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

## Abstract

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

The central result is that this picture is not free to be adjusted after the fact. Once one accepts the finite-capacity medium, the three founding postulates, and a specific minimal model for the smallest cell of space, the weak-field coefficients are fixed by counting the possible configurations of that cell. A single chain of calculation, with nothing left to tune galaxy by galaxy, then gives Newton's law, the acceleration scale a0 at which galaxies begin to depart from Newtonian expectations, the tight observed relation between galaxy rotation and ordinary matter, and the leading no-slip lensing structure.

The electron, the lightest charged particle, plays a double role. It fixes the exchange rate between committed entanglement and mass, and it also fixes the absolute size of the smallest cell. The second step uses Many-Pasts, one of the three founding postulates: the claim that the medium's present state is supported not by a single microscopic past but by many admissible histories, weighted by how well each leads to the present. In the operational branch used here this leaves all ordinary quantum-mechanical predictions intact - Born-rule statistics and no-signaling are preserved - while making the electron's microscopic dressing memoryless, and that memorylessness fixes the substrate length.

The length fixed in this way - from the electron Compton scale and the tetrahedral entropy, with no value of G entering the chain - implies a gravitational scale within about one percent of the measured Newton constant. We treat this as a calibrated coherence check, not as a clean independent prediction, because the construction's form was selected with that target partly in view. Appendix K records that provenance and the associated fork accounting.

Beyond the static weak-field sector, the framework extends to time-dependent transport, clusters, cosmology, the saturated early universe, black holes, and the charged-lepton spectrum, at varying and explicitly labeled levels of closure. The completed claim of the paper is the chain from the microscopic construction to ordinary weak gravity; the later sectors are presented as conditional extensions, frontier completions, or open tests.

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## Full Text

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

Jacob Chinitz

June 21, 2026

Abstract

We propose that empty space is not a passive backdrop but a physical medium with a
finite budget of quantum entanglement: the linking structure that allows parts of a quan-
tum system to share state. Matter forms when some of that capacity becomes locked into
stable, localized defects of the medium. A particle’s mass measures how much entanglement
is committed to such a defect. Gravity is the surrounding capacity-strain field: near matter,
slightly less entanglement capacity is freely available, and in the weak-field limit the frac-
tional shortfall gives the gravitational potential. The excess acceleration seen in galaxies,
usually attributed to particle dark matter, is treated here as the large-scale continuation of
the same capacity response rather than as a new unseen substance.
The central result is that this picture is not free to be adjusted after the fact. Once one
accepts the finite-capacity medium, the three founding postulates, and a specific minimal
model for the smallest cell of space, the weak-field coefficients are fixed by counting the
possible configurations of that cell. A single chain of calculation, with nothing left to tune
galaxy by galaxy, then gives Newton’s law, the acceleration scale a0 at which galaxies begin
to depart from Newtonian expectations, the tight observed relation between galaxy rotation
and ordinary matter, and the leading no-slip lensing structure.
The electron, the lightest charged particle, plays a double role. It fixes the exchange
rate between committed entanglement and mass, and it also fixes the absolute size of the
smallest cell. The second step uses Many-Pasts, one of the three founding postulates: the
claim that the medium’s present state is supported not by a single microscopic past but by
many admissible histories, weighted by how well each leads to the present. In the operational
branch used here this leaves all ordinary quantum-mechanical predictions intact — Born-rule
statistics and no-signaling are preserved — while making the electron’s microscopic dressing
memoryless, and that memorylessness fixes the substrate length.
The length fixed in this way — from the electron Compton scale and the tetrahedral
entropy, with no value of G entering the chain — implies a gravitational scale within about
one percent of the measured Newton constant. We treat this as a calibrated coherence check,
not as a clean independent prediction, because the construction’s form was selected with
that target partly in view. Appendix K records that provenance and the associated fork
accounting.
Beyond the static weak-field sector, the framework extends to time-dependent transport,
clusters, cosmology, the saturated early universe, black holes, and the charged-lepton spec-
trum, at varying and explicitly labeled levels of closure. The completed claim of the paper
is the chain from the microscopic construction to ordinary weak gravity; the later sectors
are presented as conditional extensions, frontier completions, or open tests.

1. Introduction: The Physical Claim
4
1.1 What Is Primitive, and What Is Closed
. . . . . . . . . . . . . . . . . . . . . . . .
5
1.2 Physical Motivation for the Primitives . . . . . . . . . . . . . . . . . . . . . . . . .
7

2. Canonical Field Content and Definitions
9

3. The Three Postulates
11
3.1 Information–Geometry Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
3.2 Mass–Entropy Equivalence
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
3.3 Many-Pasts Hypothesis
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12

4. Relativistic Continuum Structure
13
4.1 Capacity budget and continuum symmetry
. . . . . . . . . . . . . . . . . . . . . .
13
4.2 Dependency Map of the Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14

5. Why a Tetrahedral Boundary Ensemble
15

6. Admissibility Closure
17
6.1 Minimal isotropic kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
6.2 Closure condition and uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
6.3 Effective sharing entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17

7. Edge Kernel and Tree-Level Coupling
18

8. Finite-Loop Renormalization
19

9. Continuum Stiffness and SI Normalization
19

10. Covariant Action
21

11. Field Equations and Bridge Law
23

12. Newtonian Gravity and the Point-Source Limit
24

13. Electron Anchor: One-Bit Mass Scale and Seven-Sector Length Scale
25
13.1 Why the electron is the anchor . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
13.2 One-bit mass anchor
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
13.3 Seven-sector length anchor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
13.4 Consistency checks
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
13.5 Composite sectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26

14. Galactic Dynamics
27

15. Lensing, PPN, and Weak-Field Consistency
30

16. Why Dynamics Requires Extension Beyond the Static Branch
31

17. Causal Transport and Telegrapher Dynamics
32

17.5 Cluster Source Projection and the Diffuse–Decoupled Channel Split
33

18. Cosmology and the Hubble-Tension Sector
39

18.5 The Saturated Phase and the Cosmic Microwave Background
40

19. Why These Sectors Belong
43

20. Strong-Field Branch and Bounded Occupancy
43

21. Many-Pasts: The History-Space Ontology
45


![Table 1](paper-70-v10_images/table_1.png)
*Table 1*

22. Microstructure Hamiltonian and Underlying Dynamics
46

23. Closure-Status Table
47

24. Falsifiability and Observational Tests
51
24.1 Static weak-field falsifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51
24.2 Dynamical falsifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
52
24.3 Cosmological falsifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
52
24.4 Correlated-constant falsifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
24.5 Many-Pasts status . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53

25. What the Theory Would Have to Get Wrong to Fail
53

26. Comparison with Other Approaches
54
26.1 Relative to ΛCDM
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
54
26.2 Relative to MOND-like interpolation programs . . . . . . . . . . . . . . . . . . . .
54
26.3 Relative to Verlinde-style emergent gravity . . . . . . . . . . . . . . . . . . . . . .
54
26.4 Relative to TeVeS and other multi-field modified gravities . . . . . . . . . . . . . .
54
26.5 Relative to AeST . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
26.6 Relative to scalar-tensor gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
26.7 Relative to quantum-mechanical interpretations . . . . . . . . . . . . . . . . . . .
55
26.8 Relative to CDT, spin foams, and group field theory
. . . . . . . . . . . . . . . .
55

27. Conclusion
56

Part I. Physical Idea and Canonical Definitions

1. Introduction: The Physical Claim

Space is not an empty container that matter sits inside. It is a finite medium of entanglement
capacity, and the particles we call matter are stable defects of that medium rather than inde-
pendent agents acting on it from outside. Gravity is then what the surrounding medium looks
like once some of its capacity is locked up in such a defect. In brief:

• Matter is a localized capacity defect of the substrate.

• Mass is the entanglement that defect commits, read in mass units.

• Gravity is the extended capacity strain — the fractional capacity deficit — the medium
carries around the defect.

• Dark-matter phenomenology is not a new substance but the same medium read in two
further regimes: the long-range reach of the capacity-strain field on galactic scales, and its
saturated phase in the early universe.

• General relativity is the low-energy geometry of this same capacity medium, not an
external stage to which it is added.

The continuum statement is a scalar EFT for a vacuum-relative entanglement field Sent(x) and
its deficit δS relative to the background capacity. The defect sector is written at continuum scale
in ordinary stress-energy variables, but its ontology is unchanged; inertial mass enters through
the mass-per-entropy map κm, and the weak-field potential is the fractional deficit δS/S∞.

The proposal replaces part of the usual dark-sector story rather than relabeling it. The stan-
dard picture keeps visible matter and Einstein gravity and adds dark components to supply the
missing gravitational response; here the vacuum already carries a finite entanglement-capacity
structure, and the same medium accounts for ordinary weak-field gravity, the galactic excess usu-
ally attributed to dark matter, and the homogeneous cosmological mode. It also goes beyond
an ordinary scalar extension of Einstein gravity: GR is not a stage to which the entanglement
field is appended but the low-energy capacity geometry of the same substrate, with the scalar
sector tracking how that geometry is depleted and redistributed by localized defects.

The paper asks whether that ontology can be made quantitative, and the answer offered
here is conditional. The finite-capacity substrate, the three postulates of Section 3, and the
minimal tetrahedral boundary ensemble are theory-defining inputs; the paper does not derive
them from a deeper Hamiltonian. Once they are fixed, the static weak-field coefficients are no
longer phenomenological parameters but are determined by a finite ultraviolet counting problem:
admissibility closure fixes the effective sharing entropy, edge transport and finite-loop dressing
fix the stiffness, source projection fixes the coupling, and the weak-field bridge relates fractional
capacity deficit to gravitational potential.

The primary closed branch is therefore the static weak-field chain

microstructure −→coefficient chain −→continuum EFT −→{G, a0, RAR, no slip}.

It recovers the Newtonian point-source limit, the galactic acceleration scale, the radial-acceleration
relation, and leading no-slip lensing without per-system tuning. The absolute substrate length is
fixed through the electron anchor and the Many-Pasts memoryless dressing branch; because the
construction was developed with Newton’s constant partly in view, the one-percent agreement
with G is treated as a calibrated coherence check rather than a clean independent prediction,
with the provenance recorded in Appendix K.

The later parts extend the framework into regimes with lower closure status — time-dependent
transport, galaxy clusters, cosmology, the saturated early phase, strong fields, and the particle
and gauge extensions — with Part VI recording the bookkeeping explicitly.

Many-Pasts belongs with the foundations. Its history weighting enters the scale-setting branch
through memoryless electron dressing, and its operational consequences for quantum probability,
branch realization, and the arrow of time are developed in Section 21 and Appendix G.

1.1 What Is Primitive, and What Is Closed

The word “closure” is used here in a specific sense. The paper does not derive the existence of a
finite entanglement substrate from no assumptions, nor does it derive the tetrahedral boundary
ensemble from a deeper microscopic Hamiltonian in the main text; those are theory-defining
ultraviolet inputs, and the closure claim begins only once they are fixed. There are five such
inputs: finite local entanglement capacity; matter as localized defects of that capacity; mass–
entropy equivalence, so that mass is committed defect entropy read in mass units; the Many-
Pasts history weighting of Postulate III; and the minimal tetrahedral boundary ensemble, in
its physical realization as the full admissibility-closed sharing entropy, serving as the ultraviolet
counting architecture.

Given those inputs, the static weak-field sector closes with no phenomenological freedom
remaining. The finite boundary count fixes the admissibility weighting and the effective sharing
entropy; a faithful sector-resolution principle fixes the substrate cell length L∗; edge kernel,
finite-loop dressing, continuum stiffness, and source projection follow in turn; and the weak-field
bridge δS ↔Φ then yields the Newtonian limit, the galactic acceleration scale a0, the radial-
acceleration relation, and leading no-slip lensing, with no per-system tuning. This is the central
result. Everything beyond the static branch extends the same ontology at a lower closure level
and is labeled accordingly: time-dependent transport, clusters, the saturated-phase cosmology,
the strong-field branch, and the particle and gauge extensions. Part VI collects this bookkeeping
in a closure-status table.

In compressed form, the central 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 the result being proven; it is the finite ultraviolet counting
problem from which the weak-field sector is derived.

Only the absolute length calibration is deferred rather than defined here. The substrate cell
length L∗is fixed by identifying the electron — the lightest clean charged defect — with the
maximally extended ground state of a history-space sector-resolution operator, supported over
its reduced Compton scale. That principle is itself derived rather than assumed: its content is
that each dressing channel carries the full admissibility entropy and the channels are mutually
independent. The first holds because the dressing is memoryless in substrate time, which is
the Many-Pasts content of Postulate III; the second holds because inter-channel correlation
would contract the defect and raise its mass, so the lightest charged defect carries none. The
electron’s two roles — mass anchor and length anchor — and the explicit formula are developed
in Section 13 and derived technically in Appendices D.4 and H. The central technical claim is
that, given the three postulates and the realized minimal ensemble, the static weak-field branch
has no remaining phenomenological freedom.

The continuum action can superficially resemble an ordinary scalar-tensor theory of the Brans–
Dicke lineage [44]. In a conventional scalar-tensor model one asks whether variation of a covariant

action containing a scalar 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 grav-
itational response once it is adopted.

Several external tasks remain: derive the same microscopic ensemble from a deeper substrate
Hamiltonian, independently audit the graph-level return operator, confront the resulting radial-
acceleration law and no-slip prediction 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 reconstruction and validation tasks outside the already-closed static
weak-field branch, not hidden fit parameters inside it.

A reader is therefore being asked to accept a small number of commitments, collected here in
one place.

First, the vacuum has finite entanglement capacity: it is not empty background space but a
medium with a bounded local capacity for entanglement.

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

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

Fourth, the entanglement network has many admissible microscopic pasts, and its present
state is governed by a probability-weighted ensemble of them, with nearer histories weighted
more heavily.
This is the Many-Pasts postulate.
Its operational branch preserves ordinary
Born-rule and no-signaling laboratory statistics, and its memoryless history weighting is the
ingredient used in the electron dressing that fixes the substrate length.

Fifth, the ultraviolet cell is the minimal tetrahedral boundary ensemble, and this is not a
continuous tuning parameter. Given fermionic face data, maximum-capacity channel selection,
injective four-face assignment, and minimality, the first admissible face alphabet has seven states,
which gives the 1680-state ensemble (Section 5, Appendix B).

The finite-capacity substrate is the primitive ultraviolet premise; the identification of geometry
with capacity, the mass–entropy equivalence, and the Many-Pasts weighting are the three pos-
tulates of Section 3; and the tetrahedral ensemble is the minimal ultraviolet architecture. Once
they are adopted, the static weak-field sector is claimed to close: the admissibility weighting, the
effective sharing entropy gshare,eff, the edge kernel, the loop dressing, the continuum stiffness, the
source projection, the weak-field bridge, the Newtonian limit, the galactic acceleration scale, the
radial-acceleration relation, and the leading no-slip lensing structure are all derived within that
construction. The single number the construction produces, gshare,eff = 7.4198, is not a fitted
parameter but a derived entropy value inside a selected construction: it has no adjustable dial
and is reproducible from the stated rules alone, while the historical selection of the construction
itself is audited and priced in Appendix K. The sectors beyond the static branch are not hidden
adjustments to that closed result; they are labeled separately as conditional extensions, frontier
completions, empirical tests, or open tasks.

In one line, the architecture is three postulates, one finite-capacity substrate premise, one
minimal ultraviolet ensemble, and a small number of explicitly labeled conditional readings —
and from those, a single derived entropy value carries the static weak-field chain.

1.2 Physical Motivation for the Primitives

These commitments are primitives of the framework, not theorems proved from deeper assump-
tions in this paper. They are nevertheless not arbitrary. Each one is motivated by a place
where established physics already strains against its own foundations, and each earns its place
by making a known difficulty look less mysterious once it is adopted. None of it is offered as
proof; it is the reason the starting points are reasonable ones to adopt before the derivation
begins.

Mainstream gravitational physics has been converging on a finite-capacity substrate for decades.
A black hole’s entropy scales with the area of its horizon rather than the volume it encloses, as
though the contents of a region were written on its boundary; the Bekenstein bound limits the
information a bounded region can hold; and holography and entanglement-based reconstructions
of geometry tie the shape of spacetime directly to patterns of entanglement. Each of these is
usually treated as a deep clue without a mechanism. The framework takes the clue literally: the
vacuum is a medium with a finite local budget of entanglement, and geometry is the large-scale
description of that budget. Several long-standing puzzles then become the ordinary behavior
of a medium that can fill up. The area law is one. A black hole is a region whose capacity is
exhausted, so there is nothing left in the interior to count, and the only live bookkeeping is at
the surface where filled medium meets unfilled, which is why the entropy sits on the area and not
in the volume. The black-hole information question softens for the same reason: if the interior
holds no independent degrees of freedom, there is no interior store of information to lose, and
what falls in is recorded in the boundary layer that later evaporation reads back out. And the
old problem that a continuum theory of gravity develops infinities at short distances does not
arise here, because below the smallest cell there is no continuum to diverge, and a finite state
space has nothing to renormalize away.

The mass–entropy identification addresses a coincidence that general relativity encodes but
does not explain. General relativity builds in the equality of inertial and gravitational mass
geometrically, through the equivalence principle, but gives no microphysical account of why the
mass that resists acceleration and the mass that sources attraction — equal to fifteen decimal
places — should be one and the same. Here they are the same quantity seen from two sides: a
defect’s committed entanglement is at once its resistance to being accelerated and its pull on the
surrounding medium, so the equality follows from the construction rather than being imposed
by hand. The same identification addresses why gravity is so extraordinarily weak. Rather
than inserting a tiny coupling by hand, the framework produces gravity’s strength by counting:
the induced gravitational scale carries the factor e−14gshare,eff, so a large entropy makes gravity
exponentially feeble, and the famous gap between gravity and the other forces becomes a matter
of arithmetic rather than fine-tuning. It also gives a concrete substrate reading of gravity: the
extended capacity-strain field around a localized deficit. This makes its universality automatic
(everything is made of substrate, so everything gravitates), its attractive-only character natural
(a deficit draws the medium inward; it does not push), and the galactic acceleration scale a
derived threshold rather than an unexplained constant. That galactic scale, a0 ≈ϵ cH0, ties the
onset of anomalous rotation to the cosmic horizon, giving a concrete form to the old Machian
suspicion that local inertia should somehow depend on the universe at large. And the tight,
nearly scatter-free way a galaxy’s rotation tracks its visible matter, the very feature that makes
an independent dark particle look so strange, follows directly when a single medium is responding
to the ordinary matter, rather than being a separate component that must be arranged to follow
it.

A second consequence, in plain terms before its technical form: because the medium never
holds still — every instant it is re-drawn from all the ways it could be — a particle is not an
object sitting in space but a pattern the medium keeps re-forming against its own restlessness,

and the entanglement it commits is a maintained quantity, re-established beat by beat. Two
different questions can be asked about such a pattern, and their answers pull apart. One is how
busy the medium is keeping the knot bound: it must hold its grip on every strand the knot
is tied into at once, and separate holds add up, so this maintenance is large — the medium
works hard and continuously to keep the defect in being.
The other is how often the knot
fully re-completes, every strand falling into alignment at the same instant; because simultaneous
coincidences multiply rather than add, that completion is exponentially rare. The particle’s
mass — which is its rest energy — follows the second question, not the first: it is fixed by
how rarely the binding fully re-closes, not by how much work the holding takes. One depth
of binding governs both, but holding adds while coinciding multiplies, so the maintenance is
large and additive while the recurrence that sets the mass is exponentially small. This is why
a particle can be vastly lighter than the natural substrate scale with no small number inserted
anywhere: it is not weakly bound but deeply bound, and deep binding makes full re-completion
exponentially rare. The electron is light not because its structure is weak but because that
structure’s coherent recurrence is exponentially delayed — light because it is heavy to hold.
Section 22 and Appendix H give this its quantitative form, with binding additive across the
sectors and recurrence multiplicative; here it is only the shape of the answer.

A memoryless update rule assumes the least: it is the rule one writes down when one declines
to invent hidden internal machinery that carries information forward from tick to tick. It earns
its place twice over. The construction that fixes the absolute size of the cell needs each dressing
pass of the electron to sample the whole admissible ensemble afresh, and the allowed local single-
label dynamics provably cannot do this: they freeze into disconnected sectors that never explore
the full space (Appendix D.4), so a genuine refresh is required, not optional. A single universal
refresh rate also lets the theory carry a smallest length without running afoul of the experiments
that ended earlier discrete-spacetime proposals: the granularity here lives in the capacity, not in
a preferred spatial lattice, so gravitational waves and light travel at the same speed and there is
no frame-dependent dispersion left to detect.

Many-Pasts holds that the present configuration of the entanglement network is supported
not by one definite microscopic past but by a whole ensemble of admissible pasts, each weighted
by how naturally it leads to the present.
Its first payment is in the quantum puzzles that
interpretations of quantum mechanics were invented to handle. The interference in a double-slit
experiment is the persistence of the unrecorded alternative histories in that ensemble; making a
which-path measurement conditions the ensemble on the new record, and the interference goes
away. The correlations of an entangled pair come from weighting the histories of the whole joint
system, which reproduces the nonclassical statistics without any signal passing between the two
wings of the experiment. Measurement adds no separate collapse law; it lays down a durable
record, after which the relevant histories are simply the ones compatible with it. In the branch
used here all of this leaves ordinary laboratory predictions exactly intact (Born-rule statistics
and no-signaling both hold), and the same weighting supplies the arrow of time. The postulate is
more than interpretation because it is also load-bearing in a measured number: the same history
weighting that satisfies the Born-rule consistency condition also makes the electron’s dressing
memoryless, and that memorylessness is one of the two conditions that fix the substrate length
and hence the strength of gravity. Elsewhere in physics the interpretation of quantum mechanics
is detachable philosophy; here it has a concrete job.

Tetrahedra are the standard building block in several independent approaches to quantum
geometry, so the cell is a familiar object rather than one invented for this paper. What the
framework adds is a set of its own restrictions that pick out one ensemble: the faces carry
fermionic data, the maximum-capacity channel is selected, the four faces must be assigned
distinct labels, orientations are counted with both parities, and the smallest admissible alphabet
is taken. Those restrictions force a seven-label face structure and a boundary ensemble of exactly

1680 states. The same machinery that fixes the count also limits how many shell excitations
the cell can carry before its closure structure degenerates, which terminates the charged-lepton
ladder at three families, a candidate answer to the Standard Model’s otherwise unexplained fact
that matter comes in exactly three generations. How this particular ensemble came to be chosen,
and how much freedom that choice had, is audited separately in Appendix K. Within the stated
construction, its entropy is computed from the rules with no adjustable dial.

Several mainstream results already point toward the finite-capacity reading of gravity: the
derivation of Einstein’s equations as a thermodynamic relation of state, the identification of
entanglement entropy with horizon area, the reconstruction of spatial connectivity from en-
tanglement, and the conjectured identity between entanglement and geometric bridges. This
framework is a concrete, finite, and falsifiable instance of that program.

The primitives are not proved before the theory begins, because no theory proves its own
starting point, and a list of mysteries explained is the easiest kind of case to assemble after the
fact, since a proposal of this scope can almost always produce one. The discriminator is not
how much the primitives explain, but whether the explanations were forced by the construction
or fitted to the target (the question Appendix K confronts directly), and whether the same
construction then survives measurements it did not anticipate. This subsection claims only the
weaker statement: the starting points are motivated by real and independent pressure points in
gravity, quantum information, black-hole physics, and quantum measurement, and once adopted
they constrain the weak-field sector tightly enough to be tested. The rest of the paper derives
those constraints and confronts them with observation.

2. Canonical Field Content and Definitions

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

We define the fundamental continuum variable as the vacuum-relative coarse-grained entan-
glement 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 capacity-strain field sourced by the

defect sector, not an independent medium acted on by matter from outside. For nonlinear work
it is useful to define the bounded occupancy fraction

q(x) ≡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.
Each is used where it is most transparent: δS for the weak-field theory, because it talks directly
to the Newtonian potential; q for the nonlinear and strong-field completion, because boundedness
is built in from the start; and Sent itself for the covariant EFT, because it is the field that appears
in the action. The operational meanings are:

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

• 0 < q < 1: partial local capacity reduction around a defect configuration;

• q = 0: complete local exhaustion of available capacity on the physical branch.

Fixed-epoch normalization.
The absolute normalization of 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 later, through the electron anchor and the Many-Pasts mem-
oryless dressing of Postulate III (Section 13, Appendix D.4); here we only record the canonical
definition that the later derivation populates. The defining relation is 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.

3. The Three Postulates

The framework rests on exactly three postulates because they answer three different questions,
and no fewer suffice to state the ontology. What is spacetime geometry? — Postulate I: geome-
try is the long-wavelength expression of the capacity substrate itself. What is matter, and what
is mass? — Postulate II: matter is localized defect capacity, and mass is the defect’s committed
entropy in mass units. What is quantum probability, and what fixes the direction of history? —
Postulate III: quantum probability is a weighting over admissible past histories of the entan-
glement network. The first two define the gravitational ontology; the third defines its quantum
and historical structure and, through memoryless dressing, also enters the scale-setting branch.
The main text treats all three as theory-defining inputs, not derived outputs.

3.1 Information–Geometry Equivalence

The first postulate states that spacetime geometry is the continuum expression of the entanglement-
capacity substrate. This is stronger than saying entanglement contributes an additional piece
of stress-energy inside otherwise standard general relativity.
The metric is not an indepen-
dent background to which Sent is appended; the geometric sector and the scalar capacity sector
are two continuum projections of one finite-capacity medium. In the weak field, gravitational
potential is the fractional deficit of available capacity.

Two consequences of this reading should be kept distinct from the start. First, absolute Sent
is not itself “the gravitational potential”; the observable weak-field potential comes from the
fractional deficit δS/S∞, which is why a fixed-epoch rescaling of the entropy units leaves gravity
unchanged (Section 2). Second, because geometry and the scalar are projections of the same
substrate, gradients and deficits of the entanglement field are not metaphorical contributions
to gravity — they belong in the gravitational bookkeeping as a matter of what the geometry
is, not as an extra source added to it. In the EFT this is realized concretely: the scalar enters
a covariant action, contributes its own stress-energy, and couples to a trace-equivalent defect
source written at continuum scale in the usual stress-energy variables.

3.2 Mass–Entropy Equivalence

The second postulate is that mass is not something added to the substrate from outside but the
inertial reading of localized capacity commitment. At scale ℓ,

m(ℓ) = κm(ℓ) ∆S.

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

For elementary fermionic sectors the canonical defect increment is

∆Sf = ln 2.

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

Two corollaries are used later. First, because mass and entanglement budget are two descrip-
tions of the same defect, and the masses of separated defects add, capacity committed in service
of one defect cannot simultaneously serve another: shared service would make the joint budget,
and with it the joint mass, sub-additive. Commitment is therefore per-defect — each committed
unit carries the label of the defect it serves. Second, the same per-defect bookkeeping makes
the saturated early phase countable: if mass is committed capacity, committed units cannot
be double-counted across separated defects, and the abundance of committed capacity can be
counted rather than fitted (Section 18.5).

3.3 Many-Pasts Hypothesis

The third postulate is one of the framework’s three defining inputs. It concerns what the present
entanglement network is supported by. The claim is that the present is not the endpoint of one
microscopic past. For any present coarse configuration, many admissible microscopic histories of
the network could have produced it, and the physical state is governed by a probability-weighted
ensemble over those histories rather than by a single distinguished one.

Three objects make this precise:

• H: an admissible microscopic history of the entanglement network — one self-consistent
way the present could have been reached;

• P: the present coarse configuration that all admissible histories must reproduce;

• D(H, P): a history-space distance measuring how far a given history lies from the present
state.

In canonical closed form the operational history weight is

P(H|P) ∝e−D(H,P),

so histories close to the present dominate and distant ones are exponentially suppressed. This
is the branch α = 1, β = 0 of a generalized weighting family.

This postulate occupies the conceptual role usually played by an interpretation of quantum
mechanics. It is not many-worlds: reality does not branch forward into many co-real macroscopic

outcomes.
It is not a collapse theory: there is no added physical collapse event.
It is not
hidden-variable ignorance: probability is not ignorance over pre-existing classical values. It is a
history-space ontology — quantum probability read as a weighting over admissible past histories
of the substrate. This is the framework’s alternative to those accounts: ontologically central,
though its laboratory predictions are deliberately conservative.

The operational branch is then fixed by two constraints. The Born rule is recovered: requiring
exact Born statistics in the projective laboratory limit selects α = 1. No-signaling is preserved:
forbidding any operational bias channel that could transmit information selects β = 0. The
substantive content is that the family admits a single branch satisfying both at once. Born
recovery is therefore stated honestly as a consistency condition the weighting must satisfy, not
as a fresh derivation of the rule from outside it; this differs from programs that obtain the
rule from noncontextuality [53], decision theory [54], or envariance [55], with no-signaling [56]
the companion constraint. Beyond these, the same weight orients the arrow of time through
conditional typicality and makes the dressing of an elementary defect memoryless in substrate
time, each pass an independent draw from the admissibility ensemble.

That memorylessness is Postulate III’s entry into the central result. The coefficient chain
that fixes the Newtonian limit, the galactic law, and lensing does not call on Many-Pasts. The
substrate-scale branch does: the memoryless dressing is one of the two ingredients that fix the
absolute substrate length in Appendix D.4, the other being the mutual independence of the
electron’s seven dressing channels, which follows from its identity as the lightest charged defect
(Appendix H). Its operational consequences and the arrow of time are developed in Section 21
and Appendix G.

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 transport within the surrounding network. Because the substrate is
isotropic, the cost of motion depends only on the rotational scalar v2 at leading order, with the
temporal rate maximal at v = 0 and vanishing when the budget is exhausted at v = c. These
endpoint conditions alone admit many interpolating functions and so do not fix the form of the
time-dilation relation. The form is fixed once the finite update speed is treated as invariant
across inertial coarse descriptions: homogeneity, isotropy, and the relativity principle then select
the Lorentz group rather than the Galilean one, giving the invariant interval

c2dτ 2 = c2dt2 −dx2,

and hence
dτ

r

1 −v2

dt =

c2 .

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

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

q = 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 language,
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, and it faces a sharp known obstacle:
a discrete structure that defines a preferred rest frame feeds dimension-four Lorentz-violating
operators into the infrared with order-unity coefficients through loops [24], against laboratory
bounds many orders of magnitude below unity. The protection here is structural. The tetrahe-
dral ensemble is combinatorial and pre-geometric: it lives in the state counting from which the
continuum is constructed, defines no embedding lattice in the emergent spacetime, and imprints
on the EFT only through frame-independent scalars (L∗, gshare,eff, η∗). Discreteness of this class
is compatible with exact low-energy Lorentz symmetry, as causal-set sprinkling demonstrates
by construction [25, 26]. The cosmological bath does select a frame, in the same environmental
sense the CMB does: a state rather than an operator, while the laboratory bounds constrain
operators. The supporting calculation this argument calls for — that substrate loop corrections
generate no dimension-four Lorentz-violating operators — is open and recorded as such in the
closure table.

4.2 Dependency Map of the Theory

The logical flow of the theory begins with the three foundational postulates — Information–
Geometry, Mass–Entropy, and Many-Pasts — on equal footing, and runs through the static
weak-field chain before reaching the conditional sectors:

{Info–Geometry, Mass–Entropy, Many-Pasts}

→finite-capacity substrate ontology →tetrahedral boundary ensemble

→admissibility closure →edge kernel / loop dressing / stiffness / source map

→weak-field EFT →{Newton, a0, RAR, no-slip lensing}.

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

Two features of this map matter. First, it is a dependency graph, not an 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, and
Part VI makes that difference explicit in a closure-status table. Second, Many-Pasts appears
among the foundational postulates rather than downstream, because its history weighting is
already active in the scale-setting branch through memoryless dressing. The later Many-Pasts
section (Section 21) develops the operational consequences of a postulate that has already done
load-bearing work upstream.

Part II. UV Coefficient Chain

Part I fixed what the theory is about. The question now is whether the local capacity-sharing
structure can actually be counted.
If the substrate has finite local capacity, the coefficients
that appear in the continuum weak-field theory should not be free continuum parameters; they
should descend from a finite local boundary problem. The next five sections follow that problem
through: the smallest boundary cell that can carry capacity and close isotropically, the weighting
that selects well-closed configurations, the cost of neighboring cells disagreeing, the local returns
that dress that cost, and the continuum coefficient they leave behind. Nothing in this chain is
matched to a gravitational observable; the observables enter only in Part III.

Several of the ultraviolet choices below may look at first like independent tunings: the tetrahe-
dral boundary cell, the seven face labels, the ordered injective assignment, the parity doubling,
the admissibility kernel, the transverse (2/3) export, and the seven-sector electron anchor. None
is a phenomenological knob in the closed branch. Section 5 and Appendix B derive the minimal
tetrahedral seven-state ensemble and its unique admissibility-closed entropy; Appendix C fixes
the transverse export from the tetrahedral bond-frame Green response; Appendices D and H
derive the faithful sector-resolution length relation from the electron anchor and the memoryless
dressing; and Appendix K records the historical fork accounting and calibration status. The line
to keep in view is between the primitive adoption of the finite-capacity tetrahedral substrate
and the downstream coefficient chain, which contains no continuous fit once that substrate ar-
chitecture is fixed. What the construction does not yet exhibit is a single microscopic action or
path-integral measure whose continuum expansion generates these coefficients simultaneously;
each is derived through its own local reduction, so the chain is a sequence of forced steps rather
than one unified continuum limit, and consolidating it into a single measure is an open task.

5. Why a Tetrahedral Boundary Ensemble

The problem is to find the smallest discrete boundary cell that can carry finite channel entropy,
close isotropically, and hand a single scalar response up to the continuum. The smallest structure
that meets all three needs is a tetrahedral cell, fixed by four 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

j0 = 1

2, 3

2, 5

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.

j0 is sometimes mistaken for a free dial; it is not. Within this minimal construction j0 is the
smallest fermionic label that satisfies injectivity; a larger j0 does not describe a fluctuation inside
the same cell but a different, larger boundary ensemble. The minimal theory therefore has no j
to tune — it has the first value that closes.

In that sense the seven-state face sector is derived from fermionic face data, maximum-capacity
channel selection, tetrahedral injectivity, and minimality. The derivation chain itself makes no
reference to the weak-field observables it later feeds; the order in which the construction was
historically found, and where the evidential weight accordingly sits, are recorded in Appendix K.
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) = 7.42654907240.

The exact K2 spectrum and multiplicities are carried in the appendices. The j-labeled tetra-
hedron used here coincides with the quantum tetrahedron of simplicial spin networks [50, 51],
whose discrete geometric spectra [52] arise from the same SU(2) representation theory; the
present construction differs in weighting these states by admissibility closure rather than by a
spin-foam amplitude, and in routing them to a capacity entropy rather than to area and volume
operators. The UV theory thus begins with a finite microscopic counting problem rather than
a free continuum ansatz.

6. Admissibility Closure

6.1 Minimal isotropic kernel

Not every boundary configuration should count equally. The raw combinatorial ensemble is
too permissive to be the whole UV story: some configurations sit close to the regular closure
pattern expected of a smooth local cell, while others are badly distorted. Admissibility closure
is the statement, in its mildest form, that more poorly closed configurations contribute less to
the coarse ensemble. The minimal rotationally invariant measure of that distortion is a single
quadratic closure-defect scalar K2, and the weighting it induces is

pη(b) ∝e−ηK2(b).

This is not chosen because it works phenomenologically; it is the minimal isotropic maximum-
entropy kernel under normalization and a fixed quadratic closure moment. Higher invariants
such as K4 carry additional UV information and so enter as subleading refinements, not as
competing leading kernels.

6.2 Closure condition and uniqueness

The admissibility precision η is not chosen externally; it is fixed by maximizing the normalized
closure evidence.
Tetrahedral closure is the vanishing of the three-component oriented-face
sum, so the closure-defect space is three-dimensional, and the quadratic family on it carries a
determinant weight η3/2. The closure-evidence functional is therefore

F(η) = ln Z(η) + 3

2 ln η,

and its stationary point gives the closure condition

⟨K2⟩η = 3

2η,

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

On that spectrum it is unique,

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.

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

With η∗fixed, the effective sharing entropy carries no remaining freedom; the exact spectrum,
multiplicities, and uniqueness proof are given in Appendix B.2.

7. Edge Kernel and Tree-Level Coupling

The admissibility closure says what one cell can carry. The edge kernel says how costly it is for
two neighboring cells to carry different things, and that cost becomes stiffness in the continuum
field: a medium whose cells resist disagreement strongly is one whose capacity deficits spread
reluctantly. The same UV closure data fix this cost. 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 and gives the bare edge smoothness
coupling

Jbare = 2

3η∗.

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

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

3
= 2η∗

Jtree
eff
= Jbare

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.

Origin of the horizon target.
The horizon target

σ∗=
π
gshare,eff
is the closure-consistency value required by the horizon-normalized field convention.
In the
admissibility-closed boundary ensemble, one active microscopic sharing unit carries effective
entropy gshare,eff. In the continuum normalization used for the weak-field scalar, the occupancy
variable is normalized by
S = πQocc,

so a coarse horizon-normalized channel with occupancy Qocc = 1 carries entropy π in the S-
field convention.
If σ∗denotes the asymptotic conditional-independence weight seen by the
rooted shell hierarchy (Appendix B.3), consistency between the boundary entropy count and
the horizon-normalized continuum field requires

σ∗gshare,eff = π,

and therefore
σ∗=
π
gshare,eff
= 0.42340665 . . . .

The role of σ∗is to match the admissibility-closed microscopic entropy normalization to the
horizon-normalized scalar-field convention; it is the closure target fixed by that choice of branch.
The rooted shell observable converges to that target rapidly enough that the nonlocal correction
is already strongly constrained by small shell depth. The tetrahedral identity keeps this bridge
controlled: the four discrete channel directions average to the correct isotropic tensor structure
in the continuum limit, so the same combinatorial data that fixed admissibility also fix the
tree-level transport. The shell hierarchy and phase-selection checks are carried 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 structure of that self-energy is not a generic loop number. The returns split into seven
sector-diagonal channels and one collective mode. The seven are the face-label channels, each
returning independently without mixing. The one is the permutation-symmetric combination
across channels, which returns as a shared closure-singlet rather than as a channel-specific loop,
and it is weighted by the same transverse projection and branch-dilution factors that define the
tree edge map,
2

 1


= 2

9.

3

3

The leading local self-energy is therefore

9 = 65

Σret = 7 + 2

9 .

Equivalently, on the seven-channel scalar return space,

9Psing,
Psing = |u⟩⟨u|,
u =
1
√

Rret = I7 + 2

7(1, . . . , 1),

with Σret = Tr(Rret).
The orthogonal six-dimensional sum-zero sector carries no net scalar
charge in the coarse branch and so adds no separate scalar return. The singlet weight is fixed
by the tree map, not introduced here, so no new loop parameter appears.

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.

The loop correction is no longer schematic: the finite renormalization 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

a,i
(Qa −Qa+L∗ˆni)2,

where the sum runs over all sites a and all four outgoing nearest-neighbor directions ˆni, so each
nearest-neighbor edge is counted twice. The microscopic four-cell has 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 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
.

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

G∗:= 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.

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

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.
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.

Definition.
Faithful sector resolution fixes the substrate cell length L∗without using G, and
the induced gravitational scale G∗= c3L2
∗/ℏfollows from the electron Compton length and the
tetrahedral entropy alone. The chain takes no value of G as input.

Comparison.
Independently of that length, the scalar bridge and Green-matched source map
fix the weak-field normalization through the invariant ratio κ/(γS∞),

G = c2

8π
κ
γS∞
,

which lets G be compared with G∗. Because both are built from the single substrate scale L∗,
this is not a second measurement from independent inputs; it is a coherence check of whether
that one scale propagates through the scalar stiffness, the source map, and the bridge to the
observed normalization. The check has content because fixing L∗does not by itself guarantee
that κ/γ, S∞, and the bridge organize around it to give the right normalization.

Status.
The two agree to about one percent. We read this as a calibrated coherence check,
not a clean independent prediction, because the construction was developed with the measured
G partly in view; Appendix K records that provenance and the associated fork accounting. The
dressing dynamics behind faithful sector resolution is derived in Section 22 and Appendix H.

This is the simplest covariant realization of the closure chain: one metric, one scalar en-
tanglement field, one trace-equivalent defect-source channel, and one renormalized background
branch.

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.

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 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)

S∞
.

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

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

G =
c2κ
8πγS∞
.

The bridge law connects the entropy description to observable gravity: the deficit field acquires
a unique weak-field normalization in terms of the ordinary gravitational potential.

Matter enters once.
The two equations are not two independent ways for ordinary matter
to source the weak-field potential, and reading them as such would double-count it. Matter
enters the weak-field branch through a single channel: the trace χ = −T µµ/c2 sources the
scalar equation, the scalar response δS is converted to the lapse by the bridge law, and the
observed Newtonian potential is that bridged lapse. The metric equation is the covariant im-
age of the same response — T (ent)
µν
is the stress-energy of the scalar field that χ has already
sourced, not a separate matter contribution — so G = c2κ/(8πγS∞) and the Newtonian limit
of Gµν = (8πG/c4)(T (matter)
µν
+ T (ent)
µν
) are the same field written two ways. Ordinary matter is
therefore not a second, independent source of Φ standing beside the bridge; it is the source of the
scalar response that the bridge reads as the potential, and Bianchi consistency enforces matter
conservation through that same trace channel rather than through a separate coupling. The
Einstein equation above is thus written in the already-bridged physical-metric frame — T (ent)
µν
records the scalar response the bridge has converted into the lapse — and not as an independent
second scalar force acting on test bodies.

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∞

r2 = GM

M

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.
This is the first point where the capacity-strain picture becomes ordinary
gravity. A point defect produces a 1/r capacity deficit; the bridge reads the gradient of that
deficit as the Newtonian inverse-square force.
Ordinary gravity is the small-deficit, weak-
curvature limit of the extended capacity strain around localized defects, so the 1/r2 law is
emergent, not fundamental.

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

The electron does two separate jobs in this framework, and keeping them apart turns the weak-
field normalization into a genuine consistency check rather than a single calibration. As the
lightest one-bit fermionic defect, it is the cleanest mass anchor for the mass–entropy map. As
the lightest coherent defect whose dressing resolves the seven face sectors, its Compton scale is
the length anchor that fixes the substrate cell size. The two uses draw on the same particle but
on different facts about it — its mass on one side, its coherent dressing support on the other.

13.1 Why the electron is the anchor

The mass–entropy map needs a clean elementary anchor because, in this framework, the el-
ementary matter sector is the localized defect sector. The electron is the natural choice: it
is the lightest simple charged fermionic defect, not a composite, and its mass is not obscured
by hadronic or QCD dressing. A single fermionic face-exclusion defect carries the canonical
increment
∆Sf = ln 2,

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

13.2 One-bit mass anchor

At the electron Compton scale ℓ= λe the mass–entropy map reads

κm(λe) = me

ln 2.

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

κm,UV =
ℏ
cL∗

1
ln 2,

with canonical running law

1+αcl
,
αcl = 0

L∗

κm(ℓ) = κm,UV

ℓ

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

13.3 Seven-sector length anchor

The same electron fixes the absolute cell length through a different fact about it. Its reduced
Compton wavelength λe is the coherent support of the lightest defect’s ground-state dressing,
and that dressing resolves the seven face sectors of the tetrahedral alphabet. Two properties

make the resolution clean. The Many-Pasts weighting of Postulate III makes each dressing pass
memoryless in substrate time, so each pass carries the full admissibility entropy gshare,eff; and the
electron, as the lightest charged defect, carries no inter-channel correlation, so the seven sectors
are resolved independently. Seven independent resolutions add, giving the support exponent
7gshare,eff and hence

ln
 λe


= 7gshare,eff −ln
3


,

L∗

2

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

This does not mean an electron is a single tetrahedral cell carrying seven simultaneous labels.
The local 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. The Compton scale calibrates the substrate length hierarchy; the
one-bit mass calibrates the source map. The two uses impose a nontrivial joint requirement on
the electron’s role in the gravitational normalization without duplicating a single input.

13.4 Consistency checks

Two cross-checks test the dual role rather than feed it.
The first is the support exponent
itself. The length relation reads the physical ratio as the first power of the dressing support,
λ/L∗∝eH with H = 7gshare,eff for the electron, and that linear power is not imposed. Letting
it float as λ/L∗∝eαH and demanding the observed Newton constant from the electron branch
requires α = 0.9999, while the same map applied to the charged-lepton ladder (Appendix I.1) is
independently solved by αµ = 1.001 and ατ = 1.001. A square-root map (α = 1

2) would misplace
mµ/me near 14 and a quadratic map (α = 2) near 4 × 104, so the spectrum excludes them by
orders of magnitude. Three determinations across two sectors that share no gravitational input
bracket α = 1 within 10−3. The dictionary itself is derived in Appendix D.4 from the recurrence
structure of the dressing: the waiting time for joint typicality of independent channels is the
exponential of the summed entropies, so the linear power is the arithmetic of joint probability.
The bracketing is the quantitative check of that derivation; identifying one phase cycle per
dressing recurrence is the single assumption that remains beneath it.

The second cross-check is the induced gravitational scale. The length anchor fixes L∗, and
hence G∗= c3L2
∗/ℏ, from the electron Compton length and the tetrahedral entropy alone, with
no gravitational input. Whether that one substrate scale also propagates through the scalar
stiffness, the Green-matched source map, and the weak-field bridge to the observed normalization
is a coherence test, not a second independent measurement (Section 10). The comparison agrees
to about one percent, and Appendix K records how the measured value of G figured in selecting
the construction.

13.5 Composite sectors

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

mhadron = κm(ℓH) Sdressed
ent,H ,

with the dressed 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 is structural compatibility between the mass–entropy map and
the standard QCD mass budget. The elementary-fermion anchor is settled in the simple sectors;
the hadronic sector remains structurally compatible but not yet coefficient-complete.

14. Galactic Dynamics

The static Newtonian branch already gives the ordinary baryonic force. The galactic excess
appears when, at low acceleration, the transverse vacuum-capacity mode adds a bosonic emission
factor on top of that force, and the scale at which this turns on is set by horizon thermality
reduced by the microstructural sharing factor. The rest of the section is the precise version of
those two sentences, and it separates cleanly into two questions: where the turn-on scale a0
comes from, and why the resulting radial-acceleration relation has the shape it does.

The galactic sector is one of the main payoffs of the coefficient chain.
Its characteristic
acceleration scale, the radial-acceleration relation, and the deep-MOND limit all follow from one
chain that uses no per-galaxy interpolation function. The chain has three structural inputs: a
stable bosonic vacuum mode (Appendix D.6), the Gibbons–Hawking thermal structure of that
vacuum at the apparent horizon (Appendix E), and the 1+2 channel decomposition that places
the cosmic horizon scale in the transverse sector.

Vacuum-state origin of the Bose–Einstein occupancy.
The Bose–Einstein statistics in
the galactic mode sector follow from the vacuum-state thermality of the entanglement field at
the apparent horizon, with the cosmological boundary normalization fixed in Appendix E. The
mechanism is therefore equilibrium horizon thermality, not dynamical thermalization.

Appendix D.6 establishes that δS fluctuations around the on-shell background constitute a
massless bosonic scalar at quadratic order with positive kinetic stiffness γ > 0. For a massless
bosonic scalar in a spacetime with apparent horizon RA, the vacuum state restricted to the
static patch is thermal at the de Sitter / apparent-horizon temperature

kBTH =
ℏc
2πRA
,

with mode occupancy nB(ϵ) = 1/(eϵ/kBTH −1). This is a Gibbons–Hawking statement of QFT
in curved spacetime, fixed by the same horizon geometry that sets S∞.

It is useful to write a horizon temperature as an acceleration scale,

aT ≡2πc

ℏkBT.

For the apparent-horizon vacuum this gives

aT = c2

RA
≡aH.

In the closed transport branch, R−1
A = H0/c at the present epoch (from §17 with τ −1
0
= H0), so

aH = c2

RA
= cH0.

The acceleration scale defined by horizon thermality is therefore fixed by the same closure chain
that fixes S∞. More generally aH(t) = cH(t) at any cosmological epoch, so the horizon thermal
scale is epoch-dependent. The present-epoch value cH0 is the one relevant to the local (z ≈0)
galactic RAR.

The MOND scale a0 is related to aH by the microstructural sharing factor that connects
gshare,eff to phase-space transverse-mode density in the 1 + 2 channel decomposition:

a0 = gshare,eff

4π2
aH = cH0gshare,eff

Hence a0 is the horizon thermal acceleration aH = cH0 reduced to the transverse scalar sector of
the 1+2 decomposition. The reduction factor gshare,eff/(4π2) is fixed, not fitted. Its (2π)2 is the
canonical two-dimensional mode density d2k⊥/(2π)2 — the number of states per phase-space cell
of a continuum two-plane, a quantization normalization rather than a Fourier convention — and
gshare,eff is the admissibility-sharing content carried per cell. The normalization is (2π)2, not the
spherical 4π, because the weak-field response samples the local transverse two-plane orthogonal
to the baryonic forcing, not the full horizon sphere. That the split exists at all is kinematic, not
a modeling choice: the baryonic gradient picks the longitudinal direction ˆn = ∇Φbar/|∇Φbar|,
and its orthogonal complement in three-space is two-dimensional by dimension count, so the
transverse two-plane and its (2π)2 cell are forced.
The one premise that is physical rather
than geometric is the assignment of the low-acceleration response to the stationary transverse
vacuum branch rather than the telegrapher transport sector of Section 17 — the step that ties a0
to cH0 rather than to a transport timescale, and that is testable through the epoch dependence
a0(z) ∝H(z). Given it, the coefficient follows, and a0 and cH0 share one origin.

1 + 2 channel decomposition and the radial-acceleration law.
The longitudinal accel-
eration scale is gbar and the transverse vacuum reference scale is a0. Each channel carries an
acceleration frequency through the Unruh correspondence [36],

c ,
ω⊥∝a0

ω∥∝gbar

c ,

with a common prefactor that cancels in the occupancy argument below. The two channels mix,
and a coupled pair of modes softens at the geometric mean of its two frequencies rather than
their average; that softened mixed mode dominates the low-acceleration response. Precisely:
the action is quadratic, so the two channels couple bilinearly, and the bilinear form fixes the
cross-mode scale. For unit-normalized modes with stiffnesses ω2
∥and ω2
⊥and cross coupling
λ, the determinant of the form is ω2
∥ω2
⊥−λ2, so the mixed mode softens, and dominates the
response, at the threshold scale
ω× = pω∥ω⊥.

The geometric mean is the spectral invariant of the bilinear cross term itself. Other exchange-
symmetric combinations correspond to no invariant of the quadratic form; an arithmetic-mean
scale would require a non-bilinear vertex the action does not contain. The cross-scale amplitude
is therefore
aeff = √gbar a0.

The bath temperature is set by the transverse vacuum scale, kBTbath = ℏω⊥in the normalization
of Appendix E, and the occupancy argument is the frequency ratio

x =
ℏω×
kBTbath
= aeff

a0
=
rgbar

a0
,

invariant under common rescaling of the frequencies and the temperature, so no Unruh normal-
ization convention survives in the law. When gbar ≫a0 the system reduces to the ordinary
Newtonian branch; when gbar ≪a0 the response crosses into the low-acceleration completion.

The resulting radial-acceleration law is

gobs = gbar

1 + nB(x)


=
gbar

gbar/a0
,

1 −exp

−
p

with the asymptotic limits

gbar ≫a0 =⇒gobs ≈gbar,
(10)

gbar ≪a0 =⇒gobs ≈√a0gbar.
(11)

The two endpoint limits follow from the occupancy factor alone; the exact interpolation between
them is conditional on kernel neutrality, F(x) = 1, established structurally below and flagged
there as the open gold-standard vertex computation. The deep-MOND branch gives the baryonic
Tully–Fisher law [37, 38]
v4 ≈a0GMb.

Division of labor between the action and the state.
The field equation of the weak-field
sector is linear and remains linear here: it carries propagation and the Newtonian channel. The
nonlinearity of the galactic law enters through the statistical state of the transverse vacuum, in
the same way that Fermi–Dirac occupancy makes electrical conduction nonlinear in temperature
while the Maxwell sector stays linear. The coefficient chain fixes the value of a0; the bilinear
form fixes the argument x; the occupancy enters as the bosonic emission factor

gobs = gbar

1 + nB(x)


.

The factor is a rate statement, and the distinction carries the physics. Static linear response of a
Gaussian theory is temperature-independent — the retarded propagator does not depend on the
state — so no static susceptibility can bring the occupancy into the force law. The occupancy
enters because the weak-field force here is the rate of a capacity-transfer process between the
channels; the entropic ontology requires exactly this, since the force is a flux of capacity and
fluxes are rates.

The rate factor then follows from first principles in three steps. First, for linear coupling to a
bosonic mode the emission rate into occupation n is exact,

Γ = Γ0
⟨n+1|a†|n⟩
2 = Γ0 (1 + n),

which on the thermal cross mode reads Γ0(1+nB(x)). Second, the absorption channel that would
restore detailed balance and cancel the stimulated term against Γ0 nB is structurally closed:
the reverse process requires the defect to partially refill its capacity deficit, and that deficit is
the quantized one-bit increment ∆Sf = ln 2 of Section 5, which admits no partially refilled
final state. The thermodynamic arrow of Part V suppresses the remaining collective reverse
fluctuations by the same conditional-typicality weighting that orients macroscopic entropy flow;
the one-bit quantization that sets the substrate length thus also permits a low-acceleration excess
to survive at all. Third, normalizing the bare rate on the Newtonian channel, Γ0 ↔gbar, gives
the law above with its two limits, since 1 + nB(x) = (1 −e−x)−1. The identification beneath
this derivation—that the static weak-field acceleration is the steady capacity-transfer rate—itself
follows from the transport sector, in two parts.

The flux identification from transport.
The static limit of the telegrapher dynamics of
Section 17 is a conservation law: with capacity current J = −D∇δS and a static source,
continuity reduces to ∇·J = σ, so the field around a mass is a steady conserved capacity current
with radial flux density J(r) ∝M/r2. Because δS is dimensionless capacity, J is a bit-number
current, and its static density carries the Green-matched normalization of the weak-field chain
with no new constant: the Newtonian field is a rate density, a statement the static transport limit
derives rather than assumes. Delivery of this flux to a test defect is an inter-channel process—the
current resides in the longitudinal channel while the defect’s dressing occupies the transverse
sector through the 2/3 export—and inter-channel transfer is generated solely by the bilinear
cross term, so it proceeds at the cross-mode frequency, inside the thermally occupied transverse
vacuum, at the field point. The stimulation is therefore local: the argument x is evaluated where
the transfer is delivered, the feature the wide-binary discriminator of Section 24.1 rests on.

Kernel neutrality.
What remains to verify is that the golden-rule kernel contributes no de-
pendence on x beyond the occupancy, i.e. that Γ = gbar F(x) (1 + nB(x)) has F ≡1. Four
observations isolate the condition the kernel must meet. The factor 1 + n is exact ladder alge-
bra, |⟨n+1|b†|n⟩|2, common to all other kernel factors. A continuum density of states ρ(ω) ∝ω2

would deform the law, but the transfer is into the per-cell transverse channel, whose state count
per admissibility cell is fixed by the same 4π2 cell structure that fixes the coefficient of a0; no
frequency power enters. An energy-per-quantum conversion ℏω× would likewise deform it, but
capacity is exchanged in the fixed one-bit unit ∆Sf = ln 2, the same quantization that closes
the absorption channel, so the transfer is number-counted. The number flux itself is the con-
served bit current above, whose static density is gbar. Together these reduce the kernel to a
single requirement: that the coarse inter-channel vertex couple to the admissibility-weighted
ensemble mean of the transition availability rather than to its spectral spread.
Under that
mean-coupling condition F ≡1 and gobs = gbar(1 + nB(x)) exactly, and the structures enforcing
it—the admissibility-cell channel and the one-bit quantization—are the same two facts that fix
a0 and forbid absorption, so the law’s shape is tied to the framework’s founding bookkeeping
rather than to a free choice.

The main text establishes this mean-coupling condition structurally; its verification from the
substrate Hamiltonian remains the audit task. A first ensemble-level computation shows why
that check is nontrivial: one-step admissible transitions across the 1680-state ensemble correlate
with K2 (Pearson r ≃0.5, class means spanning 1 to 6 of 8 moves), so the raw hopping kernel
is not spectrally flat, and neutrality requires the coarse vertex to see only the ensemble mean of
this availability—a single number—with the static field unable to re-weight the local ensemble
and leak the spectral dependence into the kernel. Establishing that mean-coupling from the
graph-level inter-channel vertex of Appendix H is the remaining gold-standard computation.
Empirically, any residual F is bounded below the few-percent level across five decades of x by
solar-system precision and the measured flatness of the galactic and lensing relations, and the
closure table records the status accordingly.

The law expresses vacuum-state mode occupancy at the apparent-horizon temperature. De-
partures from it would require either (a) an excitation mechanism beyond the vacuum, supplied
by the causal nonequilibrium transport sector of Appendix E for mergers and transients, or
(b) a different transverse reference, which would require modifying the cosmological boundary
normalization that fixes aH. Neither route belongs to the closed stationary weak-field branch;
both would be treated as extensions or departures from it.

The galactic branch is therefore fixed by the same channel-identification structure already
used elsewhere in the weak-field EFT, with the radial-acceleration law read off from one chain
rather than interpolated per galaxy.

15. Lensing, PPN, and Weak-Field Consistency

A modified-gravity account can reproduce galaxy rotation curves and still fail: if the field sup-
plying the extra dynamics does not bend light the same way ordinary mass does, the two metric
potentials slip apart and lensing comes out wrong. The weak-field branch here avoids that fail-
ure at leading order, and for a structural reason. 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. Light bending and dynamical mass estimates are there-
fore 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.

The same response that governs the dynamics governs light deflection, so galactic support
does not come at the cost 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.
At leading post-Newtonian order,
γPPN = βPPN = 1.

Scalar-induced corrections to the slip Φ −Ψ enter only at quadratic weak-field order, schemat-
ically O(Φ2/c4). Thus the leading weak-field EFT reproduces galactic phenomenology without
introducing gravitational slip or solar-system-scale pathologies.

The closed prediction here is the absence of leading gravitational slip, Φ = Ψ with γPPN =
βPPN = 1. The corresponding open test is empirical: stacked weak-lensing measurements of
the dynamical-to-lensing mass ratio, together with PPN and fifth-force searches, confront that
prediction directly. At leading weak-field order the sector is closed; higher-order precision con-
frontation is an audit task, not an architectural gap.

The trace coupling is not a fifth force.
The action’s trace term −κχSent could be read
as a Brans–Dicke-like scalar charge that test bodies feel in addition to the metric, which solar-
system fifth-force bounds would then constrain severely. That is not the operative reading. The
trace channel sources the capacity deficit δS; the bridge law converts that deficit into the lapse
N = exp[−δS/(2S∞)]; and test bodies follow geodesics of the resulting physical metric. There
is no second, independent scalar force beside geodesic motion — the scalar’s entire effect is
already carried by the metric the bridge produces. The leading post-Newtonian values follow
from that single bridged metric: writing U = −Φ > 0 and N = e−U/c2 gives g00 = −N2 =
−1 + 2U/c2 −2U2/c4 + O(c−6), so βPPN = 1, while the absence of leading scalar anisotropic
stress gives γPPN = 1. The remaining formal audit is higher-order: if the canonical T (ent)
µν
is
retained as an independent Einstein-sector source, one should check explicitly that its quadratic
pieces reproduce this bridge expansion rather than adding to it. That confrontation is the flagged
audit task; the leading calculation returns the general-relativistic values.

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

16. Why Dynamics Requires Extension Beyond the Static Branch

The static weak-field branch answers one question: what the capacity-deficit profile looks like
once a source has settled. That is enough for ordinary galaxies, which are quasi-static. It is not
enough for clusters and mergers, which ask a different question — how that profile develops:
how it propagates when a source moves, how it lags behind a fast disturbance, how it saturates,
and how it responds when matter separates into distinct dynamical phases. None of those is
contained in a static Poisson law.

If the entanglement-capacity medium is physical, it must answer that second question too,
with propagation and causal response built in. The time-dependent sector is therefore not an
optional add-on but the natural dynamical completion of the medium that produces the static

EFT, and the cluster and cosmological sectors draw on it. The next section gives its minimal
causal form; Section 17.5 then uses it for clusters and mergers.

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 set-
tled, 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 depen-
dence becomes negligible. The static capacity-strain field is not being replaced; the telegrapher
equation describes how that same field propagates and settles when sources change. “Relax-
ation” in this section means genuine time-dependent settling, distinct from the static strain of
the weak-field branch.

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.
The cluster and merger phenomenology, which sets the source weights this transport then evolves,
is developed in Section 17.5.

17.5 Cluster Source Projection and the Diffuse–Decoupled Chan-
nel Split

Clusters are where the simple galactic radial-acceleration relation stops being enough, and they
fail it in two distinct ways.
Relaxed clusters show a hook-shaped residual — near unity in
the stellar-dominated center, peaking at intermediate acceleration, converging back in the deep
outskirts. Merging clusters show lensing peaks that stay with the collisionless galaxies while
the dominant baryonic mass, the X-ray gas, is displaced. A viable account must produce both
without granting galaxies hidden baryonic mass and without altering the galaxy law itself.

The mechanism proposed here is that the long-range entropic-excess channel does not couple
equally to every baryonic phase. A diffuse, phase-averaged medium couples only through the
transverse projection ϵ already fixed by the galactic sector; a dynamically decoupled collisionless
component recovers the full projection; and a virialized coherent bath can be lifted above it.
The rest of the section develops that one distinction and confronts it with data, and its status
is kept explicit throughout: the projection coefficient is derived, the trend direction and hook
shape are supported, the lift amplitude is not yet derived, and the decisive resolved-map test
has not yet been performed.

The residual is not a single number but an acceleration-dependent profile: lensing and kine-
matic analyses of relaxed clusters [4, 15, 16] find the ratio of observed to galaxy-RAR-predicted
acceleration near unity in the stellar-dominated centers, rising to a peak of roughly 3–5 at inter-
mediate accelerations (gbar ∼10−11–10−10 m s−2), and then apparently converging back toward
the galaxy relation at the lowest probed accelerations, subject to gas-extrapolation caveats in
the outskirts [16]. The older integrated value of 1.5–2 from hydrostatic analyses at higher ac-
celerations is the high-gbar edge of this hook-shaped profile, not its amplitude. Merging clusters
sharpen the problem. In systems of the Bullet type the dominant baryonic component—the
intracluster plasma—is ram-pressure slowed and displaced from the collisionless galaxies, while
the lensing peaks remain with the outgoing collisionless components rather than the X-ray gas.
A viable cluster sector must explain both the relaxed normalization and the merger gas/lensing
separation without assigning galaxies additional hidden baryonic mass and without altering the
galactic mass anchor.

The mechanism is not transport lag and it is not extra galaxy mass; both are excluded.
Transport lag is excluded because, in the canonical τ −1
0
= H0 branch, disturbances propagate
at c and communicate changes across a cluster-scale configuration on a light-crossing time of
order 1 Myr, three orders below the merger timescale, so the field tracks the moving source;
an underdamped configuration retains memory of the pre-merger centroid, not of the outgoing
collisionless component, and a retarded forward solution sourced equally by all baryons therefore
does not by itself keep the lensing peak on the outgoing collisionless component. Extra galaxy
mass is excluded because the galactic acceleration scale and the gas-dominated-dwarf RAR fix
the deficit per unit baryonic mass with no freedom to enhance it. The viable mechanism is
instead a phase-dependent source projection for the long-range entropic-excess channel, built
from a distinction the framework already contains.

The projection coefficient as the galactic reduction factor.
Section 14 fixes the galactic
acceleration scale as the transverse reduction of the horizon thermal scale,

a0 = gshare,eff

4π2
aH,
aH = cH0,

where the (2π)2 is the two-transverse-mode phase-space cell of the 1 + 2 split and gshare,eff is
the admissibility-sharing content per cell. The same construction fixes the ratio between the

transverse static projection and the full horizon projection,

aH
= gshare,eff

ϵ ≡a0

4π2
≃0.188.

This is not a new coefficient. It is the already-fixed reduction factor of the galactic sector, read
as a statement about source projection: a source that accesses only the transverse static two-
plane couples to the long-range channel at strength ϵ of a source that accesses the full horizon
projection.

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

The unsuppressed projection is accessed by matter that is not part of the phase-averaged
continuum: a collisionless overdensity that is spatially separated from, and dynamically decou-
pled from, a surrounding diffuse medium. The criterion is relational, not intrinsic compactness.
Compactness alone would misclassify: stars in ordinary galaxies, isolated ellipticals, and globu-
lar clusters are compact and bound yet must remain on the galaxy RAR, and they do, because
none is a collisionless node decoupled from a distinct diffuse continuum. A galaxy in a cluster
is different only because it is embedded in, and decoupled from, the intracluster medium. The
suppression is the property of participating in the continuum; matter that has decoupled from
the continuum escapes it. So decoupled collisionless matter sits at the baseline weight, and
incoherent diffuse gas is suppressed to ϵ, with ϵ = gshare,eff/4π2.

A virialized bath is the third state. Once the diffuse atmosphere relaxes into a coherent,
extended phase it is no longer a sum of uncorrelated elements, and it opens a collective long-
range response that lifts it above the baseline, with a bath weight Wbath > 1 that grows with
how completely the atmosphere has virialized, gated by Bbath ∈[0, 1] below. This super-baseline
lift is a distinct premise from the suppression rule above—suppression and its absence span
only [ϵ, 1]. The lift carries a derived ceiling: the transverse channel holds 4π2/gshare,eff = 1/ϵ
of capacity per admissibility cell relative to a single incoherent element, so no source weight
can exceed Wbath = 1/ϵ ≃5.32—the reciprocal of the derived suppression coefficient, and
not a new constant. A fully developed coherent bath saturates this bound, occupying every
transverse cell of the channel it is otherwise suppressed within. The lift function therefore has
fixed endpoints, Wbath = 1 at zero bath and Wbath = 1/ϵ at saturation; the minimal linear
candidate Wbath = 1 + (1 −ϵ)Bbath and the growth profile between the endpoints are evaluated
against cluster data below.

The effective entropic-channel source χent is therefore regime-dependent. In a relaxed cluster
the gas is a virialized bath,

χrel = ρdec + Wbath ρbath

with ρdec the decoupled collisionless substructure and ρbath the virialized continuum; in a non-
equilibrium merger the central gas is shocked and incoherent while only a residual atmosphere
stays virialized,

χmerge = ρdec + ϵ ρshock + Wbath ρvir

which in the Bullet limit, where the displaced gas is shocked and little virialized bath remains on
the cores, reduces to χBullet ≃ρdec+ϵ ρshock. Ordinary matter continues to gravitate through the
usual metric coupling; the projection rule concerns only the long-range entropic-excess channel.

The bath gate as a measured input.
Decoupling requires something to decouple from.
The gate is therefore tied to a measured property of the diffuse atmosphere: the fraction of the
halo’s cosmic baryon allotment that has become an extended virialized hot phase,

Mhot,vir(< r500)



fb,cos = Ωb

Ωm
≃0.156,

Bbath = clip[0,1]

fb,cos M500

with fb,cos fixed by Planck values [39] and Mhot,vir, M500 read from X-ray/SZ and total-mass
estimates. The bookkeeping of the sector is then: ϵ is fixed and derived, fb,cos is fixed cosmologi-
cally, and the per-system inputs (f⋆, fgas, Mhot,vir, M500) are all measured; the one undetermined
object is the coherence-growth profile of the lift, Wbath(Bbath) ∈[1, 1/ϵ], whose endpoints are
fixed and whose interior form is not yet derived. In merging systems Mhot,vir is to be evaluated
from the pre-merger virialized atmosphere rather than the mid-collision gas state, since the
shocked, displaced gas is not a relaxed continuum.

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

fcont = Mhot,vir

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

Rrel = 1 +

Wbath −1


fcont

and the minimal linear candidate for the lift,

Wbath = 1 + (1 −ϵ)Bbath,
1 −ϵ = 1 −gshare,eff

4π2
≃0.812,

uses the amplitude (1 −ϵ), the part of the full horizon channel that the suppressed transverse
branch is missing—not a new coefficient. The physical reading is that as a virialized diffuse bath
forms, the continuum itself opens a collective cluster response whose amplitude scales with how
complete the bath is (Bbath) and how much of the baryon budget sits in it (fcont).

This linear form has the qualitatively correct mass trend: both Bbath and fcont increase with
halo mass—hot-gas fractions rise toward clusters [14, 13] and the atmosphere becomes more fully
virialized—so their product rises monotonically from groups to massive clusters, reproducing the
observed direction with no fitted parameter. Its amplitude, however, is excluded. For CLASH-
scale clusters the gate gives Bbath ≃0.83, fcont ≃0.9, hence Rrel ≃1.6; but evaluating the
source weight required to reproduce the published CLASH relation [4] across its data-supported
acceleration window gives R ≃3.7 at gbar = 10−10 m s−2 rising to R ≃7.8 at 10−11 m s−2. The
linear lift (1 −ϵ)Bbathfcont is short of the observed peak residual by a factor of roughly 2.5–4,
and no escape through missing baryons (a multiple of the X-ray gas mass would be required),
hydrostatic bias (the masses are lensing-based), or sample heterogeneity is available at that
magnitude.

The linear candidate is excluded on amplitude.
Because Bbath ≤1 and fcont ≤1, the
linear candidate bounds the relaxed residual by Rrel < 1 + (1 −ϵ) ≃1.81. The measured peak
residual of relaxed clusters is 3–5 [4, 15], well above this bound, so the linear lift is excluded.
The exclusion is specific to the lift function: the channel-split ontology itself makes a structural
prediction about the residual’s shape that the data support, taken up next.

The hook morphology.
Independently of the lift amplitude, the channel split makes a radius-
resolved prediction about the shape of the cluster residual. In cluster centers the baryons are
dominated by the BCG stellar component, which is decoupled and carries weight 1, so the local
residual starts near unity. Moving outward, the virialized gas continuum comes to dominate and
the residual rises toward the bath-weighted value. At the lowest accelerations the deep branch
compresses any bounded source weight W toward
√

W in acceleration terms, pulling the residual
back down. The predicted profile is therefore a hook: near unity in the stellar-dominated center,
peaking where the bath dominates at intermediate acceleration, and converging back toward
the galaxy relation in the deep outskirts. This is precisely the morphology class reported by
current measurements [15, 16]—a shape that neither a total-baryon modified-gravity law (which
predicts no relaxed-cluster excess at all) nor a constant offset produces.

The framework therefore gets the kind of residual right and, with the linear candidate, its
amplitude wrong: the predicted peak is ≃1.5 for CLASH-like parameters against the observed
3–5. The constraint on the lift function is thus two-ended: Wbath must grow faster with bath
development than the linear candidate, reaching peak residuals of 3–5 at the developed-bath
cluster scale, while remaining small enough at the group scale that X-ray-faint groups stay near
the galaxy relation.

At the saturation weight Wbath = 1/ϵ the predicted mass residual for developed-bath param-
eters is Rsat = fdec + fcont/ϵ ≃4.7–4.9, inside the required window and at the upper edge of
the measured peak band, and the radius-resolved profile then peaks at ≃3.9 in acceleration
terms for CLASH-like parameters against the observed 3–5. The same weight applied uniformly
at the group scale overshoots by a factor of ∼2, so saturation must be gated by coherence
development: massive relaxed clusters sit at or near the bound, while X-ray-faint groups remain
far below it, with the group-scale data requiring Wbath ≲1.4 there.

One caveat accompanies the saturated profile: the predicted central residual depends on the
stellar/gas decomposition and on excluding multiphase cool-core gas from the coherent bath.
The deep-outskirt question—power-law continuation [4] versus convergence toward the galaxy
relation [16]—is adjudicated directly below.

The ceiling against cluster data.
Inverting the X-COP hydrostatic measurements [57]
into source-weight terms gives twenty-four direct tests of the bound: for each of the twelve
clusters, at R500 and at R200, the weight S solving ν(S gbar/a0) S gbar = gobs with non-thermal-
corrected masses, where ν(x) = [1 −e−√x]−1 is the response factor of the radial-acceleration
law (Section 14). Every point respects S ≤1/ϵ; the sample maximum is S = 3.96, at R500
of the merging cluster A2319, the most model-dependent point, and the relaxed systems span
S ≃1.8–2.7 at R500 and 1.4–2.2 at R200.

The same inversion adjudicates the deep end: the power-law continuation requires S ≃6–8
at gbar ≃1–2 × 10−11 m s−2, precisely the accelerations of the R500–R200 points, which sit at
S ≃1.5–2.7; within this sample the deep end converges rather than continuing, and the strongest
published challenge to the bound is not borne out.

The methodological caveat is that the continuation was fit to lensing-based masses of higher-
redshift systems while the inversion here uses local hydrostatic masses; breaching the bound at
R500 would require the non-thermal-corrected masses to be low by a factor of ≃2.3, well beyond
any claimed hydrostatic bias. The measured radial run of the source weight—S ≃3.7–4.9 in the
hook-peak window, where the saturation band is reached, declining to ≃2.2 at R500 and ≃1.5
at R200—is the quantitative target the coherence-growth profile must reproduce.

Direct and fluctuation-based turbulence measurements find low non-thermal support in re-
laxed systems at all probed radii [58, 59], so the decoherence agent gating the lift cannot be the

cluster-to-cluster turbulence level: it must grow with radius even in fully relaxed atmospheres.
The coherence-growth profile between the fixed endpoints, so constrained, is the object the
channel-selection theorem must deliver, with the group end requiring Wbath ≲1.4.

The decoupled-fraction form is excluded by the mass trend.
A second candidate class
lets the residual ride on the decoupled stellar fraction, R = 1 + 4.32 Bbathfdec with fdec =
f⋆/(f⋆+ fgas). This class is excluded by the mass trend rather than the amplitude. Because
fdec falls with mass while Bbath rises, their product peaks at the group/poor-cluster scale and
declines toward massive clusters, predicting the largest anomalies at intermediate mass; the
observed residual instead rises monotonically from groups to massive clusters, where it is most
firmly established. A decoupled-fraction form passes only at the intermediate scale where its
spurious hump crosses the observed band. The residual therefore rides on the continuum, not
the decoupled fraction—the trend direction settles which component carries it, even while the
lift amplitude remains underived. The broader lesson is that the relaxed-cluster amplitude and
the Bullet morphology are distinct observables and must not be collapsed into a single scalar
law.

Bullet-type mergers.
The relaxed residual above and the Bullet morphology are governed by
the same fixed coefficient ϵ but by different source expressions, because the gas is in a different
physical state. In a relaxed cluster the gas is a virialized continuum and drives the residual
through the bath-lift term (Wbath −1)fcont. In a Bullet-type merger the central gas is shocked,
displaced, and out of equilibrium—not a relaxed bath—so it has small effective Bbath and couples
to the long-range channel at the suppressed weight ϵ, while the outgoing collisionless cores remain
decoupled. The two regimes share one coefficient; they are not one scalar law, and we do not
present them as such.

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

Σedge

Agas
Aedge
,
fdec
2ϵfgas
≃0.47,

Σgas
∼
fdec
2ϵfgas

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

Relation to the transport sector.
The telegrapher sector of Section 17 is not the cluster
mechanism; it governs how the field propagates and relaxes once the source weights are set.

The source-projection rule supplies the static weights χent; the transport sector then evolves
them. This division avoids the failure mode of a transport-only account, in which a field sourced
equally by all baryons cannot hold a lensing peak on the outgoing collisionless component. The
full merger observable is obtained by evolving χent(x, t) = ρdec(x, t) + [ · · · ] through the causal
equation with the observed geometry as input.

Falsifiers and open status.
The rule makes sharp predictions beyond the relaxed normal-
ization. (i) The linear candidate’s ceiling Rrel < 1.81 lies below the measured peak residual of
relaxed clusters [4, 15], which excludes that form of the lift. The shape prediction of the channel
split is the hook: residual near unity in BCG-dominated centers, a single peak at intermedi-
ate acceleration where the virialized bath dominates, and convergence back toward the galaxy
relation in the deep outskirts. A relaxed cluster with a residual profile that is monotonic in
acceleration, or that peaks in the stellar-dominated center, would falsify the channel split itself.
The capacity bound supplies the sector’s hard ceiling: Rrel ≤fdec + fcont/ϵ ≃5 for developed-
bath parameters, with developed baths reaching it in the hook-peak window and all twenty-four
X-COP source-weight inversions respecting it; a robust relaxed-cluster mass residual well above
this bound would falsify the capacity bound, the sector’s central commitment.

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

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

κobs(x, y) = A
h
Wbath Σbath(x, y) + ϵ Σshock(x, y) + Σdec(x, y)
i
+ b

with ϵ fixed at gshare,eff/4π2 ≃0.188 or fit freely, and Wbath fit per system within [1, 1/ϵ], the
saturated value 1/ϵ ≃5.3 predicted for developed baths.
Recovering both ϵ ≃0.19 on the
shocked channel and saturation on the bath channel would be a two-coefficient parameter-free
success. In the relaxed limit this reduces to the residual law Rrel = 1 + (Wbath −1)fcont; in the
Bullet limit the central gas is shocked (Σshock at weight ϵ) and the peaks fall on Σdec. A best-fit ϵ
near 0.19 across Bullet, MACS J0025, the Musket Ball, a relaxed CLASH sample [4], and X-ray-
bright versus X-ray-faint groups would be a parameter-free success that neither ΛCDM nor total-
baryon modified gravity predicts, while a best fit near 1 or near 0 would falsify the sector. This
map fit is the principal outstanding empirical task and has not yet been performed. It is decisive
only if it removes the escape hatch that makes existing reconstructions non-discriminating: ϵ
must be floated against the channel-weighted baryonic maps (Σbath, Σshock, Σdec) with free
dark haloes disallowed, since free haloes absorb the gas weight and leave ϵ unconstrained—
which is why current Bullet reconstructions neither recover nor exclude ϵ ≃0.19. The three-
component decomposition makes the fit more demanding than a two-component one, but also
more distinctive.

Three open items remain at the theory level.
First, the relaxed residual and the Bullet
morphology are governed by one fixed coefficient ϵ but by two regime-specific source expressions
(virialized bath versus shocked non-equilibrium gas). This is justified by the differing physical
state of the gas, but it is a weaker unification than a single scalar law, and the boundary between
the regimes—how Bbath interpolates as a shocked atmosphere re-virializes—is not yet derived.

Second, the channel-selection rule rests on two distinct physical premises, not one.
The

first is a suppression premise: participation in a phase-averaged diffuse continuum restricts a
source to the transverse projection (weight ϵ), so that a collisionless overdensity decoupled from
that continuum recovers the full horizon projection (weight 1). The second is a collective-lift
premise: once a diffuse continuum virializes into a coherent extended phase it opens a collective
long-range response that raises it above the decoupled baseline, to a weight Wbath > 1. These
two are logically independent—escaping the suppression returns a source to weight 1 but does
not by itself carry it past unity—and it is the collective-lift premise, not the suppression premise,
that carries the relaxed-cluster residual. The coefficient ϵ is derived; both premises are plausible
within the ontology but neither is yet derived from the microscopic source map, and supplying
those derivations is the channel-selection theorem. The collective-lift premise is the weaker of
the two: the coherence/phase-averaging argument that motivates the suppression does not on its
own generate super-baseline coupling, and the linear form of the lift is quantitatively excluded by
the measured peak residual (above). The channel-selection theorem must therefore deliver the
coherence-growth profile between the fixed endpoints, reproducing the measured radial decline of
the source weight under uniformly low turbulence, constrained at the group end to Wbath ≲1.4
and saturating at 1/ϵ in the hook-peak window.

Third, the identification of the decoupled component with stellar/galaxy mass and of the
continuum with the gas is a coarse split; intracluster light and tidally stripped stars blur it at
a level the resolved map fit would expose. Abell 520, whose reported gas-coincident dark core
is itself disputed, is a phase-state stress case rather than a clean test: a re-cohering or quasi-
bound central component would raise its effective Bbath and return lensing toward the gas,
while a confirmed young merger with a robust gas-centered, galaxy-free lensing peak—where
the re-coherence escape is unavailable on time grounds—would be a serious challenge.

This sector is therefore presented as a structured, falsifiable proposal, not as closed physics:
the projection coefficient ϵ is derived; the bath gate is a measured input; the trend direction
(residual riding on the continuum, rising from groups to clusters) and the hook morphology
of the residual profile are supported by current data; the linear lift candidate is excluded on
amplitude; the saturation amplitude is fixed by the capacity bound at the reciprocal of the
derived projection coefficient, and the saturated residual lands inside the measured peak band;
the coherence-growth profile, the resolved-map test, and the channel-selection theorem are the
named open work.

18. Cosmology and the Hubble-Tension Sector

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

The cosmological sector uses the same field split,

S(x, t) = 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

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.

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.

The direction of the shift matters here, and so does 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 required
direction; what remains open is the full perturbation propagation and likelihood-level confronta-
tion. The homogeneous mode modifies the background history; the inhomogeneous branch con-
tinues to govern the local weak-field observables already fixed earlier in the derivation. That
separation 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

18.5 The Saturated Phase and the Cosmic Microwave Background

In galaxies the substrate has slack: a source’s demand for capacity is met by a developing
local response, and the radial-acceleration law of Section 14 follows. The early universe has
no slack.
The capacity bath is saturated across the linear perturbation regime: wherever a
perturbation-scale source demands a response, the available transfer channel has already reached
its ceiling, so there is no room for a developing response field. What gravitates is then not a field
relaxing toward a source but a conserved count of committed capacity channels: cells locked into
transferring one bit of capacity per tick on behalf of a defect. That committed count is carried
along with the expansion and dilutes as the volume grows, so it redshifts as a−3 and gravitates
as pressureless dust. This saturated phase plays, in the microwave-background sector, the role
conventionally assigned to cold dark matter — not a new particle species but a phase of the
same medium.

Saturation is forced at recombination.
At z ≃1100 the accelerations of linear density
perturbations satisfy x = gpert/a0(z) ∼10−3, where the response demand ν(x) ∼30 far exceeds
the capacity ceiling 1/ϵ = 5.32. The bound derived for the cluster bath (Section 17.5) is therefore
mandatory in the early universe: the response channel for the cosmological perturbation bath

is saturated. The background itself sits permanently at the ceiling acceleration, since a0(z) =
ϵ cH(z) implies the invariant cH(z)/a0(z) = 1/ϵ at every redshift.

The pinned reading and the dust theorem.
The pinned reading is the statement that the
saturated bath holds exactly at the ceiling and stays there. The transfer law quantizes channel
transport at one bit per tick as an upper bound; saturation is the attainment of that bound,
so on the pinned reading every committed channel advances by exactly one unit per tick, and
while the phase remains pinned no channel can decommit, since there is no slack for it to relax
into. The committed count is then conserved and dilutes only with the expanding volume. The
coarse-grained field ϕ of the committed phase then carries a constraint rather than a generic
kinetic term: X = −1

2gµν∂µϕ ∂νϕ = 1

2, with ∂µϕ a unit timelike covector. The leading action
consistent with the constraint is

Scomm =
Z
d4x √−g λ

X −1

2


,

whose variation gives ∇µ(λ ∂µϕ) = 0 and, on the constraint surface,

Tµν = ρ uµuν,
ρ = E0 n,
p = 0,
c2
s = 0,
ρ ∝a−3,

with uµ = ∂µϕ and n the conserved spatial density of committed cells. This is the constrained-
scalar (mimetic) class [60]: conditional on the pinned reading, the committed phase is pressure-
less dust, and the perturbation carrier is the conserved committed-capacity density itself. The
gravitating component is therefore a phase of the medium, not an added species; throughout,
“dark matter” names the conventional attribution this phase replaces. The known caveats of
the class — caustic formation and the need for a small-scale completion — are inherited and
recorded; they parallel the standard cold-dark-matter small-scale account. The continuous dy-
namical passage itself — how the generic unconstrained kinetic term of the weak-field branch is
driven onto the constraint surface X = 1

2 as the channel saturates, rather than the constrained
action being posited at the ceiling — is asserted here rather than derived; supplying it, together
with the commit/decommit energy accounting recorded below, is an open task within this con-
ditional sector. The constrained route also parallels the mechanism by which the one relativistic
completion of Milgromian dynamics known to meet this test does so [61].

Uniqueness of the carrier.
The constraint coupling is the unique survivor of the dynamics
classes examined (Appendix L). Relaxational response kernels are excluded twice over: a growth
clock calibrated on the cluster radial decline misses cosmological development by a factor of
order thirty, and the precision of the measured acoustic peaks bounds any oscillatory leakage
of a lagged response below one part in five hundred, a rejection no causal filter achieves over
the few oscillation periods available before recombination. Bound-type rail readings sit at the
sound-speed pole and carry no perturbation; plateau approaches have c2
s = −1/(2n+1) < 0 and
are gradient-unstable; the released branches have c2
s = 1

2 and 1 and free-stream. A general no-go
for relaxation-plus-cap carriers is recorded in Appendix L; the conserved committed density is
precisely the additional integrating variable that the no-go requires and the constraint reading
supplies.

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

ϵ =
4π2

Ωc
Ωb
= 1

gshare,eff
= 5.321
against the measured 5.364 ± 0.065,

a 0.7σ agreement with the abundance counted rather than fitted [39]; equivalently Ωm/Ωb =
1 + 1/ϵ = 6.32. A standard Einstein–Boltzmann computation [62] with the cold component tied

to this value and the acoustic scale θ∗held fixed reproduces the Planck best-fit temperature
spectrum with a root-mean-square residual of 0.15% over ℓ= 2–2500 and 0.08% in the third-
peak region. At the likelihood level, the tied model carries one fewer free parameter than ΛCDM
and sits at ∆χ2 = +2.15 against the six-parameter best fit on the compressed Planck TT,TE,EE
likelihood [30] — within noise of the full fit, and an upper bound on the constrained optimum
since the remaining parameters were not re-optimized.
The comparison must hold θ∗fixed:
tying the densities at fixed H0 instead breaks the acoustic scale and misreports the residual.

That the committed density attains the ceiling exactly, rather than a development-dependent
fraction of it, follows from the recruitment structure of commitment. Commitment is source-
driven: relaxation serves the demand of specific defects, and cells receiving no demand commit
nothing. Each source’s allocation is capped at 1/ϵ per unit source mass, the same per-source
bound that governs the cluster bath (Section 17.5). At recombination the demand side is fixed
by the arithmetic above: perturbation-scale sources demand ν ∼30, far above the cap, while
the homogeneous background sits at the invariant acceleration cH/a0 = 1/ϵ, where the demand
ν(1/ϵ) = 1.11 lies below it. With supply abundant and commitment irreversible in the pinned
regime, every source therefore recruits to its cap and the background recruits nothing. Alloca-
tions add, because committed units are per-defect: a unit serving two defects would make their
joint entanglement budget, and hence their joint mass, sub-additive, contradicting the additivity
of separated masses under Postulate II (Section 3.2). The committed density is then

ρc = 1

ϵ ρb

pointwise at the commitment epoch, which delivers the abundance above and hands the per-
turbations adiabatic, baryon-tracking initial conditions, δc = δb — the conditions the Einstein–
Boltzmann computation assumed. The derivation is conditional on the same pinned reading as
the dust theorem and on the per-defect bookkeeping just invoked; the energy accounting of the
committed budget remains the sector’s open completion, and conservation of the perturbations
is taken up directly below.

Conditional conservation: release at the caustic.
The committed component must be-
have as conserved dust while the linear bath remains pinned, yet the galactic branch (Section 14)
requires collapsed systems to follow the baryonic response law with no surviving collisionless halo,
so the completion is a conditional-conservation law: ∇µJµ
commit = 0 in the pinned regime, with
decommitment upon release. The constrained phase fixes the release mechanism itself. Because
the committed flow is the gradient of a single clock field, uµ = ∂µϕ, it is exactly irrotational
and single-valued, and a multi-stream or vortical velocity field admits no such potential; gravi-
tational collapse therefore terminates the phase at first shell-crossing. The caustic — the known
breakdown locus of the constrained-scalar class — is here the decommitment event, converting
committed capacity into the local response branch. Under the synchrony definition of commit-
ment, one counter per cell ticking with the local phase, decommitment at first stream-crossing is
forced rather than chosen: a single fixed cell cannot remain synchronized with two distinct phase
branches, so conversion is event-like at the first caustic, a result holding at the same conditional
grade as the pinned reading itself. Re-entry is forbidden on entropic grounds: recommitment of a
virialized region would require shedding vorticity and re-synchronizing with the advanced global
clock, a spontaneous recoherence, so decommitment is irreversible, with the door’s orientation
resting on the same history weighting that fixes the arrow of time (Section 21). The resulting
partition is the one the data require: the linear cosmological field never shell-crosses and remains
committed dust, while collapsed systems have shell-crossed and retain none. The alternative,
commitment tracking the instantaneous local acceleration, is excluded directly by rotation-curve
data: below gc the binned radial-acceleration residuals lie within 0.05 dex of the response law [1],
against the +0.40 dex excess that locally regrown dust at the cosmic ratio would produce. One

completion remains open and is recorded, the energy bookkeeping of conversion at the caustic,
together with the consequence that late nonlinear structure growth proceeds on the response
branch. The sharp geometric prediction is registered as a falsifier: infalling material between
turnaround and first shell-crossing is single-stream and still committed, so clusters should carry
a surviving committed component on their infall streams, terminating at the splashback surface
[63], with an excess gravitating rim in that shell, a sharp edge at the outermost caustic, and
none inside it. The recruitment count fixes the rim’s amplitude as well as its geometry: along
a stream of which a fraction frel has already shell-crossed, the surviving committed density is
(1 −frel) ρb/ϵ, so the prediction specifies location, edge, and magnitude together.

Domain structure.
The release threshold ν(xc) = 1/ϵ gives xc = 0.0433, an acceleration
gc = xc a0 ≃5.2 × 10−12 m s−2 today. The linear cosmological field remains below threshold at
all epochs (x ≲2 × 10−3 today, smaller as √a earlier), so linear-regime observables — baryon
acoustic oscillations, the linear power spectrum, microwave-background lensing — coincide with
the ΛCDM form at the abundance above; in particular the framework inherits, and does not
relieve, the S8 tension. Departures are confined to released, collapsed structures, which is the
galactic and cluster phenomenology of Sections 14 and 17.5.

Distinction from horizon saturation.
The committed cosmological phase — channels ad-
vancing at capacity, a running clock — is distinct from the terminal saturation of the strong-field
sector (Section 20), where capacity is exhausted and transactions cease. Whether the two termini
are stages of one process is open and carries a stated consistency burden: the gravitating energy
of committed capacity must coincide with the mass already accounted at infinity, with no dou-
ble counting. This is recorded as an open question shared by the strong-field and cosmological
sectors.

19. Why These Sectors Belong

The most directly constrained chain — microstructure to static weak-field observables — is now
in hand, and the sectors on either side of it form one program rather than a list of add-ons. Each
asks what the same finite-capacity substrate does once it leaves the static weak-field regime.

Transport asks how the capacity-strain field propagates and settles in time once a source
moves (Section 17). Cosmology asks how the homogeneous capacity mode behaves on horizon
scales (Section 18). The saturated phase asks what happens when the bath has no slack left and
the carrier becomes a conserved committed density (Section 18.5). The strong-field branch asks
the opposite-extreme question, what happens where local capacity falls to zero (Section 20).
Many-Pasts asks what history and probability structure the substrate carries, and supplies the
memoryless dressing the scale-setting already used (Section 21). The microstructure Hamiltonian
asks what dynamics underlies the whole construction (Section 22).

These sectors share the static weak-field ontology but not its evidential status. The static chain
is the closed result; the others are conditional extensions, frontier completions, or structural
realizations, marked where each appears.

20. Strong-Field Branch and Bounded Occupancy

The weak-field theory used small capacity deficits. Black holes are the opposite regime: they
ask what happens when the deficit is no longer small, and ultimately where the medium runs
out of available capacity altogether. The branch therefore begins with the same variable as the

weak-field bridge, written in the form that makes boundedness 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. 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: horizon regularity at q = 0 selects the ingoing mode at leading order, so the
leading reflectivity vanishes. Echoes are correction-level rather than generic; they arise only if
microscopic boundary channels carry finite UV reflectivity or if the effective reflection surface
is displaced outward to a stretched layer q = ϵ > 0. The entropy result follows from substrate
inputs already used elsewhere: Postulate II identifies the per-channel cut entropy as ln 2 via
fermionic face exclusion, while the transverse bulk graph response fixes the channel density, and
their product is the Bekenstein–Hawking 1/4 as an exact identity. The geometric identification

of q with the areal-radius gradient invariant |∇R|2 in spherical symmetry further shows that
the substrate q equals 1 −2GMMS/(c2R) on the exterior, so the q = 0 saturation surface coin-
cides with the apparent horizon and is inherited from ordinary trapped-surface formation in GR
collapse rather than from a separate substrate-transport postulate.

The strong-field branch is therefore closed at the level of universal predictions. Static and sta-
tionary asymptotically flat vacuum exteriors reduce to Schwarzschild, Kerr, Reissner–Nordström,
or Kerr–Newman through the standard vacuum Einstein and Einstein–Maxwell branches; the
horizon location is fixed as the marginal-capacity surface; the area-law coefficient is closed under
Postulate II and the bulk graph response; and the leading reflectivity vanishes by horizon regular-
ity. What remains is nonuniversal: the microscopic relaxation spectrum, possible stretched-layer
corrections, transient response on the boundary, and the explicit graph-side consistency check
on substrate transport during dynamical collapse.

21. Many-Pasts: The History-Space Ontology

Many-Pasts is the most conservative part of the theory to test and the most radical to adopt.
In the selected operational branch, it changes nothing a laboratory measures: it reproduces
Born-rule statistics and no-signaling exactly. Ontologically it replaces the entire interpretive
layer of quantum mechanics — branching worlds, collapse events, or hidden variables — with
a single object, a weighting over the admissible past histories of the entanglement substrate
(Section 3.3).

The operational weight.
The history weight P(H|P) ∝e−D(H,P) returns ordinary quantum
statistics in the projective laboratory limit through the identity

e−D(H,P) = Tr(ΠP ρH→now) ,

the history-space distance reducing to the overlap of the present projector with the state evolved
from H. Born recovery is a consistency condition on the weighting family, not a fresh derivation
of the rule: requiring exact Born statistics selects α = 1, and forbidding any signaling-sensitive
operational bias selects β = 0. In that branch the rule and no-signaling both hold exactly, and
nothing a laboratory measures distinguishes the framework from standard quantum mechanics.
The conservatism is deliberate: the ontological content carries no operational cost.

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

Familiar quantum examples.
In the standard puzzles the weight changes the ontology, not
the calculation. In the double-slit experiment the particle is not taken to have secretly travelled
one slit while the other path was unreal: before any which-path record exists, the detection event
is supported by the admissible histories through the apparatus, including the alternatives whose
amplitudes interfere in ordinary quantum mechanics, and making a which-path record conditions
that ensemble, removes the cross terms by decoherence, and erases the fringes. In an EPR or
Bell experiment the present is a joint record of the pair and the detectors, and the histories
are weighted as histories of the whole entangled system, which reproduces the nonclassical
correlations while leaving the local Born marginals — and so the impossibility of signaling —
untouched; the “spooky” element is the global conditioning of the history ensemble on a shared

record, not a faster-than-light influence. Measurement is read the same way: it adds no collapse
law but creates a durable record, and the operative ensemble is then the histories compatible
with that record. The probabilities are the usual Born probabilities throughout; Many-Pasts
supplies only the history-space reading of why those are the numbers being counted. The worked
versions are in Appendix G.10.

The arrow of time.
The same weight orients time through conditional typicality. Among the
histories consistent with the present macroscopic records, overwhelmingly many show entropy
increasing toward the future direction those records define. This is not an added law of laboratory
probability; it is a statement about which histories dominate the weight, and it places the
substrate’s account of temporal asymmetry on the same footing as its account of quantum
statistics.

Where the weight is used.
Many-Pasts enters the gravitational chain through the electron
dressing. Its memorylessness in substrate time is one of the two conditions that fix the absolute
substrate length: each dressing pass of the electron is an independent draw from the admis-
sibility ensemble, so the seven channels carry the full sharing entropy and add (Sections 13,
22; Appendix H). The same conditional-typicality weight that orients time also supplies the
orientation of decommitment in the saturated phase, forbidding the spontaneous recoherence
that re-entry into the committed phase would require (Section 18.5). The same weight there-
fore appears in three places: the scale-setting branch, the cosmological release law, and the
quantum-foundational layer.

Status.
Operational closure is exact: the laboratory sector is standard quantum mechanics,
and the technical reduction is collected in Appendix G. The ontology is central rather than
secondary, but its distinctive claims — branch realization and the arrow of time — are structural
and cosmological, not new laboratory predictions, and they should be read at that grade.

22. Microstructure Hamiltonian and Underlying Dynamics

The UV closure chain has a microscopic dynamical realization with two sides: the emergence of
the continuum geometry from the substrate, and the defect dynamics that fixes the scale-setting
principle of the weak-field chain. The first is the condensate-side compatibility check; the second
closes the one microscopic principle the static branch was conditional on. Section 13 stated the
electron anchor in physical terms; this section gives the microscopic dressing dynamics behind
it, and Appendix H carries the technical derivation.

On the geometry side, the realization is a GFT/condensate picture [48, 49] 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.

On the defect side, the same substrate supplies the dynamics behind the faithful sector-
resolution principle that sets the absolute cell length L∗.
The elementary fermionic defect
is governed by a dressing Hamiltonian whose single-channel terms make the lightest charged
defect bind all seven face sectors once — the one-pass ground state that supplies the exponent

7 — and whose inter-channel term controls whether the seven-channel dressing cloud factorizes.
Factorization turns the seven equal sector contributions into the additive 7gshare,eff, and needs
two things: each channel must carry the full admissibility entropy, and the seven must be
mutually independent. The first is memorylessness in substrate time — the Many-Pasts content
of Postulate III, with the substrate relaxing to a fresh draw long before the electron is read out.
The second follows from the electron’s own lightness: inter-channel correlation would contract
the defect and raise its mass, so the lightest charged defect carries none. Each channel at full
entropy and the channels independent give dimeff = e7gshare,eff and λe/L∗= 2

3e7gshare,eff, worked
out in full in Appendix H.

Reading the support ratio λe/L∗as a physical Compton length used one further identification:
that the defect advances one phase cycle per dressing recurrence.
Appendix H.6 gives that
recurrence an operator form and removes the identification as a free assumption. The mass is
the exported gap of the memoryless refresh super-operator, mec2 = 3

2(ℏ/τ∗) e−7gshare,eff (the 3

2 the
transverse export of the length relation), so it is read as a recurrence rate, not a binding-energy
sum, and the phase-cycle claim decomposes into the recurrence rate, a unit committed charge
per pass from fermionic one-pass occupation, and the standard condensate phase relation, none
with a free constant. What is not forced is the identification of the winding phase with the
condensate U(1) (Fork A); Appendix H.7 derives the three dressing functionals from the mean-
field condensate and shows that Fork A and the rank-one inter-channel coupling reduce to a
single premise — a single scalar field on that condensate — which is the founding premise of the
framework and is tested in the charged-lepton ladder (Appendix I.1). The hierarchy then rests
on one asymmetry. Binding across the seven sectors is additive, but their coherent recurrence is
multiplicative, so the continuous maintenance of the defect is large and linear in the commitment
while the rest energy, fixed by the rare joint return, is exponentially small. The mass scale is
the rare-return gap of the refresh generator, not the bare binding cost: the electron is light not
because it is weakly bound but because its completed seven-sector dressing recurs rarely.

Together these two sides do more than check compatibility. The defect dynamics derives the
faithful sector-resolution principle from the mass–entropy ontology and the electron anchor, so
the substrate length and the induced gravitational scale follow from commitments the theory
already makes. What remains microscopically open is twofold: the first-principles derivation of
every inhomogeneous continuum coefficient from the full kernel, and — conditional on Fork A —
the GFT substrate actually producing the single-scalar mean-field condensate, with its emergent
four-geometry, on which the defect spectrum and the geometry side both depend (Appendix H.7).

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.

The status column uses a fixed vocabulary.
Closed means derived within the stated pos-
tulates and ensemble. Fixed means no phenomenological freedom remains once the branch is
adopted. Conditional means the result follows if a named reading or completion holds. Fron-
tier or open means the piece is structured but not yet complete. Empirical support means a
comparison with data rather than a derivation, and audit task means an independent derivation
or check is still required. Rows follow the order of the text, grouped by sector from the UV
counting through the foundational, weak-field, galactic, transport, cosmological, strong-field,
and quantum-foundational sectors.


![Table 2](paper-70-v10_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
stationary normalized closure evidence
ln Z + 3

Part II, App. B

2 ln η maximized on the exact
K2 spectrum; the 3/2 is the
determinant weight of the three
closure-defect components

gshare,eff
UV entropy
Closed
exact weighted evaluation
Part II, App. B
Substrate length L∗
(faithful sector
resolution /
support-to-length)

3 e7gshare,eff from the
recurrence time of the memoryless
seven-channel dressing (waiting time for
joint typicality); support exponent
bracketed at unity within 10−3 by G∗
and the lepton ladder; phase-cycle
identification discharged in App. H
(Γ ⊕∆Q=1⊕Josephson), residual is
Fork A (winding phase = condensate
U(1)) and the unit-winding integer,
both tested by the lepton ladder

λe/L∗= 2

scale setting
Derived from
dressing
recurrence;
identification
discharged under
Fork A

Part I, App. D,
H

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,
K

Conditional
derivation at
percent level;
calibration, with
provenance
recorded

Matched G
weak-field
gravity

bridge law + Green-matched κ/γ +
fixed-epoch normalization of κ/(γS∞);
comparison with G∗tests coherent
propagation of the same L∗, not a
disjoint-input second derivation of G

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

Hseven = 7gshare,eff from subadditivity
plus mass minimization: full
per-channel entropy from the
memoryless dressing, channel
independence forced because correlation
would raise the defect mass (∆= 0 for
the lightest one-bit defect)

Seven-sector
additivity

UV scale
theorem

Closed conditional
on the
support-to-length
identification

App. D, H

UV entropy × horizon scale cH0, with
4π2 = (2π)2 the canonical
two-transverse-mode phase-space cell of
the 1 + 2 split

a0
galactic EFT
Fixed in the closed
weak-field
realization

Part III, App.
C


![Table 3](paper-70-v10_images/table_3.png)
*Table 3*

Quantity / Claim
Sector
Status
Type of Support
Where
Established
RAR law
galactic EFT
Closed at the
rate-bookkeeping
level

a0 fixed by the coefficient chain;
x =
p

Part III, App.
C

gbar/a0 forced by the spectral
invariant of the bilinear form; 1 + nB(x)
from exact bosonic emission statistics,
with absorption closed by the quantized
one-bit deficit; flux identification
derived from the static transport limit
(conserved bit current, Green-matched
normalization), with inter-channel
delivery fixing the locality of x; kernel
neutrality F ≡1 from per-cell channels
and bit quantization; graph-level
substrate computation of the
inter-channel vertex is the remaining
gold-standard check

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

Saturated-phase
committed dust

cosmology
Conditional on the
pinned transfer-law
reading

saturation forced at recombination;
constrained-scalar dust theorem (p = 0,
c2
s = 0, ρ ∝a−3); relaxational carriers
excluded

Part IV, §18.5,
App. L

Committed-capacity
abundance
Ωc/Ωb = 1/ϵ

cosmology
Derived from
forced recruitment,
conditional on the
pinned reading and
per-defect
bookkeeping

over-demanded sources recruit to the
per-source cap, the background sits
below it, and per-defect allocations add
(Postulate II corollary), giving
ρc = ρb/ϵ with adiabatic seeding
δc = δb; temperature-spectrum check at
fixed abundance; counted, not fitted

Part IV, §18.5

Caustic release /
conditional
conservation

cosmology
Conditional —
forced under the
synchrony reading

committed dust decommits at first
shell-crossing of the irrotational clock
flow; splashback-rim falsifier; conversion
bookkeeping open

Part IV, §18.5

Diffuse source
projection
ϵ = gshare,eff/4π2

cluster sector
Derived (no new
coefficient)

same transverse reduction factor that
fixes a0/aH (Section 14)

Part IV, §17.5

Bath gate Bbath
cluster sector
Operationally
closed; microscopic
origin open

Mhot,vir/(fb,cosM500) from X-ray/SZ;
no per-system fitted knob

Part IV, §17.5

linear candidate’s ceiling 1.81 lies below
measured peak residual 3–5 [4, 15, 16];
capacity bound fixes saturation at 1/ϵ,
giving Rsat ≃4.7–4.9 and a
radius-resolved peak ≃3.9 inside the
measured band; the bound holds against
all twenty-four X-COP source-weight
inversions, which also exclude the deep
power-law continuation within that
sample; coherence-growth profile open

Cluster residual
Rrel =
1 + (Wbath −1)fcont

cluster sector
Hook morphology
and trend direction
supported; linear
lift candidate
excluded; lift
function open

Part IV, §17.5

Relaxed vs. merger
source expressions

cluster sector
Conditional — two
regimes, one
coefficient

residual rides on virialized bath; Bullet
on shocked gas + decoupled clumps;
regime boundary not yet derived

Part IV, §17.5


![Table 4](paper-70-v10_images/table_4.png)
*Table 4*

Quantity / Claim
Sector
Status
Type of Support
Where
Established
Channel-selection
rule (suppression +
collective lift)

cluster sector
Conditional —
suppression open;
linear lift excluded

transverse-suppression premise
(participation ⇒weight ϵ) not yet
derived from the source map; linear
form of the lift excluded by the
measured peak residual; saturation
endpoint fixed at 1/ϵ by the channel
capacity bound; coherence-growth
profile between the endpoints not yet
derived

Part IV, §17.5

ϵ brings gas/galaxy to near-parity,
compactness completes the inversion;
existing flexible reconstructions
non-discriminating because gas weight
is halo-degenerate

Bullet gas/lensing
inversion

cluster sector
Structurally
supported;
consistent but
untested (ϵ not yet
measured)

Part IV, §17.5

three-component κ ∝
(1+(1−ϵ)Bbath)Σbath +ϵΣshock +Σdec;
predicted best-fit ϵ ≃0.19, floated on
baryonic maps with free dark haloes
disallowed; not yet performed

Resolved cluster
lensing-map test

cluster sector
Open — principal
empirical task

Part IV, §17.5

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
Closed at leading
EFT order

unchanged Schwarzschild exterior;
absorbing boundary at q = 0 fixed by
horizon regularity; echoes only as UV /
stretched-layer corrections

App. F

per-channel cut entropy ln 2 from
fermionic face exclusion + channel
density from G⊥= (2/3)Gtet(0);
product nhorScell
∞
= 1/4 as exact
identity

Bekenstein–Hawking
area-law bridge

strong field
Closed within the
constrained-
capacity EFT
under Postulate II
and the transverse
graph response

App. C, F

q = |∇R|2 = 1 −2GMMS/(c2R) in
spherical symmetry, with q = 0 at the
marginal-trapped-surface inherited from
standard apparent-horizon formation;
remaining work covers relaxation
spectra, stretched-layer corrections, and
transient response

Horizon formation
and boundary
microphysics

strong field
Formation closed
geometrically;
nonuniversal
spectroscopy open

Part V, App. F

Rotating / charged
stationary exteriors

strong field
Closed within the
constrained-
capacity EFT

vacuum Einstein / Einstein–Maxwell
reduction on Mq gives Kerr /
Reissner–Nordström / Kerr–Newman
exteriors; q = 0 identified with outer
Killing horizon; inner Cauchy horizons
are non-physical analytic continuations
into q < 0

App. F

exact Born statistics selects α = 1
within the weighting family; the content
is the existence of a branch meeting
Born and no-signaling simultaneously

Many-Pasts Born
compatibility

quantum
foundations

Consistency
condition

Part V, App. G

No-signaling in
operational branch

quantum
foundations

Consistency
condition

forbidding signaling-sensitive bias
selects β = 0

Part V, App. G

Arrow-of-time
account

quantum
foundations

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


![Table 5](paper-70-v10_images/table_5.png)
*Table 5*

Quantity / Claim
Sector
Status
Type of Support
Where
Established
Microstructure
Hamiltonian

UV realization
Defect side closes
scale-setting;
geometry side
coherent

dressing Hamiltonian with one-pass
ground state, memoryless dressing, and
minimal-mass channel independence
(scale-setting); mass as the
refresh-operator gap with the
phase-cycle identification discharged
into Γ ⊕∆Q=1⊕Josephson (App. H.6)
and ε0, Fm, V from the mean-field
condensate with rank-one V as one
scalar (App. H.7); GFT/condensate
origin of the kinetic term (geometry)

Part V, App. H

three-generation termination from K2

Charged-lepton
spectrum

particle-sector
extension

Derived;
coefficients forced,
one map-form
input

App. I

permutation-invariance at N = 3;
mN/me = 720N(2/7)N2 with 720 = 6!,
2/7 the singlet projection, N2 the
ordered-pair count; matches PDG at
0.5–0.65% with no fitted parameters

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.

Cosmological row.
The saturated phase enters the ledger as follows: the dust form of the
committed phase is a theorem of the constrained-scalar class, conditional on the pinned reading
of the transfer law; the abundance 1/ϵ = 5.321 stands against the measured 5.364±0.065, derived
from forced recruitment conditional on the same pinned reading and the per-defect bookkeeping
of Postulate II; the energy accounting of the committed budget is open; the linear regime is
closed by the domain structure.

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, 34], the baryonic Tully–
Fisher scaling in systems where the EFT should apply, and the weak-field lensing sector, where
stacked galaxy–galaxy lensing now probes the relation two decades below the rotation-curve
regime [35]. 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, 43], also require
the leading no-slip branch to survive at high precision, and laboratory fifth-force searches [45]
bound any residual static scalar channel in the same regime.

Wide binaries supply an independent solar-neighborhood discriminator. The occupancy ar-
gument of Section 14 is local, a feature its transport derivation fixes rather than assumes: the
acceleration entering x is the total local baryonic field, so a binary embedded in the Galactic field
gext ≃1.4–1.9 × 10−10 m s−2 has its internal dynamics amplified once the pair’s own field falls
below the Galactic term. The predicted internal-force boost rises from unity at small separa-
tions to 1+nB(
p

gext/a0) ≃1.4–1.5 beyond a few thousand au. This lands close to the AQUAL
external-field value through a different mechanism — locality of the occupancy argument in the
total baryonic field, with no nonlinear field equation — so the wide-binary regime separates

the framework from Newtonian gravity while leaving it degenerate with AQUAL-class theories.
Current analyses divide: a low-acceleration boost of about 1.4 is reported in Gaia DR3 samples
[27], and consistency with Newtonian gravity is reported in an independent analysis of the same
mission’s data [28]; the disagreement is methodological. The framework’s commitment is fixed
in advance: a settled Newtonian verdict falsifies the locality of the occupancy rule.

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. Cluster
data accordingly split into two tests of distinct parts of the framework: source-projection tests,
which ask whether the relaxed-cluster residual profile follows the predicted hook morphology—
near unity in BCG-dominated centers, peaked where the virialized bath dominates, converging
in the deep outskirts [4, 15, 16]—and whether resolved merger maps prefer the projection coef-
ficient ϵ ≃0.19 (Section 17.5); the linear lift candidate is excluded on amplitude (Section 17.5);
and transport tests, which ask whether the causal propagation law D/τ0 = c2 correctly evolves
those source weights through a merger. Systems such as the Bullet Cluster [2] probe both, and
should not be treated merely as larger versions of the static galaxy problem.

24.3 Cosmological falsifiers

Cosmology presents a different kind of test. The question there is whether a full Boltzmann
treatment allows the trace-coupled homogeneous mode to reduce the sound horizon without
spoiling the CMB or structure-growth observables. If it cannot, the cosmological extension fails
on its own terms. The empirical target is set by the current measurement spread: early-universe
inferences near 67.4 [39], distance-ladder determinations ranging from ≃70 [41] to 73 [40], and
a tension whose proposed resolutions are reviewed in Di Valentino et al. [42].

The epoch dependence a0(z) = cH(z) gshare,eff/4π2 is the framework’s most exposed kinematic
prediction, and high-redshift rotation curves already bear on it. At z ≃2.3 the predicted scale is
3.5 a0; a massive star-forming disk with gbar ≃2×10−10 m s−2 at 8 kpc is then predicted to show
gobs/gbar ≃2.0, against 1.36 for an epoch-independent scale. Observed outer rotation curves
of such disks decline and indicate strong baryon dominance within the disk scale [29, 33], the
direction opposite to the predicted enhancement. Pressure-support corrections at these redshifts
are large and the stacked-profile normalization is disputed, so the confrontation is unsettled.
The framework requires that massive z ≃2 disks, once those corrections are controlled, show
stronger low-acceleration boosts than matched z = 0 systems; the opposite outcome falsifies the
identification of a0 with cH(z) while leaving the static z = 0 sector intact.

Direct measurements at intermediate redshift have since entered.
A MUSE sample of 79
star-forming galaxies at 0.33 < z < 1.44 finds the radial-acceleration relation persisting with
a characteristic acceleration that rises systematically with redshift [31], and a resolved low-
redshift H i sample independently reports tentative evolution in the same direction [32]. The
sign is the framework’s: an epoch-independent a0 predicts no evolution at all. The magnitude is
not yet a test, since absolute normalizations at these redshifts carry mass-to-light and pressure-
support systematics of the same order as the effect; the clean confrontation is the survey-internal
ratio a0(zhigh)/a0(zlow) against E(zhigh)/E(zlow), a comparison with no free parameter that the
prediction must pass.

Saturated-phase falsifiers.
Three tests bind the saturated phase: a full Einstein–Boltzmann
implementation of the constrained committed fluid must reproduce the measured tempera-

ture and polarization spectra with the abundance fixed at 1/ϵ; the recruitment derivation
must survive scrutiny of its two named conditions, the pinned reading and the per-defect
bookkeeping; and departures from the ΛCDM form must be confined to accelerations above
gc ≃5.2 × 10−12 m s−2, so canonical behavior failing below that boundary, or non-canonical
behavior appearing in the linear regime, would falsify the domain assignment. The release law
adds a fourth: a surviving committed component on cluster infall streams terminating at the
splashback surface; its robust absence, or a halo-like committed component persisting inside
collapsed systems, would falsify the caustic release mechanism.

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.

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

The failure modes are not all equally severe, and it helps to order them by how much each would
bring down.

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

Kills the static weak-field closure.
A persistent gravitational slip where the scalar stress
predicts none; an RAR transition shape that systematically departs from the derived bosonic
occupancy law in systems well described by the static branch; a source projection contradicted
by the data; or a weak-field UV coefficient chain that cannot be reconciled with an independently
validated microscopic derivation — any of these breaks the central completed result while leaving
the ontology in principle intact.

Kills an extension only.
If the cosmological trace-coupled homogeneous mode cannot survive
a full Boltzmann likelihood confrontation, the cosmology sector fails while the static weak-field
branch stands.
If the saturated phase fails any of its commitments — the dark-to-baryonic
abundance departing from the capacity ceiling, linear-regime observables departing from the
inherited form below the release threshold, or no committed component surviving on cluster infall
streams out to the splashback surface — the committed-dust reading fails in the same contained
way. If the constrained-capacity branch cannot retain the tested Schwarzschild exterior, standard
absorbing-boundary ringdown, or the required horizon thermodynamics once its boundary action
is derived, the strong-field completion fails. None of these touches the weak-field core.

Requires modification, not death.
Some results could be wrong without bringing down
a sector: an adjustment to the finite-loop self-energy from a fuller graph-level computation, a
revision of the strong-field boundary microphysics, or a failure of the charged-lepton ladder,
which is a cross-check on the mass–entropy ontology rather than part of the closed gravitational
chain.

One scale-setting commitment cuts across these tiers and deserves its own line. The percent-
level G∗rests on the seven-fold additivity, which rests in turn on the electron being the lightest
one-bit charged defect, since inter-channel correlation in its dressing would raise the defect
mass. If that identification or the mass–entropy ontology failed, the scale-setting derivation of
Appendix H would not go through.

26. Comparison with Other Approaches

Because the framework reattributes the dark sector (the galactic excess to medium response, the
cosmological abundance to a saturated phase of the same medium) and partially reorganizes the
usual dark-energy story, the sections below set out 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 [17, 47, 19] 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 en-
tropic [20, 46, 21, 22], but they are usually formulated at the level of thermodynamic rea-
soning or horizon-inspired force laws. The present framework is narrower and more explicit:
finite tetrahedral boundary counting, admissibility closure, edge coupling, finite renormaliza-
tion, 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 [18] 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
continuum 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

attractive if the branch survives confrontation with data, and immediately vulnerable if future
observations demand persistent slip or extra weak-field structure.

26.5 Relative to AeST

The closest modern comparator is the AeST theory of Skordis and Złośnik [23], which reproduces
MOND-scale galaxy phenomenology and fits the CMB power spectrum using a vector field and
two scalar degrees of freedom. AeST establishes that galaxy-scale modified dynamics and a
viable linear cosmology can coexist in a single relativistic theory, and any framework in this
space is measured against it. The present construction differs in starting point and in risk profile:
the acceleration scale, the interpolation argument, and the source projections descend from a
finite UV counting problem with coefficients fixed in advance, so nothing can be adjusted where
AeST retains functional freedom, and the cosmological sector is correspondingly less developed
— the full Boltzmann treatment AeST has passed is named open work here (Section 24.3).
The two programs fail differently: AeST through accumulating parameter pressure, the present
framework through any single fixed coefficient meeting a contrary measurement.

26.6 Relative to scalar-tensor gravity

The action of Section 10 can be mistaken for a Brans–Dicke-type scalar-tensor theory [44], since
it carries a metric, a scalar, and a coupling between them. The difference is ontological, and it
removes the usual scalar-tensor freedom. In a scalar-tensor model the scalar is an independent
field added to general relativity, and its kinetic and coupling functions are free to be chosen or
fit to data. Here the metric and the scalar are two continuum expressions of one finite-capacity
substrate: the scalar is not added to gravity but the same medium read in a different variable,
and its stiffness, source coupling, and bridge normalization are fixed by the UV counting chain
rather than left as functions. The resemblance is at the level of the written action; the content
is that the coefficients are not adjustable.

26.7 Relative to quantum-mechanical interpretations

Because Many-Pasts occupies the role of an interpretation of quantum mechanics (Sections 3.3,
21), it should be placed against the standard options. Unlike many-worlds, it posits no forward
branching into co-real macroscopic outcomes; the multiplicity is over admissible pasts of a single
realized present, not over futures. Unlike collapse theories, it adds no physical collapse event
and no modification of unitary evolution. Unlike hidden-variable accounts, its probabilities are
not ignorance over pre-existing classical values. It is closest in spirit to consistent-histories and
decoherence-based reasoning, but it differs in carrying an explicit weight e−D(H,P) over histories
and in using that weight to do physical work outside the foundations — the memoryless dressing
that sets the substrate length, and the orientation of cosmological decommitment. It does not
claim to out-derive the Born rule; it recovers Born statistics and no-signaling as consistency
conditions on the weight (Section 21).

26.8 Relative to CDT, spin foams, and group field theory

The construction uses discrete tetrahedral boundary data, and the j = 3/2 tetrahedron coincides
with the quantum tetrahedron of simplicial spin networks [50, 51, 52], so it sits near the causal-
dynamical-triangulation, spin-foam, and group-field-theory programs. The way it uses those
structures differs on three points. It does not sum over arbitrary triangulations or evaluate a
spin-foam vertex amplitude as the primary weighting; it solves a finite local boundary-counting
problem and weights configurations by admissibility closure, routing the result to a capacity
entropy rather than to area and volume operators. Matter is not added on top of the geometry
but is a fermionic capacity defect of the same substrate. And the history weighting is supplied

by Many-Pasts rather than by a path integral over geometries. The GFT/condensate picture
enters only as a compatibility check on the continuum kinetic term (Section 22), not as the
engine of the derivation. The relation is therefore one of shared combinatorial vocabulary with
a different selection principle and a different target.

27. Conclusion

One physical picture runs through the paper. The vacuum is a finite medium of entanglement
capacity; matter is that capacity tied up in stable, localized defects; a particle’s mass measures
the entanglement its defect commits; and gravity is the capacity strain the surrounding medium
carries once that commitment is made. General relativity is not assumed underneath the picture
but recovered as its low-energy geometry, and the extra gravity seen around galaxies, usually read
as dark matter, is the long-range reach of the same medium rather than a separate substance.

The central result is that this picture is not free to be adjusted. Once the finite-capacity
substrate and the three postulates are granted, the static weak-field sector follows from a finite
counting problem with nothing left to tune. The minimal tetrahedral ensemble fixes the sharing
entropy; the electron anchor fixes the substrate length; the edge kernel, loop dressing, continuum
matching, and Green-matched source map fix the coefficients of the scalar EFT; and the weak-
field bridge turns the capacity deficit into the gravitational potential. The rest of the paper rests
on that chain, the only part offered as closed.

Several quantities then fall out of the one chain without separate tuning. Newtonian gravity
is its point-source limit. The galactic acceleration scale is the horizon thermal scale reduced
by the sharing factor, and the radial-acceleration relation follows from it with no per-galaxy
fitting.
Lensing and dynamics share a single potential at leading order, so galactic support
does not come at the cost of a lensing inconsistency. The same substrate length implies a value
for Newton’s constant within about one percent of the measured one; we read this as a check
that a single length propagates correctly through the whole chain, not as a second independent
measurement, and Appendix K records its provenance. The charged-lepton mass ratios come
out to better than a percent from the same shell algebra, as a cross-check on the mass–entropy
ontology rather than as part of the gravitational chain.

The medium reaches further, into territory the paper holds more tentatively. Transport gives
the field a finite propagation speed and the lag that clusters and mergers need. The cluster sector
reads the lensing anomalies as a phase-dependent projection of an already-fixed coefficient. The
cosmological mode pushes the sound horizon in the direction the Hubble tension wants. In the
saturated early universe the medium becomes a conserved committed density that gravitates
as pressureless dust, with an abundance that is counted rather than fitted and lands on the
measured dark-to-baryonic ratio. The strong-field branch ends the continuum where capacity
runs out, recovering the Schwarzschild and Kerr exteriors and the horizon thermodynamics.
These are extensions and frontier completions, not closed results, and the closure table marks
the distinction.

Many-Pasts carries more weight here than an interpretation of quantum mechanics usually
does. It is the claim that the present state of the medium is supported by many admissible
microscopic pasts, weighted by how well each leads to the present. In the operational branch
used throughout, it changes nothing a laboratory measures: Born-rule statistics and no-signaling
both hold, and it supplies the account of the arrow of time. But the same weighting also makes
the electron’s dressing memoryless, and that memorylessness is one of the two conditions that
fix the substrate length. The postulate therefore does real work in the scale that sets gravity,
not only in the foundations of quantum theory.

The remaining work is specific and limited. The finite-loop self-energy should be confirmed
by an independent graph-level calculation. The cosmological sector should be carried through
a full Boltzmann likelihood. The cluster lift should have its coherence-growth profile derived
and its lensing-map test performed. The strong-field boundary should have its non-universal
spectroscopy worked out. And the substrate length’s recurrence derivation, which rested on
identifying one phase cycle per dressing recurrence, is now discharged in Appendix H into a
recurrence rate, a unit committed charge per pass, and a condensate phase relation; the residual
that remains is the sharper Fork A identification — that the winding phase is the condensate’s
— together with the single integer fixing the electron as the unit-winding defect, both exposed
to the charged-lepton ladder. The paper states these plainly, because its value, if it has one, lies
in making them sharp enough to settle.

The case for the picture is its economy: one finite-capacity medium, counted once in the
ultraviolet, fixes the coefficients in advance rather than fitting them one at a time. That same
economy is its exposure — with nothing left to adjust in the closed sector, one clean contrary
measurement would settle it. The aim of the paper has been to make that confrontation possible.

Appendix A: Symbol Dictionary and Canonical Conventions

Appendix A gathers the conventions used throughout the technical material that follows, fixing
the units, field definitions, and couplings in one place before the denser calculations begin.

Plain-language terms.
Several physical words recur throughout the paper and are collected
here in plain form before the symbols:

• Capacity — the entanglement support locally available in the vacuum medium.

• Defect — a stable, localized commitment of that capacity; coarse-grained, a particle.

• Deficit — capacity no longer freely available near a defect, δS = S∞−Sent.

• Capacity strain — the extended deficit profile whose fractional value gives the weak-field
potential and whose gradient gives the gravitational field.

• Committed capacity — capacity locked into transferring on behalf of a defect; the conserved
carrier of the saturated phase (Section 18.5).

• Dressing — the ground-state cloud an elementary defect builds by resolving the boundary
sectors; its coherent support fixes the defect’s length scale.

• Saturation — the regime in which the capacity bath has no slack left, with the available
transfer channel at its ceiling.

• Pinned reading — the statement that the saturated phase holds exactly at that ceiling
and stays there.

• Admissibility — the weighting that favors boundary configurations close to a regular,
isotropic local cell.

• Closure — the condition that the four oriented face data sum to zero, so the cell closes
into a regular volume; K2 measures the failure of closure.

• Many-Pasts — the postulate that the present is supported by a probability-weighted en-
semble of admissible microscopic pasts, with weight e−D(H,P).

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 make those
two descriptions genuinely comparable.

A.2 Core scalar variables

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

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 B: UV Boundary Ensemble and Admissibility Closure

Appendix B records the finite ultraviolet counting problem in its explicit form. It supports Sec-
tions 5–6 by showing that the seven-state tetrahedral ensemble and the admissibility weighting
are finite and auditable rather than phenomenological dials: the theory begins from a discrete
boundary ensemble and ends with a unique admissibility-closed entropy, not with an uncon-
strained 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 that the counting 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.

This also settles the status of j0 as a parameter: it has none to tune. Within the minimal
construction j0 = 3/2 is forced as the smallest fermionic label meeting injectivity. A larger
j0 does not describe a fluctuation inside the same cell; it specifies a different, larger boundary
ensemble, with a different state count Ωtet and a different closure spectrum. The minimal theory
therefore fixes j0 rather than leaving it open, and the seven-state sector is not one option among
a family but the first that closes.

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 admissibility precision η is fixed by stationary normalized closure evidence. The closure
constraint is the vanishing of the three-component oriented-face sum c(b) = P
i ˆnimi ∈R3,
of which K2 is the quadratic invariant; marginalizing the three continuous closure components
against the family at precision η contributes a determinant factor η3/2, so the normalized closure
evidence is η3/2Z(η) and its logarithm is

F(η) = ln Z(η) + 3

2 ln η.

With ∂η ln Z = −⟨K2⟩η, the stationary condition F′(η) = 0 is the closure relation

⟨K2⟩η = 3

2η,

so the factor 3/2 is the determinant weight of the three independent closure components, not an
equipartition rule imported onto the bounded discrete spectrum; the discreteness, multiplicities,
and positive floor of the spectrum remain inside the exact finite sum Z(η). The second derivative
is F′′(η) = Varη(K2) −3/(2η2), and on the exact spectrum F′ has a single interior zero,

η∗= 0.0298668443935,

at which Varη∗(K2) = 15.69 is dwarfed by 3/(2η2
∗) = 1681.6, giving F′′(η∗) = −1665.9 < 0: η∗
is the unique local maximum of the normalized closure evidence. Because the parity-symmetric
ensemble is finite, both Z(η) and ⟨K2⟩η are exact finite sums over the spectrum. The distinct
closure-defect values and their degeneracies are

K2
122

3
134

3
142

3
146

3
152

3
154

3
mult
96
96
96
288
192
144

K2
158

3
54
164

3
166

3
170

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
interior maximum of an exact finite-spectrum functional, 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

Ccl := η∗⟨K2⟩η∗= 3

2,
C−1
cl
= 2

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.

The admissibility parameter stops being free here. 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 C: Edge Kernel, Finite Renormalization, and Contin-
uum Matching

Appendix C carries the local boundary ensemble of Appendix B into edge transport, loop dress-
ing, and 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

z −1 = 2η∗

Jtree
eff
= Jbare

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

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

a,i
(Qa −Qa+L∗ˆni)2,

2

where the sum runs over all sites a and all four outgoing nearest-neighbor directions ˆni (each
nearest-neighbor edge counted twice). The microscopic four-cell is

∆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

iso = ln7

∆Sdef := −ln
DZdef

E

6.

Zvac

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

i
da
i db
i = Gtet(0)

Gab
loc = Gtet(0)1

X

3
δab.

4

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

so
G⊥= P ab
⊥Gab
loc = 2

3Gtet(0).

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

π
= ln(7/6)

δQdef = ∆Sdef

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.

The three cancellations, made explicit.
The passage from the dimensionless lattice in-
sertion sdef/J(ren)
eff
= ln(7/6)/(πGtet(0)) to the source theorem turns on three reductions, none
of which leaves a free constant. (i) The factor π cancels against the field normalization. The
insertion is written for the occupancy field, where one defect shifts δQdef = ln(7/6)/π. Con-
verting to the entropy field through S = πQocc multiplies by π, so the S-field shift is δSdef =
π δQdef = ln(7/6) and the explicit π does not survive into the source theorem. (ii) ln(7/6) is
carried by the entropy density, not the coefficient. A defect contributes entropy ln(7/6) at the
isotropic benchmark, so a defect number density sources an entropy density proportional to it;
writing that source as σdef = ρ/κm(L∗) folds the benchmark and the mass–entropy unit into
the single density σdef. The geometric coefficient 3L∗/(4Gtet(0)) that multiplies it is fixed by
the tetrahedral 4/3 projection and the cell length alone and contains no ln(7/6). (iii) J(ren)
eff
cancels in the ratio. The source amplitude sdef and the stiffness γ each carry one power of the
renormalized edge coupling, so it cancels in the source-to-stiffness ratio κ/γ, leaving the lattice-
geometric constant 3L∗/(4Gtet(0)κm(L∗)). The renormalized loop coupling therefore drops out
of the observable normalization.

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 non-
degeneracy 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. Two features of this
number matter for the source theorem in C.5. First, aUV is finite and positive, confirming that
the closed branch at η∗has a non-degenerate zero-mode response rather than a critical singularity
that would invalidate the linear Green-function matching used to derive κ/γ. Second, the same
local susceptibility controls the stability of the closure mode that is being propagated into the
shell and loop calculations, so the residual fractional discrepancy between the local benchmark
and σ∗in C.2 is genuinely loop-sector rather than a local-closure failure.

The actual closure of κ/γ in the canonical branch is the Green-matched theorem in Ap-
pendix C.5. The role of aUV here is restricted to confirming non-degeneracy of the local mode
being matched.

C.7 UV-to-IR payoff

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

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 substrate-induced scale

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 tests that this normalization propagates the substrate scale
coherently; it is not a disjoint-input second derivation of G, since the matched route carries L∗
within it.

The coefficients this appendix supplies to the main weak-field chain are Jbare, Jtree
eff , Σret,
J(ren)
eff
, γ, and the Green-matched source projection κ/γ. 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 collects the weak-field derivations that are central but too dense for the main line:
the bridge law, Newtonian recovery, the electron anchor, and the EFT consistency checks.

D.1 Bridge-law 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,
and this subsection makes the link explicit. The task is to turn a dimensionless count — the
effective number of dressing configurations the electron’s seven channels resolve — into a physical
length ratio λe/L∗, and hence into the induced gravitational scale G∗. The argument has three
parts: a recurrence derivation that fixes the linear, first-power map from the seven-channel
sharing entropy to the length; a demonstration that the allowed local single-label dynamics
cannot reach the full ensemble, so the memoryless refresh of Postulate III is required; and
an explicit transfer-operator realization that makes the count concrete. One identification sits
beneath the map — one phase cycle per dressing recurrence — which Appendix H.6 gives an
operator form and discharges into a recurrence rate, a unit charge per pass, and a condensate
phase relation, conditional on the Fork A identification settled in H.7; the linear power itself is
then cross-checked three independent ways.

The recurrence derivation of the dictionary.
The support-to-length identification follows
from the recurrence structure of the dressing. By Postulate III the dressing is memoryless in
substrate time: each tick, each of the seven channels draws independently from the admissibility
ensemble. By the asymptotic-equipartition property — used here only in its elementary form,
that a distribution with entropy g has an effective typical support of size eg, so a specified typical
configuration appears with probability e−g per independent draw — a single channel realizes its
typical dressing configuration with probability e−gshare,eff per draw, and the seven mutually inde-
pendent channels (Appendix H) realize the complete dressing jointly with probability e−7gshare,eff.
The defect therefore re-completes its dressing once per e7gshare,eff substrate ticks. Identifying the
defect’s internal phase clock, τe = ℏ/mec2, with this recurrence period—one phase cycle per
complete re-dressing—and applying the transverse export factor of Appendix C.5 gives

τe
τ∗
= 2

3 e7gshare,eff,
τ∗= L∗

c ,

the length-setting relation below in temporal form. The linearity of the support exponent is
forced rather than chosen: probabilities of independent rare events multiply, so their exponents
add, and the waiting time for the joint event is the exponential of the sum. The three-way

bracketing of the exponent at unity within 10−3, summarized in Section 13 and tested again
in Appendix I.1, is the quantitative check of this derivation. One identification sits beneath
it—that the defect’s phase advances one cycle per dressing recurrence—which Appendix H.6
decomposes into the recurrence rate, a unit committed charge per pass, and the condensate
phase relation, relocating the residual to the Fork A identification (the winding phase is the
condensate’s) and the single integer the lepton ladder tests.

The transfer-support dual.
The same relation follows from a spatial reading. The entropy
of the one-sector transfer distribution fixes an effective support size N (1)
eff = egshare,eff, and on
the lightest charged branch the seven sectors carry no mutual information (Appendix H), so the
joint support is N (7)
eff = e7gshare,eff. A coherent transfer orbit over N cells of step length a has first
phase mode of wavelength λ1 = aN; with the transverse export fixing the step at a = 2

3L∗(Ap-
pendix C.5), the first mode over the typical support is λ = 2

3L∗e7gshare,eff, the recurrence relation
above in spatial form. The two readings are reciprocal faces of one asymptotic-equipartition
fact—a designated configuration within a typical set of size eS is reached with probability e−S

per draw, and traversed at unit step in eS cells—and they share a single residual identification:
the temporal form assumes one phase cycle per dressing recurrence, the spatial form that the
Compton mode is the first phase mode over the typical support. These are the same statement
in two grammars, so the branch carries a single residual identification rather than two — the
one Appendix H.6 discharges into the recurrence rate, a unit committed charge per pass, and
the condensate phase relation, conditional on Fork A.

Locality cannot replace the refresh kernel.
The ensemble entering both readings is
reached only by the memoryless refresh of Postulate III, and the requirement is topological.
Under any local dynamics—single-label shifts preserving admissibility—the state graph of one
parity copy fractures into 4! = 24 disconnected ordering sectors of
7
4


= 35 states each, because
a unit shift can never reorder two slots without passing through the collision injectivity forbids.
Slot ordering is a frozen charge of local motion: no local transfer operator has pη∗as its equilib-
rium, the strength-weighted walk measure saturates at entropy 7.374 against gshare,eff = 7.4198,
and the dominant transfer mode localizes within a single sector. The full count is available only
to a kernel that redraws from the ensemble, which is the content of Postulate III; the postulate
is therefore load-bearing in the scale chain by necessity.

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

Ccl := η∗⟨K2⟩η∗= 3

2,
C−1
cl
= 2

3.

The faithful sector-resolution principle is


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



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 metrolog-
ical. The residual carries no statistical interpretation within the chain: gshare,eff is an exact
weighted evaluation, G∗∝e−14gshare,eff, and matching the CODATA central value would require
δg/g ≃7.5 × 10−4, a shift the 1680-state combinatorics does not admit. The residual therefore
measures a physical correction omitted at this order. The charged-lepton ladder residuals of
−0.51% and −0.65% (Section 13, Appendix I) cannot share an algebraic origin with it, because
the ladder contains no factor of gshare,eff; the framework carries two independent systematic
residuals, each assigned to dressing physics beyond leading order. All three residuals are neg-
ative, a regularity that constrains candidate corrections without establishing a common origin.
A candidate correction enters the closure ledger only by predicting the sign and magnitude of
the residual it addresses in advance of any fit. That standard is met at the level of sign by
the framework’s most natural correction channel. Once the dressing is read as a recurrence of
independent draws (Appendix D.4), the dressing exponent fluctuates about its ensemble mean,
and convexity fixes the direction of every induced shift: for fluctuating exponent h, ⟨eh⟩≥e⟨h⟩

and ⟨e−h⟩≥e−⟨h⟩, so the combinatorial mass ratios shift upward and, through the lengthening
of L∗, the induced G∗shifts upward as well. The channel admits no opposite-signed member,
and all three measured values exceed their combinatorial predictions: three deficits, one forced
sign, three matches. The magnitudes are not yet derived; the branch variances are computable
from the ensemble statistics with no free parameters, and the resulting ratio predictions δτ/δµ
and δln G/δµ are the most direct test this sector can state in advance of measurement.

The factor 3/2 is fixed structurally by the tetrahedral transverse-export geometry. The tetra-
hedral identity gives the transverse export fraction

4
X

i=1
(ˆni · ˆu)2 = 1

3,
f⊥= 1 −1

3 = 2

1
4

3,

so f−1
⊥
= 3/2. The admissibility fixed point returns the same value,

C−1
cl
=

η∗⟨K2⟩η∗

−1 = 2

but it is built in. The condition defining η∗is ⟨K2⟩η = 3/(2η), i.e. η⟨K2⟩= 3/2, so Ccl = 3/2
holds by construction — the fixed point returns the value the condition put in. The 3/2 is fixed
once, by the transverse-export geometry, and the closure saturation only reproduces it. 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 admissibility 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%
fixed by transverse tetrahedral ge-
ometry; mirrored by the closure-
saturation normalization
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

ln 2.

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 electron does not simultaneously occupy seven
mutually exclusive face labels in one tetrahedron; the labels are sector channels in the dressing

history of the one-bit defect. The appropriate support object is therefore a history space, with
one admissibility-closed sector layer for each m = −3, . . . , 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 .

The justification of this relation is given in Appendix H. An explicit dressing Hamiltonian (Ap-
pendix H.2) makes the lightest charged fermionic defect the one-pass ground state that resolves
each of the seven sectors once. The refresh kernel Pη∗used above is the memoryless dressing
of Postulate III: with the substrate relaxed long before readout, each pass is an independent
draw from the admissibility ensemble, so every channel carries the full gshare,eff. The seven-
fold additivity then needs the channels to be mutually independent, and they are, for a reason
internal to the anchor: inter-channel correlation would contract the defect and raise its mass,
and the electron is the lightest one-bit charged defect, so its dressing is the product and corre-
lated dressings describe heavier excitations (Appendix H.5). With the 2/3 factor the tetrahedral
transverse projection of Appendix C.5, the faithful sector-resolution principle is a consequence
of the mass–entropy ontology and the electron anchor, and nothing in the chain remains free.

The status of the relation should be named precisely. The effective dimension eH is the per-
plexity of the coherent dressing — the effective number of mutually resolvable support positions

in the defect’s history, not a count of cells in a spatial volume. Reading it as a physical length
ratio is the recurrence identification derived above: the dressing re-completes once per e7gshare,eff
ticks, and one phase cycle per recurrence converts that waiting time into the Compton scale, so
the linear, first-power map is the arithmetic of joint probability rather than a chosen dictionary.
The map is cross-validated by the induced gravitational scale, by the charged-lepton ladder,
and by the mass-minimization argument above, which together bracket the support exponent
at unity within 10−3. One identification remains beneath the derivation — one phase cycle per
dressing recurrence, equivalently that the Compton mode is the first phase mode over the typical
support — and a one-defect transfer operator T whose coherence length obeys λ = CclL∗eH[pT ]

is the object whose spectrum would certify it; exhibiting that operator is the branch’s remaining
microscopic completion.

This is the microscopic content of the length formula above: the elementary one-bit fermionic
defect is supported over one complete transverse-exported dressing block. The matched weak-
field gravitational constant then follows from the gauge-invariant bridge

G = 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 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
same scalar medium propagating causally, relaxing toward its static limit, and supporting a cos-
mological background mode without losing contact with the weak-field structure already derived.
The status differs across these pieces: the transport relation is closed at the preferred-branch
level (D/τ0 = c2 with τ −1
0
= H0); the homogeneous Hubble-tension mechanism is directionally
supported but not Boltzmann-closed; and the full perturbation likelihood treatment remains
open.

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.

The transport relation and preferred branch are closed; the cosmological sector remains struc-
turally 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.

The static exterior can therefore 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.

Geometric identification of the capacity variable.
The static rule N 2 = q together with
the bulk reduction to vacuum Einstein on Mq forces a geometric reading of the substrate variable.
In a spherically symmetric foliation with areal radius R, the normalization-independent gradient
invariant on the orbit space is
q = hµν∂µR ∂νR = |∇R|2,

which in Schwarzschild gives

q = 1 −2GM

c2r
= N2,

matching the substrate definition q = Sent/S∞on the exterior. In spherical dynamical collapse
with Misner–Sharp mass MMS(R, t),

q(R, t) = 1 −2GMMS(R, t)

c2R
.

The substrate variable q and the geometric areal-radius gradient invariant therefore agree as a
theorem of the framework’s structural commitments, not as an additional axiom. The marginal-
trapped-surface condition θ+θ−= 0 on a closed two-surface coincides with q = 0 independently of
the null normalization used to define θ±: q > 0 is the untrapped exterior, q = 0 is the marginal /
apparent horizon, and q < 0 would be the classical trapped interior. The capacity domain Mq =
{q > 0} is the untrapped exterior, and the saturated boundary ∂Mq = {q = 0} is the apparent-
horizon two-surface. Standard apparent-horizon formation theorems for gravitational collapse
therefore produce the q = 0 saturation surface inside the framework, without an additional
substrate-transport postulate; the bounded transport law of Appendix F.7 is a substrate-side
consistency check on this geometric formation rather than its primary mechanism.

Rotating and charged stationary exteriors.
The same vacuum-domain reduction extends
to the rotating and charged cases. On Mq the bulk equations reduce to vacuum Einstein, or
to Einstein–Maxwell when a conserved gauge sector hosted by the substrate (Appendix I.2) is
present. The standard no-hair theorems then identify the stationary asymptotically flat exterior
branches as Kerr, Reissner–Nordström, or Kerr–Newman, and the saturation surface q = 0
coincides with the outer Killing horizon. The Bekenstein–Hawking entropy

S = kBA

4L2∗
retains its form with A the appropriate horizon area, and the Hawking temperature follows from
the surface gravity as usual,

TH =
ℏκsg
2πkBc.

The inner Cauchy horizons of Reissner–Nordström and Kerr are not physical substrate interiors;
they are analytic continuations into the q < 0 region the substrate does not enter.

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 substrate inputs already in the framework, rather
than from a stipulated channel-counting rule. The derivation uses two independent ingredients,
each of which is independently closed within the bulk EFT before any horizon machinery is
introduced.

Per-channel cut entropy from fermionic face exclusion.
At a saturation surface q = 0,
the physical continuum domain terminates at ∂Mq.
A tetrahedral cell on the exterior side
whose outward face would, in the unsaturated bulk, be paired with a neighboring cell across
that face instead has an unfilled pairing slot, because the would-be partner lies outside Mq.
By Postulate II the elementary face slot is fermionic and admits only the occupied (paired)
state or the excluded (unpaired) state. A horizon cut face is therefore an instance of the same
elementary face-exclusion defect that anchors the one-bit fermionic sector in the bulk.
The
seven-state jeff = 3 channel belongs to the paired bulk link j0 ⊗j0 →jeff, not to the cut
face itself: with the partner cell absent, the paired representation is never formed. The seven-
state structure enters the boundary count through the bulk graph response and hence through
the channel density below, not through the per-channel entropy. The entropy carried by an
elementary cut defect is consequently the same primitive fermionic increment that anchors the
electron at κm(λe) = me/ ln 2,
∆Sf = ln 2.

Channel density from the transverse bulk graph response.
Because the local graph
Green tensor is isotropic,

Gab
loc = Gtet(0)

3
δab,

a codimension-one horizon cut with local normal ˆn exports only the transverse two-plane com-
ponent,

G⊥= (δab −ˆnaˆnb)Gab
loc = 2

3Gtet(0).

The horizon channel density is therefore not a new boundary response coefficient; it is the
transverse projection of the same bulk Green response already fixed in Appendix C. The number
of active channels per outward angular direction is G⊥/ ln 2, and integrating over the horizon
two-surface (the spherical angular measure in Schwarzschild, the smooth axisymmetric horizon
for Kerr, with isotropic transverse response in either case) gives

ln 2 = 8πGtet(0)

nhor = 4π G⊥

3 ln 2
.

Closure as an exact identity.
Using the cell-normalized capacity baseline from Appendix C,

Scell
∞=
3 ln 2
32πGtet(0),

the product of the two substrate inputs is

nhorScell
∞= 1

4,

an exact identity in which the Joyce diamond-lattice constant Gtet(0) and the fermionic incre-
ment ln 2 cancel between the two factors. A horizon of area A therefore carries the dimensionless
entropy
Shor

kB
=
A
4L2∗
,

and the Bekenstein–Hawking coefficient is recovered after the gravitational scale is matched so
that LP (G∗) = L∗. The area coefficient is closed within the constrained-capacity EFT under
Postulate II and the transverse bulk graph response, both of which are independent of the horizon
construction. The remaining strong-field task concerns nonuniversal boundary spectroscopy —
the explicit microscopic Hamiltonian behind the relaxation spectrum, stretched-layer corrections,
and transient response — rather than the universal area coefficient.

F.6 Absorption, ringdown, and echoes

Exterior wave propagation is unchanged.
With q = 0 identified as the marginal-capacity /
apparent-horizon surface (Appendix F.4), the standard near-horizon tortoise coordinate r∗∼
rh ln q pushes the surface to r∗→−∞, and the wave equation reduces to (∂2
t −∂2
r∗)ψ ≃0 near
the boundary. Horizon regularity at the future horizon then selects the purely ingoing mode

ψ ∼e−iω(t+r∗),

so the leading reflectivity is
R = 0.

Perturbations outside the capacity-exhaustion surface obey the usual Schwarzschild Regge–
Wheeler/Zerilli scattering problem [11, 12], with the absorbing boundary fixed by regularity
rather than stipulated. The greybody factors, absorption cross sections, and quasinormal ring-
down agree with the standard Schwarzschild exterior calculation.

Echoes are therefore correction-level rather than generic. They arise only if microscopic bound-
ary channels carry finite UV reflectivity or if the effective reflection surface is displaced outward
to a stretched layer
q = ϵ > 0,

in which case the region between the exterior potential barrier and the partially reflecting layer
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 geometric identification of Appendix F.4 supplies the primary horizon-formation story:
standard apparent-horizon formation in gravitational collapse produces the q = 0 saturation
surface as the first marginally outer trapped surface, with no additional substrate-transport
postulate required. What remains is a substrate-side dynamical consistency check — writing a
bounded causal transport law for q and verifying that its evolution agrees with the geometric
formation, while bounding q in its physical range [0, 1] throughout the collapse. 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, at leading post-Newtonian order,

γPPN = βPPN = 1,

with scalar-induced corrections to Φ−Ψ appearing only at quadratic weak-field order, schemat-
ically O(Φ2/c4), and 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 geometric identification of q with the marginal-
trapped-surface function fixes horizon location and inherits apparent-horizon formation from
standard GR collapse. The area coefficient closes under Postulate II and the transverse bulk

graph response, and the leading reflectivity vanishes by horizon regularity. Stationary rotating
and charged exteriors reduce to Kerr, Reissner–Nordström, and Kerr–Newman through the same
vacuum-domain bulk reduction. What remains open is nonuniversal boundary spectroscopy:
the explicit graph-derived microscopic boundary Hamiltonian behind the relaxation spectrum,
stretched-layer corrections, and the substrate-side transport check on the dynamical evolution
of q during collapse.

Appendix G: Many-Pasts, Operational Closure, Branch Realiza-
tion, and the Arrow of Time

Appendix G is the technical support layer for the Many-Pasts postulate (Sections 3.3, 21). It
defines the three objects the postulate weights — a history, the present, and the distance be-
tween them — then derives the operational consequences (Born recovery, no-signaling), explains
branch realization and the arrow of time, connects the same weight to the scale-setting branch
through memoryless dressing, and locates the proposal among the standard interpretations.
The operational core is conservative: it reproduces laboratory quantum mechanics exactly. The
ambition is in the ontology built on that core.

G.1 What is a history of the entanglement network?

A history H is a coarse-grained trajectory of the substrate’s microstate: a sequence of admissibility-
closed configurations of the entanglement network, one per substrate tick, each obtained from
the previous by an allowed update. “Admissible” carries the same meaning as in the UV ensem-
ble of Appendix B — a configuration close to a regular, isotropic local cell. A history is not a
path through an external spacetime, since there is none at this level; it is a path through the
network’s own configuration space. The set of histories compatible with a fixed present is large,
and the postulate is a rule for weighting them.

G.2 What is the present coarse configuration P?

The present P is the macroscopic, record-bearing coarse-graining of the network now.
It is
the quantity held fixed: every admissible history must terminate in a microstate consistent
with P. Operationally P is represented by a projector ΠP onto the subspace of microstates
carrying the present records. The coarse-graining is the ordinary one of statistical mechanics
— macroscopically distinguishable configurations — not a new structure introduced for this
purpose.

G.3 What is the distance D(H, P)?

The history-space distance is

D(H, P) = −ln Tr

ΠP ρH→now


,

where ρH→now is the state obtained by evolving the history H forward to the present. It is the
negative logarithm of the overlap between the present projector and the state that history would
produce. A history leading naturally to the present records sits close, with small D and large
weight; a history that would require an improbable conspiracy to reproduce them sits far, with
large D and suppressed weight. The weight P(H|P) ∝e−D(H,P) is then exactly that overlap,
normalized over the admissible histories.

G.4 Born recovery in the projective laboratory limit

In the ordinary projective laboratory limit the evolved state is the prepared state and the present
projector is the measured outcome, so the distance collapses to a single overlap,

e−D(H,P) = Tr

ΠP ρ


,

which is the Born probability of that outcome. This is the branch α = 1 of the generalized
weighting family. The recovery is a consistency condition rather than a fresh derivation: requir-
ing that the weight reproduce exact Born statistics selects α = 1, and the substantive content is
that the family contains a branch for which this holds at the same time as no-signaling. Nothing
in the laboratory is changed; the operational theory is standard quantum mechanics.

G.5 No-signaling

The companion condition forbids any operational bias channel whose statistics could depend on
a spacelike-separated setting. Imposing it selects β = 0. With β = 0 the weight depends only on
the local present records, so a choice made at one location does not alter the marginal statistics
at another, and no-signaling holds exactly in the operational branch. Born recovery and no-
signaling are therefore the two consistency conditions α = 1, β = 0; the postulate’s nontrivial
claim is that they are simultaneously satisfiable, not that either is derived from outside.

G.6 Branch realization

The postulate makes one realized present, not many. There is no forward branching into co-real
macroscopic worlds, and no collapse event selecting among them. What is real is the present
with its records; the multiplicity the weight ranges over is the admissible pasts of that single
present, not parallel futures. “Which outcome occurred” is therefore not a question about which
branch the world fell into but a statement about which present records obtain, with the Born
weight giving their statistics. Branch realization is thereby answered without the ontological
cost of many worlds or the dynamical cost of collapse.

G.7 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 the dominant record-compatible histories are those with entropy increasing toward the present.
The arrow of time appears as a counting-dominance effect among record-compatible histories,
not as a new dynamical coupling or a new law of laboratory probability.
It is conditional
typicality: given the records, overwhelmingly many admissible pasts show entropy rising toward
them.

G.8 Memoryless dressing and the connection to L∗

The same weight does work outside the foundations. Because e−D weights whole histories with
no memory term, the dressing of an elementary defect is memoryless in substrate time: between
one readout and the next the substrate relaxes to a fresh draw from the admissibility ensemble,

so each dressing pass is statistically independent of the last. That independence is exactly the
input the scale-setting recurrence of Appendix D.4 requires: it lets the seven channels each carry
the full sharing entropy gshare,eff and multiply. The mixing-time hierarchy that justifies treating
each pass as a fresh draw is the ∼1022 ratio of the electron Compton time to the substrate tick
(Appendix H.4). Postulate III therefore enters the gravitational chain directly: its memoryless
dressing is load-bearing in the length that sets G∗.

G.9 Relation to many-worlds, collapse, hidden variables, and decoherence

Placed among the standard interpretations, Many-Pasts is a form of history-space realism. Un-
like many-worlds, it posits no co-real forward branches; the only realized macrostate is the
present. Unlike objective-collapse models, it adds no stochastic collapse term and leaves uni-
tary evolution intact. Unlike hidden-variable theories, its probabilities are not ignorance over
pre-existing sharp values. It is nearest to the consistent-histories framework and to decoherence-
based accounts of classicality, but it differs from them in two ways: it carries an explicit normal-
izable weight e−D(H,P) over admissible histories, and it puts that weight to physical use outside
the measurement problem, in the memoryless dressing of G.8. The proposal does not claim to
out-derive the Born rule from more primitive axioms; it recovers Born statistics and no-signaling
as the consistency conditions α = 1 and β = 0 on the weight (G.4, G.5), and offers in addition
a history-space account of branch realization and the arrow of time.

G.10 Familiar quantum examples: double-slit, EPR/Bell, and measurement

In the familiar quantum examples, Many-Pasts changes the ontology rather than the laboratory
calculation. In a double-slit experiment the particle is not treated as having secretly chosen one
classical path while the other was unreal. Before a which-path record exists, the present detection
event is supported by an ensemble of admissible microscopic histories through the apparatus,
and the histories compatible with the final record include the alternatives whose amplitudes
interfere in ordinary quantum mechanics. When a which-path record is made, the admissible
history ensemble is conditioned on that record, the cross terms are removed by decoherence,
and the interference pattern disappears.
The probabilities are the usual Born probabilities;
Many-Pasts supplies the history-space reading of why those are the probabilities being counted.

The same point applies to entangled pairs. In an EPR or Bell experiment the two detector
outcomes are not produced by a signal sent from one wing of the experiment to the other. The
present record is a joint record of the pair and the apparatus, and the admissible microscopic
histories are weighted as histories of that whole entangled system. This gives the standard non-
classical correlations while preserving the local Born marginals, and because the local marginal
statistics are unchanged, no observer can use the correlations to send a signal. The “spooky”
feature is therefore not a superluminal force in spacetime; it is the global conditioning of the
history ensemble on the shared entangled record.

Measurement is read the same way. A measurement does not add a new physical collapse law;
it creates a durable record, and the relevant history ensemble is then the ensemble compatible
with that record. Operationally this is ordinary quantum mechanics. Ontologically, the realized
branch is not selected by a forward-propagating collapse event but by conditioning the present
on its admissible microscopic pasts.

None of this is a new derivation of the quantum phenomena.
In the operational branch
the calculations and the probabilities are exactly those of standard quantum mechanics; what
Many-Pasts adds is the ontology under that branch — a history-space account of what is being
counted and why — with the double slit reading as unrecorded alternatives that remain in the
admissible ensemble until a record conditions it, Bell correlations as the global weighting of

a shared entangled record rather than a faster-than-light signal, and measurement as record-
conditioned history selection rather than new collapse physics.

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

γ ∼Zσnbg

2α2
> 0.

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. The microscopic appendix there-
fore supports the EFT without replacing it: the continuum kinetic term and source channel
can arise from a concrete substrate realization, while a full first-principles derivation of every
inhomogeneous continuum coefficient from the underlying kernel remains to be done.

It does not replace the explicit coefficient derivation given earlier, but it shows that the
ontology and sign choices of the EFT are compatible with a concrete microscopic picture.

The condensate picture addresses the emergence of the continuum geometry. The comple-
mentary microscopic question — the defect dynamics that fixes the faithful sector-resolution
principle of Appendix D.4 — is taken up in the remainder of this appendix, where the principle
is shown to follow from the physically realized admissibility ensemble of Part II together with
the Many-Pasts postulate.

H.2 The dressing Hamiltonian and the one-pass ground state

The faithful sector-resolution relation of Appendix D.4 was written there as a support-scale
identity on the history space Hhist = N3
m=−3 H(m)
B , using the refresh kernel Pη∗(b, b′) = pη∗(b′)
on each sector layer. Its dynamical content is carried by an explicit defect Hamiltonian, which
we now write down so that the choice of that kernel can be derived.

The seven face-label channels of the admissibility-closed ensemble define the orthogonal pro-
jectors Em, m = −3, . . . , 3, of the face algebra A7. By Postulate II the elementary defect is
fermionic, so each channel carries an occupation number nm ∈{0, 1} with fermionic creation
and annihilation operators c†
m, cm and nm = c†
mcm. When channel m is occupied it is dressed

by a cloud state described by a density operator ρm on the boundary ensemble HB. The defect
Hamiltonian is
H =
X

+
X


ε0 nm −nm Fm(ρm)



m<m′
Vmm′(ρm, ρm′)

,

m

|
{z
}
single-channel terms

|
{z
}
inter-channel coupling

with ε0 the bare cost to occupy a channel, Fm the free energy released by dressing channel m
with its cloud, and Vmm′ the residual coupling between the clouds of distinct channels. Two
properties of the ground state of H supply the two ingredients of the sector-resolution relation.

One-pass occupation and the exponent seven.
A channel is occupied in the ground state
whenever its dressing free energy beats the bare cost, Fm(ρm) > ε0. The lightest charged defect
is the configuration that minimizes ε0. Because the seven channels are symmetry-equivalent,
each binding with the same free-energy gain, that configuration is exactly the one for which all
seven bind: any unbound sector is an excitation above the ground state. “Lightest” and “resolves
each of the seven sectors once” are then the same statement: the elementary fermionic defect fills
each face sector exactly once, with fewer sectors leaving unresolved label memory and repeated
sectors describing excited or further-dressed states. This one-pass occupation supplies the factor
seven in 7gshare,eff, the seven being the channel count |M| = 7 fixed in Part II.

Factorized cloud and the additive support.
The dressing dimension multiplies across
channels — so that the seven equal contributions gshare,eff add rather than merge — precisely
when the joint cloud state is a product, ρ = N
m ρm.
Whether the ground state of H has
this product form is controlled by the inter-channel term Vmm′. Beyond the one-pass count,
then, faithful sector resolution turns on the seven-channel cloud being a product of full-entropy
channels: each channel must sample its whole ensemble, and Vmm′ must leave the channels
uncorrelated.

The dressing Hamiltonian thus reduces the sector-resolution principle to that product struc-
ture in its ground state. The next three subsections establish it.

H.3 Slot coupling, layer factorization, and the additivity theorem

Two distinct correlation structures appear in the closure data, and separating them is essential:
conflating them leads to the false conclusion that the cloud cannot factorize.

The closure invariant is a pure pair coupling.
Expanding S2 =
P
i mi

2 = Σ2 +
2 P
i<j mimj in K2(b) = 48 −1

3(S2 −Σ2) cancels the self-terms exactly and leaves the identity

K2(b) = 48 −2

X

i<j
mimj.

3

The admissibility weight e−η∗K2(b) therefore contains only cross-terms between distinct face slots
of a single boundary state; it does not factorize over those four slots. This is the exact origin of
the residual correlation between face slots on the closed ensemble,

I(slot0; slot1) = 0.1545 nats,
I
H(slot) = 0.079,

so the four faces of one tetrahedron are about eight percent correlated, a structural feature of
the closure invariant.

Slot coupling does not obstruct layer factorization.
The factorization the support re-
lation requires is over the seven sector layers b(−3), . . . , b(3) of Hhist, each layer being a full
boundary state drawn from the entire 1680-state ensemble. The slot coupling −2

3mimj lives
inside a single layer’s boundary state: it relates the four faces of that one tetrahedron and never
couples layer m to layer m′. The two structures act on different objects:

Structure
What it couples
Role
slot coupling −2

3mimj
the
four
faces
within
one
boundary state b
the eight-percent mutual infor-
mation; lives inside each H(m)
B
layer coupling Vmm′
the seven sector layers b(m) of
Hhist
controls factorization; acts be-
tween the factors

The substrate’s intrinsic correlation, the most natural candidate obstruction to factorization,
therefore acts at the wrong level to obstruct it: it is internal to a layer, not between layers.

Additivity of the history distance.
The Many-Pasts history weight of Appendix G.1 is a
log-overlap, D(H, P) = −ln Tr(ΠP ρH→now). If the dressing state and the resolution projector
factorize over channels, ρ = N
m ρm and ΠP = N
m Πm, the trace factorizes and the logarithm
converts the product into a sum,

Tr(ΠP ρ) =
Y

m
Tr(Πmρm)
=⇒
D =
X


−ln Tr(Πmρm)

=
X

m
Dm.

m

Additivity of D over the seven channels — and hence the multiplicativity dimeff = Q
m egshare,eff =
e7gshare,eff used in the length relation — is therefore a theorem once the product structure holds.
The implication needed downstream runs in one direction: a product cloud gives additive D and
the full 7gshare,eff. Whether the electron’s dressing is that product is decided by two conditions,
established in the next two subsections.

The ceiling and the two conditions.
Subadditivity bounds the support from above. With
each channel marginal fixed to the admissibility weight, H(Bm) = gshare,eff, the joint entropy of
a single readout obeys

3
X

H

B−3, . . . , B3


≤

m=−3
H(Bm) = 7gshare,eff,

and the deficit
∆=
X

m
H(Bm) −H

B−3, . . . , B3


≥0

measures the total correlation among the seven channels. The full support dimeff = e7gshare,eff
is reached precisely when each channel carries its full entropy gshare,eff and the channels are
mutually independent, ∆= 0. The first is a condition on each layer in substrate time; the
second is a condition on the seven layers at a single readout. The next two subsections establish
them.

H.4 Each channel at full entropy: memorylessness in substrate time

A channel reaches its full entropy gshare,eff only if its dressing keeps no memory of its own past.
To see what memory would cost, consider the one-parameter family of per-channel kernels

Ka(b, b′) = a δ(b, b′) + (1 −a) pη∗(b′),
a ∈[0, 1],

which repeats the current state with probability a and otherwise redraws from the stationary
weight. Every member has pη∗as its stationary distribution and the same single-time marginal,

so the equilibrium ensemble cannot say which one governs the dressing. The per-channel con-
ditional entropy H(b′ | b) = P
b pη∗(b) H

Ka(b, ·)


, evaluated on the exact 1680-state ensemble,
nonetheless slides with the memory a:

a (memory)
per-channel H(b′ | b)
status
0.0 (refresh)
7.41980 = gshare,eff
full entropy
0.1
6.99953
reduced
0.3
5.80153
reduced
0.5
4.40051
reduced
0.9
1.06642
reduced

Only the memoryless endpoint a = 0 — the refresh kernel Pη∗(b, b′) = pη∗(b′) of Appendix D.4
— returns the full gshare,eff. The kernel carries more than its stationary marginal, so this first
condition is settled by the dressing dynamics, not by the equilibrium ensemble. The requirement
is in fact topological: single-label local moves conserve slot ordering, fracturing each parity copy
into twenty-four sectors of thirty-five states, so no local kernel reaches the full ensemble at any
parameter value (Appendix D.4).

The dynamics settle it cleanly. Take any ergodic generator Q with pη∗as its detailed-balance
stationary weight — a continuous-time single-label-swap generator built from the same admissi-
bility weights serves, with no memorylessness assumed. The kernel over substrate-time τ is eQτ,
and the mutual information between its endpoints decays to zero,

I(τ=0.1) ≈3.4 nats,
I(τ=1) ≈0.030,
I(τ=2) ≈1 × 10−4,

so eQτ relaxes to the refresh kernel. The readout sits far inside that limit: the substrate time
is τ∗= L∗/c ≈5.4 × 10−44 s, the electron’s Compton time is τe = ℏ/(mec2) ≈1.3 × 10−21 s,
and their ratio ≈2.4 × 1022 is the same hierarchy λe/L∗the length relation expresses. The
ground state is read out some 1022 mixing times after relaxation, so each channel draws afresh
and carries its full gshare,eff. This is the Many-Pasts weight P(H|P) ∝e−D(H,P) seen on the
dressing: a sum over pasts that keeps no trace of the particular one realized.

H.5 Independent channels: the electron as the lightest defect

The second condition is that the seven channels are mutually independent at one readout, so
that the deficit ∆of H.3 vanishes. This needs no separate dynamical postulate; it follows from
what the electron is.

Keep the deficit explicit in the support relation. With each channel at its full entropy, the
joint entropy is 7gshare,eff −∆, and the support reads

λ
L∗
= 2

3 e 7gshare,eff−∆.

At a fixed substrate scale L∗, a one-bit charged defect whose dressing carries correlation ∆has
spatial support λ ∝e−∆, and through m = ℏ/(cλ) a mass

m ∝e∆.

Correlation among the dressing channels contracts the defect and raises its mass. Because ∆≥0,
the lightest one-bit charged defect is the one with ∆= 0: the product dressing. The electron is
that defect, so its seven channels are independent and its support is the full dimeff = e7gshare,eff.
Correlated dressings are still allowed; they describe heavier, additionally dressed excitations, not
the elementary anchor.

The selection is physical, not an appeal to ignorance. Independence is not read off from the
absence of an inter-channel constraint in the ensemble; it is forced because correlation carries a

mass, and the electron is defined as the lightest charged one-bit defect. The argument uses only
the mass–entropy identification at the centre of the theory: the same relation m = κm ∆S that
anchors the electron makes inter-channel correlation an excess of defect entropy, and so of mass.

With both conditions met, the scale-setting branch reads end to end without a free step.
One-pass occupation fixes the seven channels (H.2); memorylessness in substrate time gives
each its full entropy gshare,eff (H.4); the electron’s lightness forces the channels independent (this
subsection). Hence

2 λee−7gshare,eff,
G∗= c3L2
∗
ℏ
,

λe
L∗
= 2

3 e7gshare,eff,
L∗= 3

with no gravitational quantity used as input.

The condensate realization gives the EFT kinetic term a natural microscopic origin, and the
dressing analysis derives the faithful sector-resolution principle from the mass–entropy ontology,
the one-bit electron anchor, and the memoryless dressing, so the substrate length and the induced
scale follow from commitments the theory already makes. The first-principles derivation of every
inhomogeneous continuum coefficient from the full kernel remains open.

H.6 Mass as a rate: the dressing operators

The scale-setting branch above fixes the support exponent 7gshare,eff and hence the ratio λe/L∗.
Reading that ratio as a physical length used one identification: that the electron advances
one Compton phase cycle per complete dressing recurrence. Memoryless sampling delivers a
recurrence rate, not by itself a Compton pole, so this subsection gives the recurrence an explicit
operator form and asks what the phase clock must be. The identification does not close to zero
residual; it decomposes into pieces that are either forced or falsifiable, with one identification
(Fork A below) remaining.

The rest mass is read as a recurrence rate, not a binding-energy sum.
Writing the
length relation as a mass,

2
ℏ
τ∗
e−7gshare,eff,
τ∗= L∗/c,
e−7gshare,eff = 2.78 × 10−23,

mec2 = ℏc

λe
= 3

shows the electron is lighter than the substrate quantum E∗= ℏc/L∗not by a small factor but by
e−7gshare,eff. That suppression is a waiting time: seven independent full-entropy channels jointly
reaching their resolved configuration, probability e−7gshare,eff per substrate tick. This rules out a
binding-energy reading. Seven channels each releasing free energy of order gshare,effE∗would give
a mass linear in gshare,eff, of order 50 E∗— heavy, and wrong by twenty-three orders. The object
that produces an exponentially light particle is the spectral gap of a stochastic generator whose
slowest mode relaxes at e−7gshare,eff per tick, not the ground-state energy of a Hamiltonian. This
rate reading sets the scale, not the content of the mass–entropy map: the electron’s committed
entanglement is the one bit ∆Sf = ln 2 of Section 13.2, while the recurrence rate fixes the
exchange rate κm at which that bit is read as a mass (the length anchor of Section 13.3). Mass
measures committed entanglement; the rate fixes how small that measure is.

The killed-refresh return rate.
By Postulate III the dressing is memoryless: the refresh
kernel P(b, b′) = pη∗(b′) redraws each tick from the admissibility Gibbs state. On the seven-layer
history space the marked configuration ⋆— the resolved dressing, each channel at its designated
typical configuration — has stationary weight Q
m pη∗(⋆m) = e−7gshare,eff. The dynamics of “at
⋆versus not” is a two-state renewal chain with per-tick return probability e−7gshare,eff; the killed

(absorbing-at-⋆) operator has leading eigenvalue 1 −e−7gshare,eff, so its gap is the raw return rate

Γraw = e−7gshare,eff

τ∗
,
⟨Trec⟩= e+7gshare,eff = 3.6 × 1022 ticks.

The physical electron Compton rate carries the transverse export factor 2/3 of the length relation,
Γe = 3

2Γraw, so mec2 = ℏΓe = 3

2(ℏ/τ∗)e−7gshare,eff, consistent with λe/L∗= 2

3e7gshare,eff. This is
the recurrence of the scale-setting branch re-expressed as a transfer-operator gap; the rate scale
was never the contested part.

The locality obstruction.
The generator that carries this recurrence cannot be local. Define
a local move as shifting one face label by ±1 while preserving injectivity. Under these moves
the 1680 states fracture into

48 components = 4! orderings × 2 orientations,
each of size C(7, 4) = 35,

verified directly on the ensemble. Slot ordering is a conserved charge of any local single-label
dynamics, so the closed seven-sector loop that winds the phase once is not reachable by a local
Hamiltonian; the generator that closes it is the non-local refresh kernel of Postulate III. This
is an honest cost of the construction: a referee asking for a local “Hamiltonian spectrum” is
owed the statement that the operative generator is the non-local refresh super-operator. It is
the content of memorylessness rather than a device introduced to evade locality.

The phase clock, by elimination.
The mass is ℏtimes a phase-turn rate, so closing “one
turn per recurrence” requires the turning clock and the refresh counter to be the same object,
not two rates set equal. Two candidates exist: (B) a clock private to the defect, or (A) the
global U(1) phase θ of the condensate order parameter σ = √n eiθ. A private clock is not forced
to complete exactly one turn per recurrence; pinning it to the refresh would require declaring
the rates equal, the identification one is trying to remove. The condensate phase is forced by
counting: one complete refresh commits exactly one defect (the lightest charged defect binds
the seven sectors once — one fermion, Postulate II), one committed defect is one unit of the
condensate U(1) charge (that charge being the count of committed cells), and one unit of charge
on a U(1) phase is one full turn. Hence

one refresh = one defect = one unit charge = one 2π turn,

each link a count or a definition, with no rate chosen. A lone defect has no conserved tally for the
refresh to increment, so the counting argument is unavailable to it: the clock is the medium’s.

The charge-tilted generator.
Tilting the renewal generator by a counting field ϕ conjugate
to committed charge gives the reduced tilted transfer matrix
 1−r reiϕ
1
0

with r = e−7gshare,eff. Its
leading eigenvalue is exactly 2π-periodic in ϕ, so committed charge is integer — the fermionic
nm ∈{0, 1} appearing as periodicity, not a tuned input — and a complete seven-sector pass is
one closed loop on the label cycle, winding number one. The identification “one phase cycle per
dressing recurrence” therefore decomposes into three pieces, none carrying a knob:

Γ = e−7gshare,eff/τ∗
|
{z
}
memorylessness+entropy

⊕∆Q = 1 per recurrence
|
{z
}
fermionic one-pass

⊕∆θ = 2π ∆Q
|
{z
}
Josephson

.

The Compton clock is now a derived winding rate rather than an asserted one.

What remains: Fork A.
Two things do not close, and are stated as such. First, the iden-
tification of the winding U(1) with the condensate phase, and its 2π with the Compton cycle
(Fork A), is consistent with these operators but not forced by them; its support is the elimination
argument above, not a computation, and it is settled in H.7, where the mean-field condensate
is shown to carry exactly one phase. Second, the residual is now a single integer — that the
electron is the ∆Q = 1 (one-loop) recurrence rather than a higher winding. Unlike the original
identification this is falsifiable: the winding-N recurrences are the muon and tau, and their rates
fix mµ/me and mτ/me with no new input (Appendix I.1). The phase-cycle identification has
thus been replaced by counting plus a condensate relation, with the residual reduced to one
integer that the lepton spectrum constrains.

H.7 The dressing functionals from the mean-field condensate

The dressing Hamiltonian named in Section 22,


ε0 nm −nmFm(ρm)


+
X

H =
X

m<m′
Vmm′(ρm, ρm′),

m

carries three named functionals ε0, Fm, V . This subsection derives them from the mean-field
condensate. The result is not that everything closes: the two residuals of H.6 collapse into a
single premise, and that premise is the one the geometry side has not yet established.

Mean-field reduction.
In the GFT condensate the single-tetrahedron expectation is the
order parameter over boundary data, ⟨ˆφ(b)⟩= σ(b) =
p

n(b) eiθ, with one global phase θ — the
Goldstone of the broken GFT number U(1). Extremizing the condensate free energy under the
closure constraint gives the admissibility Gibbs state |σ(b)|2 = n pη∗(b), so the closure weight is
a Boltzmann factor at inverse temperature η∗with K2 as energy. The substrate temperature
and closure condition are then thermodynamic,

T∗= 1

η∗
= 33.48,
⟨K2⟩η∗= 3

2T∗= 50.22,

the 3/2 being equipartition over the three quadratic closure components, each at 1

2T∗; this is
the identity η∗⟨K2⟩= 3/2 read as a temperature.

Two generators: H selects, L rates.
The mass scale does not live in the spectrum of H.
H is combinatorial: its ground state fixes which configuration the lightest defect is. The rate
is the gap of the dissipative refresh super-operator L whose stationary state is pη∗and whose
memorylessness is Postulate III, and the physical mass is that gap carrying the transverse export
2/3 of the length relation, so

2
ℏ
τ∗
e−7gshare,eff,
gap(L) = e−7gshare,eff

2 ℏgap(L) = 3

mec2 = 3

τ∗
,
e−7gshare,eff = |σ(⋆)|2 =
Y

m
pη∗(⋆m),

the exponential smallness being the condensate density at the resolved configuration, not a
fine-tuned eigenvalue. This is the operator content of H.6.

ε0, the bare cost.
ε0 is the condensate chemical potential µ0 = ∂E/∂N, the energy to commit
one cell against the mean field, of order E∗= ℏc/L∗. Deriving it from the substrate names it as
µ0: it is the single irreducible dimensionful input, relocated rather than removed, since a mass
cannot come from combinatorics alone. Energies in H are measured in units of E∗, so ε0 is the
one place the physical scale is attached, and the substrate temperature T∗= 1/η∗entering Fm
below is the dimensionless closure temperature (K2 as energy); the physical mass is set by the
rate L, not by the spectrum of H.

Fm, the dressing free energy.
A bound channel carries a cloud ρm, a distribution over the
boundary ensemble; the unbound channel carries none. The released free energy is the cloud
entropy at the substrate temperature, Fm(ρm) = T∗S[ρm]. Extremizing it at fixed mean closure
returns the Gibbs state ρm = pη∗with S[ρm] = gshare,eff: the equilibrium cloud is the condensate
density itself. The binding test Fm > ε0 is met for all seven channels, and the lightest defect
binds each sector once with a full-entropy cloud — one-pass occupation, total bound entropy
7gshare,eff. This last equality bridges the two generators: the cloud entropy that H maximizes in
binding is exactly the rarity exponent that L carries in its gap, since the joint recurrence needs
all seven full-entropy clouds at once, probability Q
m e−S[ρm] = e−7gshare,eff. Binding and lightness
are one extremum: the most bound defect, carrying maximal cloud entropy, is the rarest and
hence the lightest.
The Fm step is otherwise a consistency loop — its form is entropy, the
condensate density is pη∗, and the maximum-entropy condition hands pη∗back — so it confirms
the one-pass structure rather than independently generating it.

Two energies, kept distinct.
Two quantities both invite the name “energy of the defect,”
and conflating them is what would reintroduce the hierarchy the rate reading removes. The first
is the separation cost: the entropic free energy needed to unbind the defect from the network.
In a medium whose only currency is shared entanglement there is no stored potential energy in
the spring sense, so the cost of any operation is the free-energy cost of changing the admissible-
arrangement count, and the natural energy of a defect is what it would take to dissolve it back
into the substrate. That cost is large and linear in gshare,eff — of order the ∼50 E∗that the
binding-energy reading of H.6 produces, a number that is correct as a separation cost and wrong
only as a mass. The second is the rest energy mec2 = ℏΓe, which is the recurrence rate and
is tiny, exponential in −7gshare,eff. Binding-equals-lightness is the statement that these are one
bound entropy gshare,eff read two ways: the deeper the binding, hence the larger the separation
cost, the rarer the full re-completion, hence the smaller the rate. They are proportional through
gshare,eff but not equal, and the distinction is load-bearing — reading the separation cost as the
rest mass would set the electron at tens of E∗and undo the exponential suppression. The cost
to separate and the rate of recurrence are two faces of one entanglement, not the same number.

V , the inter-channel coupling.
This is the one piece with new content. Channels couple
by exchanging the field they share, and the framework has exactly one infrared field: the scalar
capacity fluctuation ϕ = δSent, coupled to matter through the trace channel −κχSent of Sec-
tion 10. That scalar is the sector singlet |s⟩=
1
√

7
P
m |m⟩, so each channel couples to it through

⟨m|s⟩= 1/
√

7, the same for every m. Integrating out the single shared scalar gives an outer
product,
Vmm′ = v ⟨m|s⟩⟨s|m′⟩= v

7
for all m, m′,

with v the zero-momentum scalar-exchange amplitude: the uniform 7 × 7 matrix is rank one.
The per-incidence overlap that enters the ladder is ⟨m|s⟩⟨s|m′⟩= 1/7, doubled by orientation
to the 2/7 of Appendix I.1, while the overall coupling v sets the mass scale and not the ratios.
One exchanged mode is one outer product, which is rank one. A second light field would add
a second outer product, raising the rank to two and splitting the uniform entries, which breaks
the single 2/7 and the (2/7)N2 ladder. So rank(V ) = 1 is not an assumption layered onto the
construction; it is the statement “one scalar field,” the same premise that gives no anisotropic
stress and Φ = Ψ in the PPN sector (Section 15). A higher-rank V is a second light field,
which that sector already excludes. This also supplies the minimum-mass content used in H.5:
rank-one is the minimal inter-channel correlation, so the lightest charged defect is forced to it,
and “the coupling is rank one” and “this is the lightest defect” are the same statement.

The collapse, and the wall.
The two residuals of H.6 — (A) the winding phase is the
condensate U(1) and its 2π is the Compton cycle, and (B) the lightest defect’s coupling is the
rank-one singlet — reduce to one fact. The mean-field condensate has exactly one global phase,
so the mode that mediates V and the mode that winds the Compton clock cannot be different:
there is no second U(1) to be the wrong one, which settles the underdetermination of Fork A.
And rank-one V is “one scalar.” A and B reduce to the same single-scalar condensate assumption
— a single scalar field on the mean-field condensate — which is the founding premise of the
framework, tested and passing in the charged-lepton sector to sub-percent. The cost is that
this premise is now load-bearing twice. Every step here is conditional on the GFT substrate
supplying that condensate, with density |σ|2 = pη∗, one U(1), one scalar infrared mode, and
the emergent four-geometry to go with it — the geometry-side compatibility problem left open
in Section 22 and H.1. The defect sector and the geometry sector are two outputs of the same
σ: the sub-percent lepton ladder cannot be banked as evidence independent of the geometry
problem, because it is the same condensate graded on its other output. The construction falls
out cleanly because it rests on the single-scalar condensate, and that condensate is the unproven
object; the two are one coin. Writing a fully explicit H past this point would return ε0, Fm, V
as they were put in, and the rank-one V is already as derived as it gets without the condensate
itself.

Reproduction.
The ensemble invariants (η∗, ⟨K2⟩η∗, gshare,eff, and the identity η∗⟨K2⟩=
3/2), the killed-refresh return rate, the 48 × 35 locality fracture, the 2π-periodicity of the
charge-tilted generator, the rank-one spectrum {1, 06} of Psing, and the charged-lepton ratios of
Appendix I.1 are reproduced end to end from the bare admissibility ensemble by a standalone
script provided as supplementary material (requiring only numpy and networkx).

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 Charged-lepton spectrum from the shell algebra

Before the algebra, the physical picture. The electron is the ground-state one-bit charged de-
fect — the N = 0 configuration whose seven-channel dressing sets the substrate length (Ap-
pendix D.4). The muon and tau are not new kinds of object but heavier shell excitations of the
same defect: each adds one radial entanglement shell, realized microscopically as the exclusion
of one face label from the seven-state alphabet of Appendix B. The number of generations is not
put in by hand; it is limited by the closure machinery, which stops having anything to weight
after the third admissible shell, so there are three charged leptons and not four. The mass of
each lepton is the exponentiated dressing free energy of its shell source, in the same log-overlap
form that carries the gravitational dressing (Appendix H.3). This sector is a structural extension
of the mass–entropy ontology, outside the closed weak-field chain. With the dressing operators
of Appendix H.6 in place it is operator-backed: the muon and tau are the winding-N recurrences
of the same generator, so the ratios below follow with nothing fitted to lepton data, conditional
on the Fork A identification of Appendix H.7. Because those ratios carry no dimensionful in-
put, the sector is a parameter-free, falsifiable test of the deeper construction rather than only a
cross-check.

With that picture fixed, the spectrum is written as a shell expansion,

me
= B0N + A0N2,
N = 0, 1, 2,

with the electron the N = 0 ground state and the muon and tau the N = 1, 2 radial entanglement-
shell excitations of the same core structure. Each shell excitation is the exclusion of one face
label from the seven-state alphabet of the boundary ensemble in Appendix B.

Three-generation termination from closure-spectrum collapse.
Combinatorial injec-
tivity alone allows up to N = 3 shells, since the reduced alphabet must retain at least four
distinct labels to support an injective face assignment (7 −N ≥4). The sharper structural
bound comes from the admissibility-closure machinery. For the reduced ensemble at shell N,
the closure invariant K2 inherits a discrete spectrum whose dispersion gives the admissibility
kernel e−ηK2 real weight. For N = 0, 1, 2 the K2 spectrum has multiple distinct values, and the
closure equation ⟨K2⟩η = 3/(2η) has a non-degenerate solution.

At N = 3 the spectrum collapses. With only four labels left, every injective assignment to the
four faces uses all of them, and K2 depends on the labels only through the permutation-invariant
sums S and Σ2, so every microstate of the reduced four-label ensemble carries the same value of
K2. With a single spectral value K2
0 the closure equation ⟨K2⟩η = 3/(2η) still fixes η = 3/(2K2
0)
formally; what fails at N = 3 is selection, not solvability. The kernel e−ηK2 acts trivially when
every microstate carries the same K2, so it weights nothing and leaves no admissibility dispersion
to define a branch. Shell N = 3 therefore does not define a non-degenerate admissibility-closed
branch in the same sense as the earlier shells: the cutoff rests on the collapse of the K2 dispersion,
not on the closure equation admitting arbitrary η.

The number of charged-lepton generations is consequently the number of shells with a non-
degenerate closure branch,

N ∈{0, 1, 2}
=⇒
three generations,

a structural bound sharper than the combinatorial one.
It is this mechanism, rather than
injectivity alone, that fixes the generation count at three rather than four.

Linear coefficient from the first-shell reduced-alphabet entropy.
The linear coefficient
is the entropy cost of adding one shell excitation. With one face label excluded, the reduced
ensemble has
Ω1 = 2 P(6, 4) = 720 = 6!,

i.e., the full symmetric group S6 with orientation degeneracy. Direct admissibility evaluation on
the reduced ensemble gives g1 = ln Ω1 = ln 720 to within 0.1%, since the admissibility correction
is small on the reduced ensemble where the alphabet symmetry is nearly intact. Hence

B0 = ln Ω1 = ln 720 = 6.579 . . . ,

compared with the empirical fit B0 = 6.586 (agreement at 0.1%).

Why the same 720 at every shell.
The per-shell factor is the same Ω1 = 720 for each added
shell, not a decreasing sequence 720·240 · · · in which each shell excludes a further label. The two
readings diverge sharply at the tau: cumulative exclusion would give mτ/me ≃1151 (−67%),
whereas the repeated factor gives 3454.6 (−0.65%), so the data require the same reduction at each
shell. The structural reason is the rank of the inter-shell coupling. The shells couple only through
the permutation-invariant singlet projector Psing, whose spectrum is {1, 0, 0, 0, 0, 0, 0}: rank one.
A rank-one projector pins exactly one shared direction, and it is the same direction however
many shells are already stacked, so each added shell surrenders only that one shared singlet
direction and retains the other six — the same direction at every shell, not a fresh label removed
cumulatively. The injective count realizes this exactly: P(6, 4)/P(7, 4) = 360/840 = 3/7, so

1680 →720 once and again at each subsequent shell. This also separates the two reduced-
alphabet notions that otherwise look inconsistent. The generation cutoff above counts 7 −N
surviving labels because it asks whether a non-degenerate closure branch still exists; the mass
factor here is the rank-one singlet pinning one shared direction per shell. They are different
operations, which is why the cutoff scales with 7 −N while the ladder carries a fixed 720.

Quadratic coefficient from singlet-projection shell algebra.
The quadratic coefficient
follows from three premises already in the framework: (i) mass arises from scalar capacity re-
sponse (Postulate II); (ii) the scalar EFT response is quadratic in the total source; (iii) the coarse
scalar branch projects onto the permutation-invariant singlet of the seven-state face alphabet.

Let

3
X

H7 = span{|m⟩: m = −3, . . . , 3},
|u⟩=
1
√

m=−3
|m⟩,
Psing = |u⟩⟨u|.

7

A shell excitation that marks the face state |ar⟩contributes to a coherent shell source

N
X

|JN⟩=

r=1
|ar⟩.

The coarse scalar field δS is permutation-symmetric on face labels and therefore couples only
through Psing. The scalar-channel response is quadratic in the total source, so the relevant object
is the source norm

N
X

⟨JN|Psing|JN⟩=

r,s=1
⟨ar|Psing|as⟩.

Because ⟨a|Psing|b⟩= 1/7 for any a, b, this reduces to N 2/7 independent of which specific face
labels are excluded. Including the binary orientation degeneracy of the boundary ensemble, the
orientation-summed scalar transfer weight per ordered shell incidence is

Mrs = 2 ⟨ar|Psing|as⟩= 2

7.

The ordered-pair count decomposes as N 2 = N +2
N
2


: N self-incidences along the diagonal and
2
N
2


directed cross-incidences. Two distinct steps enter here, and only the first is the quadratic
EFT response. The quadratic singlet response fixes that there are N 2 ordered incidences, each
carrying the per-incidence overlap Mrs = 2/7; it does not by itself make the mass multiplicative.
The passage from these N 2 additive incidences to a multiplicative factor is the log-overlap mass
map: the mass is the exponential of a dressing free energy whose pairwise term is P
r,s ln Mrs =
N2 ln(2/7), the same log-overlap form that carries the gravitational dressing (Appendix H.3)
and stated explicitly below. With that map, the N-shell dressing is

N2
,

N
Y

r,s=1
Mrs =
2

7

so
A0 = ln 2

7 = −1.253 . . . ,

compared with the empirical fit A0 = −1.255 (agreement at 0.15%).

The ordered-bilinear structure is not a free assumption. It is the unique scalar-channel re-
sponse to a multi-source state under the three premises just stated: the source norm ⟨JN|Psing|JN⟩
runs over both indices independently because the integrated-out scalar contribution is quadratic
in JN.

Mass ladder and numerical accuracy.
Combining the two coefficients,

N2
,
N = 0, 1, 2.

me
= 720N
2

mN

7

Equivalently,

log mN

mN−1
= ln 720 + (2N −1) ln 2

7,

so each added shell contributes ln 720 plus an odd number of new scalar-overlap incidences
(1, 3, 5, . . ., summing to N 2). The second difference is fixed at

∆2 log mN = 2 ln 2

7 = −2.506 . . . ,

compared with the empirical −2.509 (0.15%). With no parameters fitted to lepton data,

me
= 720 · 2

mµ

7
= 1440

7
= 205.71,
PDG: 206.77
(0.5%);

4
= 3454.56,
PDG: 3477.23
(0.65%).

mτ
me
= 7202
2

7

The dressing map, explicitly.
The three factors follow from one construction. Write the
N-th charged lepton as a coherent source raised on the electron ground dressing by exciting N
shells, SN = PN
k=1 sk, where each sk excludes one face label from the seven-state alphabet. Its
dressing free energy is a log-overlap functional of the kind used for the gravitational dressing
in Appendix H.3, and it separates into a per-shell term and a pairwise term. Each shell carries
the reduced-alphabet entropy ln Ω1 = ln 6! = ln 720, the permutation count of the six surviving
labels, which builds the linear factor 720N. Each ordered pair of shells contributes a singlet
log-overlap: a single label projected on the closure singlet u =
1
√

7(1, . . . , 1) of Section 8 has
weight |⟨u|m⟩|2 = 1/7, and orientation summation doubles it to 2/7. The N shells form N 2

ordered pairs, so the pairwise term is N 2 ln(2/7), and exponentiating the free energy returns

N2
= 720N
2

N2
.

2

mN

me
= ΩN
1

7

7

The coefficients are fixed, not chosen: 720 = 6! and the N 2 pair count are combinatorial, and
2/7 is the orientation-summed singlet projection behind Ωtet = 2 P(7, 4) = 1680. What remains
an input is the form of the map — that the lepton mass is the exponentiated log-overlap free
energy of the coherent multi-shell source — a log-overlap structure of the kind that carries the
gravitational dressing, taken here as the form of the mass map.

The support exponent, tested against the spectrum.
The ladder also tests the exponent
in the support-to-length identification of Appendix D.4, which reads the physical ratio as the
first power of the entropy support, λ/L∗∝eH. Letting that power float, λ/L∗∝eαH, turns
each lepton ratio into a measurement of α, since mN/me =

720N(2/7)N2α:

αN = ln(mN/me)PDG

ln

720N(2/7)N2,
αµ = 1.0010,
ατ = 1.0008.

Read the same way, the electron-to-G∗branch needs α = 0.9999 to reproduce the observed
Newton constant, so three determinations across two sectors land within 10−3 of unity and
bracket it. What the leptons settle is the power, not its final digit: a square-root map (α = 1

2)
would put mµ/me near 14 and a quadratic map (α = 2) near 4 × 104, and only the linear rule
reproduces the spectrum with the natural shell count 720 = 6!. The identification that sets the
substrate scale is therefore not a single assumption tuned to gravity; the same linear exponent
is preferred in a sector that never enters the G∗chain.

Status and remaining audit.
The three-generation termination, the linear coefficient, and
the quadratic coefficient are thus fixed from machinery already native to the framework, with
no parameters fitted to lepton data: the collapse at N = 3 is the permutation-invariance of
K2, and the 720, 2/7, and N 2 are the reduced-alphabet, singlet-projection, and ordered-pair
counts of the map above. The form of the map is not a separate posit. Mass is ℏ/(cλ) with
λ/L∗= eH, the same support-to-length identification that fixes L∗in Appendix D.4; the ladder
is its N > 0 continuation, with the electron the N = 0 entry, and the equality of the support
exponent across the two sectors is the α = 1 check above. The per-shell term is the log-overlap
additivity theorem of Appendix H.3, and the pairwise term is the singlet overlap of shells sharing
the seven-state alphabet. Beyond that one identification, the only lepton-specific input is the
structure of the excited source — a product across the six spectator channels, coherent in the
shared singlet — which produces the 2/7 pairings.

The electron anchor remains the clean weak-field entry point; the heavier charged leptons are
conditionally derived outputs of the same shell algebra, not closure-defining ingredients of the
gravitational chain. Composite hadrons remain part of the dressed bound-state entropy program
rather than a completed output of the present construction. The neutrino and quark sectors are
not treated here. The same closure-spectrum-collapse argument should apply universally if the
framework is right — the Standard Model has three generations across all fermion sectors —
but verifying that requires extending the shell construction to the relevant defect classes.

I.2 Gauge-structure extension and Standard Model ontology

The framework’s primary target is the gravitational and dark sector. Standard Model gauge
structure is hosted on the substrate as external content rather than derived from it. This subsec-
tion specifies which gauge components have substrate-natural analogues, which are substrate-
compatible but external, and how the baseline-redundancy template accommodates the full
Standard Model gauge group SU(3)c × SU(2)L × U(1)Y .

Baseline-redundancy template.
The same logic that organizes the gravity sector extends
to any conserved-charge sector. For a conserved charge Q, introduce an entropy-like potential
SQ(x) and require that physical observables depend only on differences of that potential, not on
its absolute baseline. Localizing that redundancy requires a compensating connection. In the
Abelian case,

DµSQ = ∂µSQ −qAµ,
SQ →SQ + α(x),
Aµ →Aµ + 1

q ∂µα,

giving Maxwell-type dynamics for Aµ. The non-Abelian generalization, for multiplet-valued en-
tropic potentials transforming under a compact Lie group G, gives Yang–Mills covariant deriva-
tives and field strengths in the standard form.

The template tells us what form a gauge interaction takes once a conserved-charge sector with
baseline redundancy is present. It does not select which specific groups are physically realized.

Substrate-natural components.
Two ingredients of Standard Model gauge structure are
naturally accommodated by the substrate.

U(1)Y . The baseline-redundancy template naturally accommodates an Abelian conserved-
charge sector of the hypercharge type. The associated entropy potential is conserved-charge
in character, and localizing its baseline redundancy produces the corresponding Maxwell-type
dynamics. The argument is independent of the gravity-sector derivation, applying the same
template to a separate conserved-charge potential.

SU(2)L. The substrate naturally carries an SU(2)-representation structure through its half-
integer face data.
The spin-3/2 commitment forced by maximum-capacity channel selection
and tetrahedral injectivity (Section 5) carries an SU(2) action on face data through the stan-
dard double-cover relation. The static K2 ensemble breaks this SU(2) to its U(1) Cartan via
the magnetic-quantum-number projection, but the full SU(2) is recoverable at the level of the
quantum face data the projection discards. Identifying this representation-theoretic SU(2) with
the internal weak group SU(2)L, including chirality and doublet assignments, remains external
to the present derivation.

Substrate-compatible but external: SU(3)c.
Color SU(3) does not emerge from the sub-
strate by any of the standard mechanisms (direct decomposition of the seven-state face algebra,
Seiberg duality, preon compositeness, topological soliton, monopole condensation).
The ob-
structions are structural: SU(3) is not a spin group, so it does not inherit from rotational
covariance; its fundamental representation requires three equivalent (non-hierarchical) states,
while the substrate’s natural three-fold structures (shell index, K2 classes, residual face po-
sitions) are all hierarchical; its rank-2 Lie algebra requires two independent quantum-number
axes, while the substrate provides only rank-1 axis structures; and its non-Abelian commutation
relations cannot be reproduced from the substrate’s scalar edge couplings.

A positive ontological reading is nonetheless available within the framework. Color is the
substrate’s name for an internal three-valued label on fermionic defects that lives in the non-
singlet complement of the face-state algebra. The scalar gravitational branch couples to defect
sources only through the permutation-invariant singlet (Appendix D), so the macroscopic scalar
field is blind to non-singlet structure by construction. Color labels and color dynamics therefore
live in a sector that gravity cannot resolve: the gravitational sector and the color sector occupy
orthogonal subspaces of the substrate’s algebra.
The strong interaction can be represented,
within this hosting picture, as the local gauge dynamics of the color label under the general
baseline-redundancy template.
The choice of SU(3) specifically is empirical input; it is the
realized non-Abelian structure that fits the template.

This account explains several structural features of color without claiming to derive them.
Gravity’s blindness to color follows from the scalar branch being the singlet projection of the
substrate. Color confinement itself is not derived here; the substrate account provides that the
macroscopic scalar-gravity branch resolves only singlet structure and is therefore blind to color
non-singlets, which is consistent with the observed fact that hadronic macroscopic signatures are
color-singlet but is not a replacement for the QCD confinement mechanism. The independence of
color from generation, charge, and spin reflects the independence of the corresponding substrate
sectors: color in the non-singlet complement of the face algebra, generation in the shell-excitation
index (Appendix I.1), electromagnetic charge in the baseline redundancy of a separate conserved
potential, and spin in the fermionic face data.

Hadronic matter and the mass–entropy bridge.
The mass–entropy bridge m = κm(ℓ) ∆S
applies to fermionic defects regardless of their color label. For charged leptons, the defect is
color-singlet and the dressed entropy is well-approximated by the shell-excitation budget of
Appendix I.1, with mass ratios matching observation at the sub-percent level. For hadrons, the
dressed entropy is dominated by the QCD-internal contributions of confinement-scale gluonic
flux, trace-anomaly structure, chiral vacuum reorganization, and quark binding. The mass–
entropy bridge then organizes the full dressed bound-state entropy budget,

Sdressed
ent,H
= Sdefect + Sbind + Sconf + SχSB,

without requiring the framework to re-derive QCD. The structural claim is compatibility: the
dressed entropy budget is the quantity QCD calculates, and the mass–entropy bridge converts

it to inertial mass through the same running κm(ℓ) used for elementary sectors.

Scope summary.
The Standard Model’s full gauge group SU(3)c ×SU(2)L ×U(1)Y is hosted
on the substrate at three distinct levels of derivational support: U(1)Y from baseline redundancy
(substrate-natural via independent argument); SU(2)L from spin-3/2 fermionic face represen-
tation theory (substrate-natural with chirality identification external); SU(3)c as a non-singlet
internal label (substrate-compatible but not substrate-derived). The framework’s target scope
remains the gravitational and dark sector, and the gauge sector is hosted accordingly.

The sectors in this appendix are structurally linked to the same entanglement logic but are
not part of the closed static weak-field chain.

Appendix J: Numerical Checks and Robustness

Appendix J is intentionally modest. It does not add new derivations, and these scripts and checks
do not establish the ontology: they audit the numerical consequences of the stated derivations,
and their role is reproducibility, not independent proof. 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, and it records, in Section J.3, a complete self-contained
script that recomputes the entire numerical spine of the manuscript from the stated definitions.

The numbers recomputed here, and where each enters the main text, are: the admissibility
entropy gshare,eff (Section 13, Appendices B and D.4); the substrate length L∗(Sections 2 and 13,
Appendix D.4); the induced scale G∗(Sections 10 and 13, Appendices D.4 and K); the galactic
scale a0 (Section 14, Appendix C.7); the diamond-lattice Green constant Gtet(0) (Appendix C.5);
the edge-kernel chain Jbare →γ and the return sum Σret (Appendices C.3–C.4); and the charged-
lepton ladder with its support-exponent cross-check (Section 13, Appendix I.1).

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.

J.3 Reproduction script

The script below recomputes every load-bearing number in the manuscript from the stated
definitions alone: the 1680-state K2 spectrum and its multiplicities, the closure solution η∗, the
admissibility entropy gshare,eff, the substrate length L∗and induced scale G∗, the Joyce constant
and the exact nhorScell
∞
= 1/4 identity, the edge-kernel chain, the projection coefficient ϵ and
horizon target σ∗, the charged-lepton ladder with its support-exponent cross-check, and the slot
mutual information. No value in the chain is entered by hand; each printed line carries the
manuscript value it must reproduce. The script runs in seconds under Python 3 with numpy and
scipy, and is maintained alongside the manuscript at jaigp.org.

# ============================================================
# Reproduction script for the Entropic Scalar EFT manuscript.
# Recomputes the full numerical spine from stated definitions:
# the 1680-state K^2 spectrum, eta*, g_share,eff, L*, G*, the
# Joyce constant, the 1/4 horizon identity, the edge-kernel
# chain, epsilon, sigma*, the lepton ladder, and the slot MI.
# Dependencies: Python 3, numpy, scipy.
# Every printed value is compared against the manuscript.
# ============================================================
import itertools, math
from math import log, exp, pi, gamma, sqrt
from collections import Counter
import numpy as np
from scipy.optimize import brentq

# 1. Enumerate 1680 states: injective 4-of-7 labels m in -3..3, x2 parity
labels = range(-3,4)
states = list(itertools.permutations(labels,4))
print("P(7,4) =", len(states), " total with parity:", 2*len(states))

def K2(ms):
S = sum(ms); Sig2 = sum(m*m for m in ms)
return 48 - (S*S - Sig2)/3

spec = Counter()
for st in states:
spec[round(K2(st)*3)] += 2
# parity doubling; key = 3*K2 to keep exact
print("Distinct K2 values and multiplicities (K2 as fraction /3):")
for k in sorted(spec): print(f"
K2={k}/3 = {k/3:.4f}
mult={spec[k]}")
print("Total multiplicity:", sum(spec.values()))

# 2. Solve closure condition <K2> = 3/(2 eta)
Ks = np.array([k/3 for k in sorted(spec)])
ns = np.array([spec[k] for k in sorted(spec)], dtype=float)
def avgK2(eta):
w = ns*np.exp(-eta*Ks)
return (w*Ks).sum()/w.sum()
f = lambda eta: avgK2(eta) - 3/(2*eta)
eta_star = brentq(f, 1e-4, 1.0, xtol=1e-16)
print("\neta* =", eta_star, " (paper: 0.02986684439352237)")

w = ns*np.exp(-eta_star*Ks); Z = w.sum(); p_class = w/Z
p_micro = np.exp(-eta_star*Ks)/Z
H = -(ns*p_micro*np.log(p_micro)).sum()
print("g_share,eff =", H, " (paper: 7.419800023570903)")
avg = avgK2(eta_star)
var = (p_class*(Ks-avg)**2).sum()
print("<K2> =", avg, " (paper 50.2229154254)
3/(2eta*) =", 3/(2*eta_star))
print("Var(K2) =", var, " (paper 15.6889750078)
a_UV = 1/Var =", 1/var)
print("C_cl = eta* <K2> =", eta_star*avg)

# 3. L*, G*
hbar=1.054571817e-34; c=299792458.0; me=9.1093837015e-31
lam_e = hbar/(me*c)
print("\nlambda_e =", lam_e, " (paper 3.861592671986303e-13)")
Lstar = 1.5*lam_e*exp(-7*H)
print("e^{7g} =", exp(7*H), " (paper 3.602860521062804e22)")
print("(2/3)e^{7g} =", (2/3)*exp(7*H), " (paper 2.401907014041869e22)")
print("L* =", Lstar, " (paper 1.607719470158885e-35)")
Gstar = c**3*Lstar**2/hbar
print("G* =", Gstar, " (paper 6.603991270884347e-11)")
G_codata = 6.67430e-11
LP = sqrt(hbar*G_codata/c**3)
print("L* vs Planck length:", (LP-Lstar)/LP*100, "% below
(paper 0.528%)")
print("G* vs CODATA:", (G_codata-Gstar)/G_codata*100, "% below
(paper 1.053%)")

# 4. Joyce constant
Gtet = 3*gamma(1/3)**6/(2**(14/3)*pi**4)
print("\nG_tet(0) =", Gtet, " (paper 0.4482203943883814)")
print("ln(7/6)/(pi*Gtet) =", log(7/6)/(pi*Gtet), " (paper 0.109472228)")
print("S_inf_cell = 3ln2/(32 pi Gtet) =", 3*log(2)/(32*pi*Gtet), " (paper 0.0461482516)")
print("n_hor*S_inf_cell =", (8*pi*Gtet/(3*log(2))) * (3*log(2)/(32*pi*Gtet)))

# 5. Edge kernel chain
Jbare = 2/3*eta_star
Jtree = Jbare/3
Sig = 7+2/9
cloop = 1/(1+Jtree*Sig)
Jren = Jtree*cloop
print("\nJ_bare =", Jbare, " (paper 0.0199112296)")
print("J_tree =", Jtree, " (paper 0.0066370765)")
print("c_loop =", cloop, " (paper 0.95426)")
print("J_ren =", Jren, " (paper 0.00633348)")

print("gamma coeff 4 J_ren/(3 pi^2) =", 4*Jren/(3*pi**2), " (paper 8.556e-4)")

# 6. epsilon and cluster numbers
epsv = H/(4*pi**2)
print("\nepsilon = g/(4pi^2) =", epsv, " 1-eps =", 1-epsv, " ceiling =", 2-epsv)
print("sigma* = pi/g =", pi/H, " (paper 0.42340665)")

# 7. lepton ladder
mmu = 720*2/7; mtau = 720**2*(2/7)**4
print("\nm_mu/m_e =", mmu, " PDG 206.7683 err%:", (206.7683-mmu)/206.7683*100)
print("m_tau/m_e =", mtau, " PDG 3477.23 err%:", (3477.23-mtau)/3477.23*100)
print("alpha_mu =", log(206.7683)/log(mmu), " alpha_tau =", log(3477.23)/log(mtau))

# 8. mixing-time ratio
tau_star = Lstar/c; tau_e = hbar/(me*c**2)
print("\ntau* =", tau_star, " tau_e =", tau_e, " ratio =", tau_e/tau_star)

# 9. slot mutual information (marginal of slot under admissibility)
joint = Counter()
for st in states:
wgt = exp(-eta_star*K2(st))
joint[(st[0],st[1])] += 2*wgt
tot = sum(joint.values())
pj = {k:v/tot for k,v in joint.items()}
p0 = Counter(); p1 = Counter()
for (a,b),v in pj.items(): p0[a]+=v; p1[b]+=v
I = sum(v*log(v/(p0[a]*p1[b])) for (a,b),v in pj.items())
H0 = -sum(v*log(v) for v in p0.values())
print("\nslot MI =", I, " (paper 0.1545)
H(slot) =", H0, " I/H =", I/H0, " (paper 0.079)")

Appendix K: Fork accounting for the scale chain

Because the substrate length induces a value of G∗near the observed Newton constant, this
appendix records the construction choices behind that result and quantifies the discrete-choice
fork space around it, so the scale chain can be weighed against the freedom available in the
construction. It is referenced wherever G∗is discussed in the main text.

Provenance.
By the author’s recollection, a target entropy near 7.42 was identified early in
the framework’s development by inverting the measured Newton constant, with the ensemble
constructed to deliver it jointly with other structural constraints; the documentary record does
not reach that genesis layer. What the record establishes is this: the realized ensemble was fixed,
and motivated in print by information-independence arguments, roughly ten weeks before the
induced-G∗chain existed; the export factor’s structural components predate the chain, its de-
ployment is contemporaneous with the first landing, and its candidate menu was published with
residuals disclosed; and both recorded routes to G land at one-percent misses rather than exact
values. The Newton constant was the sole observational target of that calibration; the galactic
acceleration scale, the capacity ceiling, the charged-lepton structure, and the committed-phase
abundance are downstream outputs of the calibrated number, compared against measurement
only after fixation. The induced G∗is accordingly treated as a calibration, not a prediction
— with the calibration understood as a discrete selection rather than a continuous adjust-
ment: within the selected construction the entropy is derived with no adjustable parameter,
is reproducible from the stated rules alone, and was never re-adjusted against any subsequent
observable, which is why the downstream misses stand in the record as written. The audit be-
low quantifies how much construction freedom the discrete selection could have consumed. The
evidential weight of the scale chain rests on results obtained after the ensemble was frozen —
the cluster capacity bound and its twenty-four-point test (Section 17.5), the committed-phase

abundance against the measured dark-to-baryonic ratio (Section 18.5), the redshift direction
of the acceleration scale, and the sub-gc rotation-curve regime — none of which entered the
construction.

The audit.
The substrate length depends exponentially on the realized ensemble, L∗∝
e−7gshare,eff, so the calibration is meaningful only relative to the space of constructions that
could have been written instead. This appendix quantifies that space. The grammar varies five
discrete choices around the realized chain: the label count M ∈{4, . . . , 9} with j = (M −1)/2;
the multiplicity rule (ordered or unordered injective selections, with and without parity dou-
bling, and the non-injective M 4); the closure coefficient d ∈{1, 2, 3, 4, 6} in ⟨K2⟩= d/(2η);
the support exponent n ∈{4, M, 2M, M(M −1)/2}; and the export prefactor in { 1

2, 2

3, 1, 3

2, 2}.
This yields 3300 candidate constructions, each evaluated end to end: ensemble, closure, entropy,
induced scale.

The results are as follows. The induced G∗scatters across the grammar with a standard devi-
ation of 137 natural-log units. Within ten percent of the measured value lie four constructions,
and all four are the realized one, differing only in the closure coefficient, which shifts gshare,eff
by less than 0.1% because the closure weighting is a small correction to ln Ωtet; the closure co-
efficient therefore carries almost no load. The nearest structurally distinct construction misses
by eleven percent, and within a factor of two of the measured value lie only two construction
families in the entire grammar. One further menu member belongs in the ten-percent set: re-
placing the export prefactor with the quantum-speed-limit coefficient π/2, the value a saturated
orthogonality bound would select, lands at +8.5%; the two-percent landing remains unique to
the transverse-export reading. At the same time, the menu density implies that roughly one
family is expected to land near any target by chance alone, so the numerical landing by itself
carries little evidential weight. Any residual weight would reside in the a priori force of the dis-
crete arguments that select the realized construction — the seven-state face sector (Section 5),
the parity doubling of the orientation classes, the ordered injective assignment over distinguish-
able faces, the seven-fold support exponent (Appendix D.4), and the transverse export factor
(Appendix C.5). Given the provenance above, these arguments are historically retrodictive, and
the framework’s evidential case is not rested on them here; it is rested on the post-fixation tests
listed in the provenance statement.

The same audit redistributes the evidential weight of the galactic scale.
Because a0 =
(gshare,eff/4π2) cH0 depends on the ensemble only logarithmically, sixty-three percent of the en-
tire grammar lands a0 within twenty-five percent of the observed value: the galactic acceleration
scale tests the coupling form, not the realized ensemble. The induced G∗, with its exponential
sensitivity, is the sharp test of the ensemble itself, with the charged-lepton ladder its weaker
corroborator. The audit script is maintained alongside the reproduction script of Appendix J.3
at jaigp.org.

Appendix L: Why the saturated CMB carrier must be committed
capacity, not a lagged response

The microwave background requires a gravitating component that carries the growing mode of
the baryon perturbations while rejecting their acoustic oscillation. This appendix records the
quantitative exclusion of every relaxational realization of that component and the no-go that
selects the constraint reading of Section 18.5.

Growth-limited kernels.
For development dynamics dW/dt = (Wmax −W)/τ with τ =
β tff(ρlocal), the measured cluster radial decline of the source weight fixes β ≃15.6, while satu-

ration of the cosmological bath by recombination requires β ≲0.56: the window is empty by a
factor of order thirty. The radial ratio between R500 and R200, a β-independent prediction of
the free-fall clock, agrees with the data at the several-percent level, so the exclusion is of the
absolute clock, not of the radial structure.

Lagged response.
A first-order lag transmits a fraction T = [1+(ωτ)2]−1/2 of an oscillation at
frequency ω. The transmitted component acts as additional effective baryon loading (1/ϵ −1) T
in the acoustic driving; Einstein–Boltzmann computation shows a 0.3% temperature-spectrum
tolerance bounds T ≤2×10−3, requiring β ≳37 against the development requirement β ≲0.56:
empty by a factor of order sixty-six. More generally, by recombination a third-peak mode has
completed only a few oscillations, so no causal filter of any order achieves the required rejection.

Sound-speed classification.
For L = f(X), perturbations carry c2
s = fX/(fX + 2XfXX).
The released branches give c2
s =
1
2 (deep) and 1 (Newtonian): free-streaming.
Plateau ap-
proaches f = fmax −A/Xn give c2
s = −1/(2n+1): gradient-unstable. Both natural cap readings
— stationarity of the canonical momentum and of the energy density — impose fX+2XfXX = 0,
the pole of c2
s: a rigid medium carrying no perturbation. The only locus with c2
s = 0 and healthy
density response is the constraint surface of the mimetic class, X pinned with a multiplier, which
is the reading of Section 18.5.

No-go for relaxation-plus-cap carriers.
Within dynamics consisting of relaxation toward
a demand together with a hard cap, with the cell weight as the only state variable, no coupling
of the weight to the demand carries a secular component of the modulation while rejecting
its oscillation: a pinned weight retains no memory of the modulation, and any instantaneous
coupling of bounded periodic inputs is itself bounded and periodic. A secular component requires
an additional conserved integrating variable. The conserved spatial density of committed cells
is that variable, and it is native to the saturated phase rather than added to it.

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