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

## 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.

---

## Full Text

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


![Table 1](paper-70-v4_images/table_1.png)
*Table 1*

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.


![Table 2](paper-70-v4_images/table_2.png)
*Table 2*

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


![Table 3](paper-70-v4_images/table_3.png)
*Table 3*

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.

References

[1] S. S. McGaugh, F. Lelli, and J. M. Schombert, “Radial acceleration relation in rotationally
supported galaxies,” Physical Review Letters 117, 201101 (2016).

[2] D. Clowe et al., “A direct empirical proof of the existence of dark matter,” The Astrophysical
Journal Letters 648, L109–L113 (2006).

[3] B. Bertotti, L. Iess, and P. Tortora, “A test of general relativity using radio links with the
Cassini spacecraft,” Nature 425, 374–376 (2003).

[4] Y.
Tian
et
al.,
“The
radial
acceleration
relation
in
CLASH
galaxy
clusters,”
arXiv:2001.08340 (2020).

[5] A. J. Guttmann, “Lattice Green functions in all dimensions,” Journal of Physics A: Math-
ematical and Theoretical 43, 305205 (2010).

[6] G. S. Joyce, “On the cubic lattice Green functions,” Proceedings of the Royal Society of
London A 445, 463–477 (1994).

[7] R. Arnowitt, S. Deser, and C. W. Misner, “The dynamics of general relativity,” in Gravita-
tion: An Introduction to Current Research, ed. L. Witten (Wiley, New York, 1962).

[8] J. D. Bekenstein, “Black holes and entropy,” Physical Review D 7, 2333–2346 (1973).

[9] S. W. Hawking, “Particle creation by black holes,” Communications in Mathematical Physics
43, 199–220 (1975).

[10] G. W. Gibbons and S. W. Hawking, “Action integrals and partition functions in quantum
gravity,” Physical Review D 15, 2752–2756 (1977).

[11] T. Regge and J. A. Wheeler, “Stability of a Schwarzschild singularity,” Physical Review
108, 1063–1069 (1957).

[12] F. J. Zerilli, “Effective potential for even-parity Regge–Wheeler gravitational perturbation
equations,” Physical Review Letters 24, 737–738 (1970).


---

*This document was automatically generated from the PDF version.*