the sector-resolution relation gives the closed form

2 λee−7gshare,eff 2 = 9

4 ℏc m2e e−14gshare,eff .

G∗= c3

3

ℏ

Numerically, G∗= 6.60399128 × 10−11 m3 kg−1 s−2,

about 1.053% below the CODATA value. The uncertainty imported from λe is many orders of magnitude smaller than this residual, so the difference is theoretical rather than metrological.

The factor 3/2 is fixed twice over inside the UV package, by independent routes. First, the tetrahedral identity gives the transverse export fraction

4 X

3, f⊥= 1 −1

3 = 2

1 4

i=1 (ˆni · ˆu)2 = 1

3.

Second, the admissibility fixed point gives

C−1 cl = η∗⟨K2⟩η∗
−1 = 2

3.

Thus the same 2/3 appears as both the geometric export fraction and the reciprocal closure- saturation factor. In the sector-resolution relation it enters once as a global subtraction ln(3/2), not seven times, because the closure condition is one scalar fixed-point condition on the admis- sibility family rather than seven independent sector constraints.

A short look-elsewhere check makes the role of this correction more precise. Write a possible log-gap correction as L∗= λe exp[−7gshare,eff + ∆].

Several framework numbers lie near the required correction, but they do not play the same structural role:

Correction ∆ Value Induced G resid- ual Structural status

ln(3/2) 0.40546511 −1.05% double-fixed as f −1 ⊥ and Ccl gshare,eff −7 0.41980002 +1.82% nearby, but not independently se- lected Gtet(0) 0.44822039 +7.78% local Green constant, not global ex- port factor 7[−ln c(ren) loop ] 0.32774720 −15.30% transport renormalization, wrong role

The preferred correction is the one already singled out by both the tetrahedral export geometry and the admissibility fixed-point saturation; the other nearby numbers play different structural roles.

The electron anchor carries three related roles. Its reduced Compton wavelength λe is the non-gravitational length used in faithful sector resolution. Its status as the lightest clean one- bit fermionic defect identifies which elementary excitation calibrates the seven-channel dressing block. Its mass fixes the mass–entropy map through

κm(λe) = me

The same elementary defect enters the two normalization channels in distinct roles: its Comp- ton length sets the UV cell scale, while its mass per one-bit defect entropy sets the source normalization.

The seven-sector scale-setting relation is not a counting heuristic; it has a concrete finite- dimensional realization on a transfer-operator history space, which we now construct. The face label algebra is

3 M

3 X

m=−3 CEm, EmEn = δmnEm,

A7 =

m=−3 Em = I7.

The admissibility-closed tetrahedral state space has 1680 oriented injective states, with station- ary weight pη∗(b) = Z−1e−η∗K2(b).

For a single sector, the refresh kernel

Pη∗(b, b′) = pη∗(b′)

has Perron stationary entropy gshare,eff. The point is not that the electron simultaneously occu- pies seven mutually exclusive face labels in one tetrahedron. The labels are sector channels in the dressing history of the one-bit defect. The appropriate support object is therefore a history space, with one admissibility-closed sector layer for each m = −3, . . . , 3.

Let

3 O

m=−3 H(m) B , dim HB = 1680,

Hhist =

and let Tm act as Pη∗on the mth factor and as the identity on the others. A representative one-pass dressing operator is D(0) e = T−3T−2 · · · T3.

The displayed order is only a representative of the symmetrized one-pass class. It does not introduce a physical ordering of the face sectors, nor does it multiply the support by an extra factor of 7!. What matters is that the ground state of the elementary fermionic defect resolves each simple face sector once: fewer sectors leave unresolved label memory, while repeated sectors describe excited or additionally dressed states rather than the elementary anchor.

A sector-resolution step should not be identified with conditioning the 1680-state ensemble on boundary states containing a given label m. Such conditioning changes the entropy and does not reproduce gshare,eff. The sector label instead specifies which simple face-algebra channel is being resolved while the admissibility cloud sampled in that step remains the full closed boundary ensemble.

With the effective dimension defined by the stationary Shannon entropy of the positive history kernel, each sector contributes gshare,eff, so

dimeff(D(0) e ) = exp(7gshare,eff) = 3.60286052 × 1022.

Only the transverse part of this support is exported into the weak-field scalar channel. The normalized export weight is

w⊥= 2

3,

the same fraction fixed by the tetrahedral transverse projection and by the reciprocal closure- saturation factor. Hence

dimeff(De,⊥) = w⊥dimeff(D(0) e ) = 2

3 e7gshare,eff = 2.40190701 × 1022.

The faithful sector-resolution principle is then the support-scale relation

λe L∗ = dimeff(De,⊥) = 2

3 e7gshare,eff .

This is the sole remaining load-bearing microscopic principle in the non-gravitational scale- setting branch: the electron is assumed to be the ground one-bit defect whose physical support is one complete transverse-exported faithful sector-resolution block. Once that identification is granted, the seven-fold additivity is product-measure entropy additivity and the 2/3 factor is the tetrahedral transverse projection of Appendix C.5; nothing else in the chain remains free.

This is the microscopic content of the length formula above. It states that the elementary one- bit fermionic defect is supported over one complete transverse-exported dressing block. A deeper Hamiltonian derivation would need to show that the lightest charged fermionic defect indeed selects this faithful one-pass sector resolution as its ground support. The matched weak-field gravitational constant then follows from the gauge-invariant bridge

G = c2

8π κ γS∞ ,

with the entropy-unit normalization handled by the rescaling convention described in Ap- pendix C.5.

D.5 EFT consistency checklist

The weak-field EFT does not rely only on successful phenomenology; it also passes a standard consistency checklist at the level claimed here.

• ‘No ghost‘: the scalar kinetic term carries positive sign because γ > 0.

• ‘No tachyon‘: the quadratic fluctuation operator contains no mass term at this order.

• ‘Correct-sign sourcing‘: the defect-source coupling lowers the available entanglement ca- pacity around positive-mass defect configurations rather than generating repulsive static behavior in the weak-field branch.

• ‘Causal propagation‘: the transport completion satisfies D/τ0 = c2, so the time-dependent sector propagates at finite signal speed.

• ‘Weak-field unitarity below cutoff‘: once the scalar sector is quantized around the weak- field branch, the absence of ghost or tachyonic modes leaves an ordinary sub-cutoff scalar EFT rather than an obviously pathological one.

• ‘Energy-condition role‘: the scalar gradient sector contributes positive local stiffness en- ergy, while cosmological acceleration enters through the background branch rather than through a ghost-like local degree of freedom.

These statements are made at the EFT level claimed here. They do not replace the need for a fuller UV derivation, but they do show that the weak-field scalar sector is not buying phenomenology by obvious field-theoretic pathology.

The checklist is intentionally modest. Its role is not to prove ultraviolet completion of the full framework, but to show that the low-energy scalar sector used in the weak-field branch passes the standard first tests of EFT health.

The one place where an explicit formula is worth recording is linear vacuum stability in the time-dependent sector. Writing a small perturbation δs about the vacuum branch, the linearized telegrapher equation is τ0 ¨δs + ˙δs −D∇2δs = 0.

For a plane-wave mode e−iωt+ik·x, this gives the dispersion relation

τ0ω2 + iω −Dk2 = 0.

With τ0 > 0 and D > 0, the corresponding mode frequencies have non-growing time dependence, so the vacuum is linearly stable. The same sign structure is what underlies the earlier no-ghost and no-tachyon statements: positive kinetic stiffness, positive transport coefficients, and no negative mass-squared term in the linearized sector.

D.6 Quadratic fluctuations and weak-field stability

Expanding the action about an on-shell background yields the quadratic fluctuation operator

I(2)[δS] = − Z d4x √−g γ

2 gµν∂µδS ∂νδS.

There is no quadratic mass term at this order, so the low-energy scalar sector contains one massless bosonic mode. Stability requires γ > 0, which is reinforced in the microscopic realization appendix by condensate hydrodynamics.

This is also the local EFT reason the bosonic occupancy language in the galactic section is natural rather than decorative. The weak-field branch genuinely contains a stable massless scalar mode whose occupation can be discussed meaningfully.

Appendix D provides the technical support layer for the weak-field bridge, Newton limit, electron anchor, substrate length branch, and EFT consistency audit.

Appendix E: Transport, Cosmology, and Hubble-Tension Imple- mentation

Appendix E collects the time-dependent and homogeneous extensions of the static branch. The common purpose of these subsections is to show that the same scalar medium can propagate causally, relax toward its static limit, and support a cosmological background mode without losing contact with the weak-field structure already derived.

E.1 Telegrapher equation and causal closure

The time-dependent deficit field obeys

τ0∂2 t δS + ∂tδS = D∇2δS + Aχ, A D = κ

γ .

Causality requires D τ0 = c2.

In the canonical no-new-IR-scale branch,

τ −1 0 = H0, D = c2

H0 .

This is the minimal causal completion of the static Poisson sector. The telegrapher form supplies propagation and relaxation, but it is chosen so that the static weak-field law remains the exact late-time limit rather than being replaced by a new phenomenological rule.

E.2 Static-limit recovery for galaxies

For a Fourier mode k, the telegrapher characteristic equation

τ0s2 + s + Dk2 = 0

has the roots s = −1

2τ0 ± iωk, ωk ≃ck

whenever 4τ0Dk2 ≫1. Galactic wavelengths are far below the critical scale

λc = 4πc

H0 ≈54 Gpc,

so galactic modes are deeply underdamped. Time-averaging the sourced solution over intervals large compared with 2π/ωk returns the static Poisson branch exactly, and the residual pondero- motive correction scales parametrically as

2 ∼10−8.

δFpond

Fstatic ∼e−T/(2τ0) ωorb

ωk

That estimate is why the transport sector does not undercut the static galactic results. The oscillatory contribution is present, but it is parametrically too small to compete with the near- stationary weak-field branch in ordinary galactic systems.

E.3 Homogeneous mode and cosmological sourcing

The cosmological split is S(x, t) = S(t) + s(x, t),

with S(t) the homogeneous mode and s(x, t) the inhomogeneous weak-field sector. The back- ground capacity is normalized by the apparent horizon,

S∞(t) = πRA(t)2

L2∗ , RA(t) = c p

H2 + kc2/a2 .

Because the field couples to the trace of the stress-energy tensor, the homogeneous mode is suppressed during radiation domination and turns on near matter–radiation equality.

This timing is the central cosmological virtue of the mechanism. The homogeneous mode is quiet when it must be quiet, then becomes relevant close to the epoch where a sound-horizon shift is most useful.

E.4 Sound-horizon shift and shear lock

In the closed cosmological branch, the trace-sourced homogeneous mode acts as a transient early-energy contribution. The qualitative payoff is a smaller sound horizon and an upward shift of the CMB-inferred Hubble constant toward the upper-68 / low-69 km s−1 Mpc−1 range. Local weak-field predictions are protected by the separation between S(t) and s(x, t): the homogeneous mode changes the background branch without rewriting the local static Poisson law.

The result is qualitative but substantial. The homogeneous mode can matter cosmologically without forcing a re-tuning of the local weak-field sector that already fixed the galactic branch.

Appendix E closes the transport relation and preferred branch, while the cosmological sector remains structurally supported but not yet Boltzmann-closed.

Appendix F: Constrained-Capacity Strong-Field Branch

Appendix F records the strong-field black-hole branch in the form used by the main text. The result is not a full microscopic theory of collapse, but it is stronger than a mere horizon mnemonic. Once the bounded capacity variable is adopted and the lapse is tied to surviving capacity, the static spherical exterior is fixed: the continuum EFT lives on the domain q > 0, reduces to vacuum Einstein dynamics there, and terminates at the q = 0 surface.

F.1 Bounded capacity and the unique lapse map

The strong-field order parameter is the surviving-capacity fraction

q(x) = Sent(x)

S∞ ∈[0, 1].

This bound follows directly from finite local channel capacity. If the vacuum channel count is finite and Sent is the logarithmic coarse entropy of the surviving local ensemble, then no physical branch can have either negative capacity or more than the asymptotic vacuum capacity.

In a static exterior, the lapse associated with the asymptotic Killing time is determined by the local surviving capacity. Let N = f(q).

The conditions are: f(1) = 1, lim q→0+ f(q) = 0.

The substrate-level composition axiom is that independent serial capacity losses compose mul- tiplicatively on the lapse: f(q1q2) = f(q1)f(q2).

This is the assumption that extends the linear weak-field match to a nonlinear lapse map. With continuity, the positive solutions on (0, 1] are f(q) = qα. Expanding near q = 1 −ϵ gives

N = qα = 1 −αϵ + O(ϵ2).

The weak-field bridge gives N = 1 −ϵ

2 + O(ϵ2),

so α = 1/2 and therefore N = √q, N2 = q.

Equivalently, if one writes N 2 = F(q), the unique continuous multiplicative completion is F(q) = q. The nonlinear lapse rule is therefore fixed by capacity composition and weak-field matching; it is not a freely chosen black-hole ansatz.

F.2 Constrained-capacity action on the physical domain

The physical strong-field domain is

Mq = {(t, x) | q(t, x) > 0}.

The minimal constrained-capacity action is the ADM gravitational action on Mq, with a mul- tiplier enforcing the lapse-capacity relation. In canonical form,

Mq dt d3x h πij ˙hij −NHGR −NiHGR i + √

Istrong = Z

h λ (N2 −q) i + Imatter + I∞+ I∂Mq.

The standard ADM constraint densities are

√

2π2  − c4

 πijπij −1

HGR = 16πG

h (3)R, HGR i = −2Djπji.

c4√

16πG

h

The same content can be written in Lagrangian form as

Mq dt d3x N √

Istrong = c4

Z

h  (3)R + KijKij −K2

16πG

Mq dt d3x √

+ Z

h λ (N2 −q) + Imatter + I∞+ I∂Mq.

Here hij is the spatial metric, N and N i are the lapse and shift, and

Kij = 1 2N

 ˙hij −DiNj −DjNi 

is the extrinsic curvature. The term I∞is the usual asymptotic boundary term required for a well-posed variational principle and finite ADM energy, while I∂Mq is the effective action carried by the q = 0 capacity-exhaustion surface.

Varying λ gives N2 = q.

The q variation gives, schematically,

− √

δq + δI∂Mq

h λ + δImatter

δq = 0.

Away from the q = 0 boundary, and in bulk vacuum with no explicit matter coupling directly to q, this gives λ = 0. The lapse variation gives

HGR + Hmatter = 2N √

h λ,

so in bulk vacuum HGR = 0.

The shift variation gives the ordinary momentum constraint,

HGR i + Hmatter i = 0.

The metric evolution equations then reduce to the vacuum Einstein equations on Mq [7]. The constrained branch is thus not a new scalar-hair exterior. It is Einstein vacuum evolution on the surviving-capacity domain, together with the algebraic statement that the lapse is the square root of local capacity.

F.3 Horizon boundary action and boundary conditions

The only new strong-field ingredient not present in ordinary exterior GR is the inner bound- ary action on the capacity-exhaustion surface. At the EFT level its variation may be written schematically as

∂Mq d3y √σ 1

2sABδσAB + χ∂δq + · · ·  ,

δI∂Mq = Z

where σAB is the induced metric on the boundary world tube, sAB is the effective boundary stress response, and χ∂is the response conjugate to the capacity variable. The universal boundary condition is not optional: q  ∂Mq = 0.

The further constitutive information–whether the boundary is absorbing or partially reflecting, how quickly its channels relax, and what microscopic degeneracy they carry–is encoded in I∂Mq.

This is why the static exterior can close before the full microscopic black-hole sector is finished. The exterior bulk equations only require λ = 0 on Mq and the algebraic condition N 2 = q. The detailed horizon-channel physics lives in the boundary action. A shape variation of the moving boundary would give a force-balance condition of the schematic form

∂Mq d3y √σ ξn (Pbulk + P∂q) = 0,

δξIstrong = Z

where ξn is the normal displacement and the two pressures encode the limiting bulk stress and the boundary-channel response. Deriving P∂q from the tetrahedral graph ensemble is one of the remaining strong-field closure tasks.

F.4 Static spherical vacuum exterior

In static spherical vacuum, the result follows immediately. On Mq, the bulk equations are the vacuum Einstein equations. The unique asymptotically flat static spherical solution is the Schwarzschild exterior,

−1 dr2 + r2dΩ2,

ds2 = −  1 −2GM

 c2dt2 +  1 −2GM

c2r

c2r

so the capacity variable is

q(r) = N2(r) = 1 −2GM

c2r , r > rh,

with rh = 2GM

c2 .

This also agrees with the weak-field capacity deficit:

δS(r)

S∞ = 2GM

c2r , q(r) = 1 −δS(r)

S∞ .

The domain carries the physical content. The exterior r > rh is exactly the standard Schwarzschild exterior. At r = rh, q = 0. Continuing the same real capacity branch to r < rh would require q < 0, which is not in the state space. In this framework the classical Schwarzschild interior is therefore not the physical continuation of the same continuum EFT. It is the GR man- ifold analytically continued beyond the surface where the entanglement-capacity substrate has no remaining local continuum channels.

This exterior-domain result does not by itself decide what an infalling observer experiences at the q = 0 surface. Whether the capacity-exhaustion boundary is locally smooth, dissipative, or anomalous is a question about the boundary microphysics encoded in I∂Mq, not a conclusion fixed by the exterior Schwarzschild closure alone.

F.5 Horizon thermodynamics and boundary capacity

Because the exterior geometry is unchanged, semiclassical quantities depending only on the exterior near-horizon saddle are unchanged. The Euclidean continuation is used here as an exterior-saddle calculation: the resulting periodicity depends on regularity of the near-horizon

exterior geometry, not on a physical continuation of the substrate past q = 0. The Euclidean regularity argument therefore gives the standard Hawking temperature [9, 10],

TH = ℏc3

8πGMkB .

The same exterior Euclidean saddle gives the Bekenstein–Hawking area law [8, 9],

SBH = kBA

4L2 P .

We now derive the 1/4 area coefficient from the microscopic graph response and a single boundary channel-counting rule, rather than taking it as input. The q = 0 surface is a satu- rated boundary of the capacity network, so the entropy is carried by boundary channel degrees of freedom rather than by an interior volume. The exact diamond-lattice Green response in Appendix C gives the transverse boundary component

G⊥= 2

3Gtet(0).

What remains is to count how many elementary active channels per microscopic area cell this transverse response supports. The horizon is a codimension-one saturated surface, so it counts exported boundary channels rather than bulk volume states. The channel density is not intro- duced as a new normalization constant; it is computed from the transverse component of the already-fixed bulk graph response, measured in elementary one-bit units. Since one elementary active channel carries entropy ln 2, the number of active channels per outward angular direction is G⊥/ ln 2, and integrating over the spherical angular measure gives the total active channel density

ln 2 = 8πGtet(0)

nhor = 4π G⊥

3 ln 2 .

Using the cell-normalized capacity baseline from Appendix C,

Scell ∞= 3 ln 2 32πGtet(0),

one obtains nhorScell ∞= 1

4.

Thus a horizon of area A carries the dimensionless entropy

Shor

kB = A 4L2∗ ,

and, after the gravitational scale has been matched so that LP (G∗) = L∗, the usual Bekenstein– Hawking coefficient is recovered. The area coefficient is therefore conditionally closed under the boundary channel-counting rule. The remaining strong-field task is not to tune the coeffi- cient, but to derive the boundary rule and the full action I∂Mq from the saturated-boundary Hamiltonian.

F.6 Absorption, ringdown, and echoes

Exterior wave propagation is also unchanged as long as the q = 0 boundary is effectively ab- sorbing. Perturbations outside the capacity-exhaustion surface obey the usual Schwarzschild Regge–Wheeler/Zerilli scattering problem [11, 12]. With an absorbing boundary condition at q = 0, the greybody factors, absorption cross sections, and quasinormal ringdown agree with the standard Schwarzschild exterior calculation.

Echoes are therefore not generic in this framework. They arise only if the microscopic bound- ary channels have finite reflectivity or if the effective reflection surface is displaced outward to a stretched layer q = ϵ > 0.

In that case the region between the exterior potential barrier and the partially reflecting bound- ary behaves as a cavity, and a typical echo delay scales as

∆techo ∼2rh

c | ln ϵ| + τch,

where τch is a boundary-channel relaxation time. Absence of echoes is the default absorbing- boundary prediction; echoes, if detected, would probe boundary microphysics rather than follow automatically from the absence of a classical interior.

F.7 Dynamical formation as a free-boundary problem

The static branch does not by itself solve collapse. A dynamical theory must explain how a q = 0 surface forms and moves. The natural completion is a bounded causal transport system for q, for example ∂tq + DiJi = −Γ(q)Σ[Tµν],

τJ(∂t + Lv)Ji + Ji = −D(q)Diq.

Here Ji is the capacity flux, Σ[Tµν] is a positive depletion source built from the collapsing stress- energy, D(q) is a bounded mobility, Γ(q) is a bounded depletion rate, and τJ > 0 is a relaxation time. Eliminating Ji gives a telegrapher-type equation with finite characteristic speed

r

D0

vcap ∼

τJ .

Choosing constitutive functions such as

D(q) = D0q(1 −q), Γ(q) = Γ0q

makes q = 0 and q = 1 invariant sets, preventing the evolution from overshooting into q < 0.

When q first reaches zero on a two-surface, that surface becomes a moving boundary

∂Mq(t) = {x | q(t, x) = 0}.

The level-set kinematics are fixed by differentiating q(t, X(t)) = 0 along the moving surface:

q→0+.

Vn = −∂tq

|∇q|

In spherical symmetry this becomes

r=rf(t) .

drf

dt = −∂tq

∂rq

During continued infall the exterior should be Vaidya-like with a slowly varying mass parameter, settling to the Schwarzschild exterior after the front stabilizes. This is a well-posed program, but not yet a closed derivation: the transport coefficients, boundary action, and channel relaxation spectrum must be computed from the graph ensemble or constrained by simulation.

The dynamical system is presumed to preserve the usual covariant conservation of the com- bined matter-plus-capacity stress-energy, with any local matter depletion balanced by flux,

boundary work, or capacity-sector stress. Showing that this conservation structure follows from a graph-derived transport action, rather than imposing it as a constitutive condition, is part of the dynamical closure work.

The concrete closure tests are correspondingly specific. A spherical collapse simulation should show formation of the first q = 0 surface without overshoot into q < 0. A coupled matter-plus- capacity run should approach a Vaidya exterior during accretion and a Schwarzschild exterior after settling while satisfying the combined conservation law. Exterior perturbation simulations with an absorbing boundary should reproduce standard Schwarzschild greybody factors and ringdown, while partial-reflectivity runs should produce controlled echo delays. Finally, a mi- croscopic boundary-action calculation or a graph-ensemble Monte Carlo of saturated boundary channels should reproduce the channel-counting rule that yields nhorScell ∞= 1/4. These are not new fit knobs; they are the numerical and microscopic tests that would close the dynamical and boundary sectors.

F.8 PPN boundary and weak-field breakdown

In the weak-field Solar-System regime, the scalar sector yields

γPPN = βPPN = 1 + O(Φ2/c4),

with the remaining PPN coefficients vanishing in the canonical covariant branch. Weak-field truncations fail only when |Φ|

c2 = O(1), δS S∞ = O(1),

which is exactly the regime where q rather than δS is the correct variable.

Appendix F therefore separates the strong-field claims cleanly. The bounded capacity variable, the lapse rule N 2 = q, the constrained-capacity exterior action, and the static Schwarzschild exterior on Mq are closed within the EFT. The area coefficient is conditionally closed under the boundary channel-counting rule. The microscopic horizon boundary Hamiltonian, collapse dy- namics, reflectivity, and rotating or charged generalizations remain the open strong-field research program.

Appendix G: Many-Pasts Operational Closure and Arrow of Time

Appendix G records the operational content of the Many-Pasts branch in compact form. The main thing to keep in view is that this branch is conservative where laboratory quantum me- chanics is concerned and ambitious only in the larger interpretive and cosmological claims built on top of that operational core.

G.1 Closed operational weight

The canonical history weight is

P(H|P) ∝e−D(H,P), D(H, P) = −ln Tr ΠP ρH→now
.

This is the operational branch with α = 1, β = 0.

Writing it this way matters because the generalized family is no longer left open in practice. The laboratory branch is fixed before any interpretive discussion begins.

G.2 Born recovery and no-signaling

In the ordinary projective laboratory limit,

e−D(H,P) = Tr ΠP ρ
,

so the standard Born structure is recovered exactly. This fixes α = 1. An independent signaling- sensitive bias channel is forbidden, which fixes β = 0. The result is ordinary operational quantum mechanics rather than a modified laboratory theory.

That is the core closure claim of the appendix. Whatever additional content the Many-Pasts sector adds, it does not do so by changing standard Born-rule laboratory predictions.

G.3 Arrow of time from conditional typicality

Let h = {Mt}t<t0 be a macrohistory conditioned on present records Mt0. If the count of compatible microhistories is Nh, then

P(h|Mt0) ∝Nh.

Under coarse-grained factorization,

ln P(h|Mt0) ≈ X

t<t0 S(Mt) + X

t<t0 ln T(Mt+∆t|Mt) + const,

so entropy growth appears as a counting dominance effect among record-compatible histories rather than as a new laboratory coupling.

The arrow-of-time claim should therefore be read as a statement about conditional counting in the space of histories, not as the introduction of a new dynamical force.

Appendix G settles the laboratory sector operationally and leaves the cosmological and arrow- of-time content as a coherent interpretive extension.

Appendix H: Microscopic Realization and Coarse-Graining

Appendix H addresses a different question from the weak-field appendices. Instead of asking whether the coefficient chain is internally closed, it asks whether a plausible microscopic real- ization exists in which the same scalar stiffness and defect ontology arise naturally.

H.1 GFT condensate realization and coarse-graining

The candidate microscopic realization is a GFT/condensate picture with bosonic tetrahedral quanta ϕ(g1, . . . , g4) and fermionic defects ψ. In the condensate regime, the coarse field may be written as σ(x) = p

n(x) eiθ(x).

The hydrodynamic identity

|∇µσ|2 = (∇µn)2

4n + n(∇µθ)2

shows that if

Sent(x) = S0 + α lnn(x)

nbg ,

then the coarse action contains a positive scalar stiffness

The coarse source channel arises from fermionic face exclusion: what is macroscopically read as matter is a localized defect of the condensate, and the surrounding reduction of available occupancy is the long-wavelength field captured by the EFT. In this sense the microscopic appendix plays one clean role: it shows that the EFT is not hanging in midair, even though a finished first-principles derivation of every inhomogeneous continuum coefficient from the full underlying kernel is not yet available.

That is why the appendix remains brief but important. It does not replace the explicit coefficient derivation carried out earlier, but it shows that the ontology and sign choices of the EFT are compatible with a concrete microscopic picture rather than merely with an abstract formalism.

Appendix H gives a coherent microscopic realization supporting the closed weak-field chain, but not yet a closure-defining first-principles derivation of every continuum term.

Appendix I: Mass and Gauge Extensions

Appendix I collects sectors that are structurally connected to the same entanglement logic but are not part of the closed weak-field core. They are kept here because they show how the framework may extend, not because the main derivation depends on them.

I.1 Mass extensions and lepton-shell sector

Beyond the electron anchor, the charged-lepton extension is formulated as a shell spectrum of fermionic defect excitations,

log mN = C0 + B0N + A0N2, N = 0, 1, 2.

The physical picture is that the electron is the ground-state fermionic defect, while the muon and tau are successive radial entanglement-shell excitations of the same core structure. In that reading, the quadratic log-mass ladder is the closure form taken by a short finite shell sequence, with the three coefficients fixed by the shell structure itself.

This shell ladder is not independent of the gravity sector. Its degeneracy structure is tied to the same sharing-entropy logic that fixes the weak-field couplings, so the particle hierarchy and the gravitational normalization are not being treated as disconnected subsystems.

Within this extension, the finite tetrahedral boundary topology also constrains the charged- lepton shell ladder to terminate after three generations. That statement should be read carefully: it is a striking structural payoff of the current shell picture, but it belongs to the extension layer rather than to the closed weak-field core. Still, if the framework really admits only three charged- lepton generations in this construction, that is a genuine and falsifiable output rather than an external Standard Model input.

This sector is treated as a constrained extension of the same entanglement closure logic rather than as a replacement for the electron anchor. Composite hadrons remain part of the dressed bound-state entropy program rather than a completed output of the current extension.

The distinction matters. The electron anchor remains the clean weak-field entry point, while the heavier mass sectors are exploratory continuations of the same logic rather than closure- defining ingredients.

I.2 Gauge-structure extension

The same baseline-redundancy logic that underlies the gravity sector can be extended to gauge sectors. For a conserved charge sector Q, introduce an entropy-like potential SQ(x) and require

that physical observables depend only on differences of that potential rather than on its absolute baseline. Promoting that redundancy to a local symmetry requires a compensating connection. In the Abelian case, local baseline redundancy is implemented by

DµSQ = ∂µSQ −qAµ,

with SQ →SQ + α(x), Aµ →Aµ + 1

q ∂µα.

This yields the standard Abelian gauge structure, with Maxwell-type dynamics for Aµ and straightforward non-Abelian generalization for multiplet-valued entropic potentials, where the same redundancy principle leads to Yang–Mills covariant derivatives and field strengths in the usual form.

This does not derive the full gauge sector. It shows that the baseline-redundancy logic used elsewhere in the framework naturally points toward familiar gauge structure rather than away from it. Gravity and gauge sectors are then aligned by a common principle: only baseline- invariant deficit information is physically meaningful.

Appendix I remains a coherent extension layer: structurally linked to the same entanglement logic, but not part of the closed static weak-field derivation chain.

Appendix J: Numerical Checks and Robustness

Appendix J is intentionally modest. It does not add new derivations. It collects the main numerical cross-checks that make it easier to see that the same coefficient chain survives repeated contact with independent benchmark calculations.

J.1 Cross-sector numerical checks

The cross-check program includes:

• the one-bit fermionic defect check ∆Sf = ln 2;

• the rooted-shell convergence check σ(2) ind ≃σ(3) ind;

• the UV closed-branch moments ⟨K2⟩η∗, Varη∗(K2), and aUV;

• cross-sector consistency among the electron anchor, the substrate length L∗, the induced scale G∗, the Green-matched Newton closure, and the galactic scale a0.

These checks do not replace the derivations, but they show that the same coefficient chain survives independent numerical scrutiny across the sectors where closure is claimed.

J.2 Reproducibility ledger for the substrate scale

The substrate-length calculation is short enough to record as a numerical ledger. Enumerating the 1680 oriented injective tetrahedral states with labels m = −3, . . . , 3, weighting them by e−ηK2, and solving

⟨K2⟩η = 3

2η gives η∗= 0.02986684439352237, gshare,eff = 7.419800023570903.

The one-pass seven-sector history support and its transverse export are then

e7gshare,eff = 3.602860521062804 × 1022,

2 3e7gshare,eff = 2.401907014041869 × 1022.

Using λe = ℏ/(mec) = 3.861592671986303 × 10−13 m gives

L∗= λe (2/3)e7gshare,eff = 1.607719470158885 × 10−35 m,

and hence

G∗= c3L2 ∗ ℏ = 6.603991270884347 × 10−11 m3 kg−1 s−2.

The source-side Green constant is obtained independently from the diamond-lattice integral in Appendix C.5. Uniform-grid quadrature with endpoint extrapolation gives

Gtet(0) = 0.448220394388 . . . ,

confirming the exact Joyce value

Gtet(0) = 3Γ(1/3)6

214/3π4 .

This is the value used in the source theorem. These numbers are not additional inputs; they are the numerical evaluation of the finite spectrum, the seven-sector history support, and the graph Green function already defined in the derivation.

Appendix J is supportive rather than closure-defining, serving as an audit layer for numerical consistency rather than an additional derivational sector.

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