Entropy Selects the Reference Geometry: Selection-Stitch Model Gravitational Phase Transitions

Entropy-Producing Crystallization Selects a Reference Geometry:

Detailed Balance on a Pre-Geometric Bond Complex

Raghu Kulkarni

raghu@idrive.com

SSMTheory Group, IDrive Inc., Calabasas, CA 91302, USA

Abstract

The Selection–Stitch Model (SSM) treats spatial geometry as the macroscopic phase of

a relational bond complex, whereas Gravity from Entropy (GfE) begins with a zero-entropy

reference metric and formulates gravity through quantum relative entropy [

]. We connect

these frameworks without identifying microscopic assembly with gradient ﬂow of the continuum

gravitational action. The microscopic state space consists of ﬁnite bond complexes reachable

from a minimal seed by reversible stitch and lift moves. A Metropolis–Hastings transition kernel

is deﬁned so that proposal asymmetries and kinetic lift suppression are separated from the

equilibrium weight. On every ﬁnite reachable state space the chain is reversible with stationary

measure

−1

exp



ϵB(C)



where

(

) is bond number. Under a removable-seed condition the chain is irreducible and

aperiodic, hence converges uniquely to

. Exact enumeration of history-resolved stitch–lift

complexes through

= 8 veriﬁes connectedness, reversibility, detailed balance, and stationarity

on state spaces up to 588 conﬁgurations, directly measures the bond-deﬁcit entropy, and conﬁrms

that informational lift suppression changes kinetics but not the stationary measure. We then prove

a conditional concentration theorem: if the number of conﬁgurations with bond deﬁcit

grows

no faster than

∗

relative to the maximally bonded sector, then for

ϵ > s

∗

the stationary

measure concentrates exponentially on maximally bonded complexes. This concentration is

established for the construction-bond count of the enumerated model, while close packing is

deﬁned by the unit-distance contact count. A second exhaustive enumeration, of embedded

stitch–lift conﬁgurations for

n ≤

12, locates the relation between the two: their maxima coincide

at 3

n −

6 for

n ≤

9, with every construction-bond maximizer also a contact-bond maximizer,

and separate from

= 10 onward, where the contact maxima 3

n −

5, 3

n −

4, 3

n −

3 reproduce

the known maximal contact numbers of hard-sphere clusters and the maximizers acquire the

octahedral cells that the construction-bond sector cannot produce. Geometry is reconstructed

only after this statistical selection, from the local bond second moment. We further formulate

an oriented-crystallization-wave theorem for an explicitly chirality-transporting Lift operator:

the layer registry obeys a ﬁxed

phase increment, forcing ABC stacking or its mirror ACB

sequence. On that FCC branch the twelve bond directions satisfy

jµ

jν

= 4

µν

, yielding

the identity reference metric, while the regular tetrahedral–octahedral cellulation has exactly

vanishing spatial Regge edge deﬁcits because 2

arccos

3) + 2

arccos

(

−

3) = 2

. Thus a ﬂat

isotropic spatial reference geometry can emerge as the equilibrium phase of an entropy-producing

pre-geometric dynamics. The relational degrees of freedom, local move rules, and bond scale

remain microscopic inputs; what emerges is spatial geometry without a prior metric manifold.

Corresp onding author. E-mail: raghu@idrive.com

1 Introduction

Two ideas motivate this work. In Gravity from Entropy (GfE), the gravitational action is expressed

through quantum relative entropy and a reference metric with unit eigenvalues [

]. In the

Selection–Stitch Model (SSM), unit relational bonds assemble into a close-packed structure whose

normalized bond tensor is the identity [

]. The natural question is whether the latter can

dynamically generate the former.

A naive statement would be that the SSM assembly follows gradient ﬂow of the GfE action. That

statement is neither required nor supported by the microscopic rules. Dissipative ordering is usually

controlled by a stochastic transition kernel, while a continuum action describes the macroscopic

ﬁeld theory near an emergent phase. The correct bridge is therefore statistical:

pre-geometric stochastic dynamics −→ equilibrium concentration −→ emergent metric.

The analysis is organized around statements whose logical status can be stated explicitly. The

main results are:

1. a complete ﬁnite-state transition kernel with exact detailed balance;

2. an ergodicity theorem on the reachable state space from a ﬁxed seed;

a low-temperature concentration theorem conditional on an explicit conﬁgurational-entropy

bound;

4. exact ﬁnite-state veriﬁcation through N = 8, including deﬁcit-sector counts;

5. reconstruction of the reference metric from the selected close-packed phase;

6. an exact FCC spatial-ﬂatness lemma from tetrahedral–octahedral dihedral closure.

Here “entropy builds spacetime” is shorthand for a narrower claim: entropy-producing micro-

scopic dynamics selects a macroscopic phase carrying a spatial metric and vanishing spatial Regge

curvature. The microscopic degrees of freedom and rules are assumed, not generated. Table 1

records the logical status of each claim.

Table 1: Logical status of the main claims. “Exact” means proved for the stated model; “veriﬁed”

means exhaustive ﬁnite enumeration; “conditional” names the remaining thermodynamic or geomet-

ric hypothesis explicitly.

Claim Status

Metropolis–Hastings reversibility and stationary bond

measure

Exact, all ﬁnite reachable

spaces

Finite-state irreducibility under the exposed-node condi-

tion

Exact; condition exhaus-

tively veriﬁed for N ≤ 8

Maximum bond count

max

= 3

N −

6 and bond deﬁcit

equals stitch count for states at the node cutoﬀ

Exact; general form

3(N − n) + s

Stationary measure independent of lift proposal suppres-

sion ρ > 0

Exact; veriﬁed for

1, 0.1, e

−3

Exponential concentration under the deﬁcit-entropy

bound

Exact conditional theorem

Concentration veriﬁed for the construction-bond count

hist

; close packing requires the contact count B

cont

Maxima coincide for

n ≤

separate at

= 10; veriﬁed

n ≤ 12 (Sec. 7)

Uniform thermodynamic deﬁcit-entropy bound Open

Maximally bonded embedded SSM states approach Bar-

low close packing

Veriﬁed for

n ≤

12; open as

n → ∞

FCC bond tensor is the identity metric; FCC spatial edge

deﬁcits vanish

Exact geometric lemmas

Lorentzian 3 + 1 continuum GfE limit Open

2 Pre-geometric state space

Deﬁnition 2.1 (Relational conﬁguration). A microscopic conﬁguration is a ﬁnite object

C = (V, E, σ),

where

is a ﬁnite set of nodes,

E ⊆ {{u, v}

u, v ∈ V }

is a set of undirected bonds, and

stores

only the local combinatorial information required to identify admissible stitch and lift moves. No

background coordinates, metric tensor, or curvature ﬁeld is included in the fundamental state.

The local rules are inherited from the SSM construction. A stitch adjoins a node along an

admissible occupied edge, and a lift adjoins a node to an admissible occupied triangular face. Their

geometric realizations at unit bond length are reconstructed after the combinatorial move has been

speciﬁed. Reverse moves delete exposed nodes while preserving the ﬁxed seed.

Deﬁnition 2.2 (Minimal seed). Let

be a ﬁxed oriented triangle. This is the smallest seed on

which both stitch and lift proposals are deﬁned. The seed is not spacetime: it is a ﬁnite relational

boundary condition that prevents the empty-state obstruction.

Deﬁnition 2.3 (Finite reachable state space). For a node cutoﬀ N ≥ |V (C

)|, let

Ω

)

be the set of conﬁgurations reachable from

by admissible stitch, lift, and reverse moves without

exceeding N nodes.

The cutoﬀ makes the state space ﬁnite. The thermodynamic question is then formulated as a

sequence of ﬁnite problems followed by N → ∞.

3 Reversible stochastic dynamics

Let B(C) = |E(C)| be the bond count and assign the microscopic energy

with bond scale

ϵ >

0. Kinetic suppression is placed in the proposal kernel rather than in the

equilibrium energy. This allows an informationally suppressed lift to remain energetically downhill.

For each ordered pair

C, C

′

∈

Ω

(

) connected by one elementary move, let

(

C, C

′

) be the

proposal probability. The function

may distinguish stitch and lift attempts and may contain the

lift-exposure suppression. We assume only that

Q(C, C

′

) > 0 ⇐⇒ Q(C

′

, C) > 0.

Deﬁne the acceptance probability

A(C, C

′

) = min



1, exp



−β



′

) − E

(C)



Q(C

′

, C)

Q(C, C

′

)



, (1)

where

= (

)

−1

. This is the Metropolis–Hastings construction [

]. The transition kernel is

P (C, C

′

) = Q(C, C

′

)A(C, C

′

)

for C = C

′

, with P (C, C) chosen so that each row sums to unity.

Theorem 3.1 (Detailed balance). The kernel P satisﬁes detailed balance with respect to

exp



ϵB(C)



, Z

C∈Ω

)

ϵB(C)

. (2)

Proof.

For adjacent conﬁgurations, Eq.

(1)

is the standard Metropolis–Hastings acceptance rule.

Direct substitution gives

(C)Q(C, C

′

)A(C, C

′

) = π

′

)Q(C

′

, C)A(C

′

, C).

For nonadjacent pairs both sides vanish. Hence detailed balance holds for every pair.

Remark 3.2. The local logistic identity

−q

) =

−β

is recovered when proposals are symmetric

and a reverse move removes one bond with

. The full kernel above is needed when the

numbers of eligible birth and deletion proposals diﬀer.

4 Ergodicity on the reachable complex

Detailed balance identiﬁes a stationary measure but does not by itself guarantee convergence. The

relevant state space is not the set of all connected graphs; it is the set reachable from the chosen

seed by the allowed moves.

Assumption 4.1 (Removable exposed node). Every conﬁguration

C ∈

Ω

(

) with

C 

contains at least one exposed nonseed node whose deletion is an admissible reverse stitch or reverse

lift and remains in Ω

This assumption is a precise version of reversibility of construction histories. It can be checked

algorithmically on ﬁnite enumerated state spaces.

Theorem 4.2 (Finite-state ergodicity). Under Assumption 4.1, if

(

C, C

)

0 for every

, then

the Markov chain on Ω

(

) is irreducible and aperiodic. Therefore

is the unique stationary

measure and the distribution converges to π

from every initial state.

Proof.

Repeated deletion of exposed nonseed nodes reaches

in ﬁnitely many steps. Reversing a

construction history reaches any target

′

∈

Ω

(

) from

. Thus any two states communicate

through the seed, proving irreducibility. A positive hold probability gives a self-loop at every state

and therefore aperiodicity. A ﬁnite irreducible aperiodic chain has a unique stationary measure and

converges to it.

5 Entropy production and local ordering

For a transition C → C

′

, deﬁne the stochastic system-entropy change

∆s

sys

= −k

log π

′

) + k

log π

(C)

and the medium entropy change

∆s

med

= k

log

P (C, C

′

)

P (C

′

, C)

Detailed balance gives

∆s

sys

+ ∆s

med

= 0

for an equilibrium trajectory pair. During relaxation from a nonequilibrium distribution

, the

relative entropy

D(ρ

∥π

) =

(C)

is nonincreasing under the Markov evolution. Hence local bond order can increase while entropy is

exported to the bath.

It is worth separating the three roles precisely, since the phase selection is often attributed

loosely to entropy alone. Detailed balance together with the bond energy

−ϵB

determines the

stationary measure

; entropy production drives relaxation toward it; and a free-energy competition

between the bond energy and the conﬁgurational entropy of the deﬁcit sectors determines which

macrostate dominates

. In that competition the conﬁgurational entropy opposes concentration

and the bond energy drives it, which is the content of the condition

ϵ > s

∗

in Theorem 6.2 below.

Entropy production is what makes the relaxation irreversible and the equilibrium phase attainable;

it does not by itself invent the microscopic degrees of freedom, the energy scale

, or the direction of

the selection.

6 Low-temperature concentration

Let

max

(N) = max

C∈Ω

)

B(C),

and deﬁne the bond-deﬁcit sectors

Ω

N,m

= {C ∈ Ω

) : B(C) = B

max

(N) − m}.

Assumption 6.1 (Deﬁcit entropy bound). There exist constants

∗

< ∞

and

≥

1 with

log a

= o(N) such that for all m ≥ 0,

|Ω

N,m

| ≤ a

|Ω

N,0

∗

. (3)

Theorem 6.2 (Concentration on maximal bonding). If Eq.

(3)

holds and

ϵ > s

∗

, then for every

M ≥ 1,



max

(N) − B(C) ≥ M



≤

−(β

ϵ−s

∗

1 − e

−(β

ϵ−s

∗

)

. (4)

Thus the tail is exponentially suppressed in the deﬁcit threshold

for each ﬁnite

. A thermodynamic

concentration statement requires control of the prefactor a

and of the scaling of M with N.

Proof. Relative to the maximally bonded sector,

(Ω

N,m

)

(Ω

N,0

)

|Ω

N,m

|Ω

N,0

−β

ϵm

≤ a

−(β

ϵ−s

∗

Summing the resulting geometric series over m ≥ M gives Eq. (4).

The theorem shows why bond energy alone is insuﬃcient: conﬁgurational entropy must also

be controlled. For ﬁnite systems the bound can be measured exactly by enumerating reachable

conﬁgurations. For the inﬁnite system it becomes a combinatorial or cluster-expansion problem.

7 Two levels of bond counting

The concentration theorem is a statement about whatever bond count enters the energy

−ϵB

Two distinct counts appear in this paper, and the passage from one to the other is assumed rather

than proved. We state the distinction explicitly here so that it can be treated as a single identiﬁed

problem rather than a series of local caveats.

Deﬁnition 7.1 (Construction bonds). For a history-resolved conﬁguration C ∈ Ω

), let

hist

be the number of bonds introduced by the stitch and lift moves of its construction history.

Deﬁnition 7.2 (Contact bonds). For an embedded conﬁguration

} ⊂ R

at unit bond

length L, let

cont

(X) = #{{i, j} : |x

− x

| = L}

be the number of unit-distance contacts, whether or not the pair was joined by a construction move.

Theorem 6.2 and the ﬁnite enumeration of Section 12 are carried out for

hist

. The geometric

conclusions of Sections 8–10 concern maximizers of

cont

, since it is unit-distance contact, not

construction history, that deﬁnes a close packing. These are diﬀerent functionals, and the relation

between their maximizers is not a matter of interpretation: it can be computed. We do so here.

7.1 Embedded enumeration

Both stitch and lift have unique apexes once a face and a side are ﬁxed, so a construction history

determines an embedding. The conﬁguration of Deﬁnition 2.1 carries no coordinates, so

cont

is not

deﬁned on it directly; placing the seed triangle in

and applying Eq.

(7)

and its planar analogue

recursively, with the overlap exclusion of Section 12 enforced at each step, yields for each history a

point set

X ⊂ R

on which it is. States are identiﬁed up to isometry, including reﬂection, by color

reﬁnement on the weighted complete graph followed by minimization over residual color classes.

The two enumerations in this paper count diﬀerent objects and their state totals should not

be compared directly. The history-resolved enumeration of Section 12 ranges over Ω

(

), which

contains every state with at most

nodes, and identiﬁes states by isomorphism of the colored

incidence graph of the abstract complex. The embedded enumeration ﬁxes the node count

imposes the overlap exclusion, and identiﬁes states by isometry of the point set. The gap between

them is informative rather than incidental: at

= 8 the combinatorial model has 464 states with

exactly eight nodes, while only 303 admit an admissible embedding, and the corresponding pairs

= 5

7 are 6

3, 21

12 and 94

55. Roughly a third of the abstract complexes are therefore

not geometrically realizable under the exclusion rule, which is a direct measure of how much the

embedding constraint adds.

Proposition 7.3 (Coincidence below the crossover). For

n ≤

9, the two functionals have the same

maximum on the embedded stitch–lift state space,

max B

hist

= max B

cont

= 3n − 6,

and every maximizer of

hist

is also a maximizer of

cont

. At

= 10 the maxima separate and the

maximizer sets become disjoint. Veriﬁed by exhaustive enumeration for 4 ≤ n ≤ 12.

Table 2 reports the enumeration. The construction-bond maximum is 3

n −

6 at every size, as

Proposition 12.1 requires. The contact-bond maximum departs from it at

= 10 and the gap opens

linearly thereafter.

Table 2: Embedded enumeration of stitch–lift reachable conﬁgurations. The two bond counts share

a maximum through

= 9 and separate from

= 10. Degeneracy of the contact maximum collapses

as the gap opens. Octahedral cells are counted as 6-subsets carrying 12 unit contacts with every

vertex of degree four.

n states max B

hist

max B

cont

form gap |arg max B

cont

| octahedra

9 1 851 21 21 3n − 6 0 31 0–1

10 12 061 24 25 3n − 5 1 3 1–2

11 80 396 27 29 3n − 4 2 1 2

12 543 827 30 33 3n − 3 3 1 2

Three features of the table bear on the argument. First, the coincidence for

n ≤

9 recorded

in Proposition 7.3 shows that the ﬁnite veriﬁcation of Sections 6 and 12 does not test the wrong

object at the sizes where it is performed: within its own range the construction-bond Hamiltonian

selects contact-bond maximizers. Second, the separation from

= 10 is genuine and grows, and the

maximizer sets are disjoint rather than merely unequal; at

= 12 the unique contact maximizer

has

hist

= 23 against a construction maximum of 30. Third, the separation is accompanied by a

change of cell type. Every contact maximizer from

= 10 onward contains octahedral cells, while no

construction maximizer contains one at any

, for the structural reason given by Proposition 12.1:

the sector Ω

N,0

consists of all-lift histories and a lift produces only tetrahedra. Such all-tetrahedral

complexes are moreover geometrically strained, since a pure tetrahedral fan around an edge cannot

close and leaves the residual deﬁcit

of Eq.

(6)

. Since the spatial ﬂatness of Lemma 9.1 rests

on the tetrahedral–octahedral honeycomb, in which two tetrahedra and two octahedra meet at

every edge, the octahedral cell is exactly the structure the geometric argument requires and the

construction-bond sector cannot supply.

7.2 Agreement with the sphere-packing literature

The crossover is not an artifact of the stitch–lift move set. The same transition was found by

Arkus, Manoharan and Brenner [

] for clusters of hard spheres with short-range attraction, by

graph-theoretic enumeration combined with geometry rather than by lattice growth. They report

that ground-state degeneracy grows exponentially for

n ≤

9 but decreases for

n >

9 through the

appearance of packings with more than 3

n −

6 contacts, and identify octahedra and half-octahedra

as the structures responsible. Their maximal contact numbers are 3

n −

5, 3

n −

4 and 3

n −

3 at

= 10

12, which are the values in Table 2. The degeneracy collapse is likewise reproduced, from

31 contact maximizers at

= 9 to a unique one from

= 11 onward. The contact-number problem

for small n is reviewed by Bezdek and Khan [7].

That the stitch–lift moves attain the known optima is itself informative: the move set of Section 2

is not too impoverished to reach the contact-maximizing conﬁgurations, so the separation in Table 2

reﬂects the choice of Hamiltonian and not a limitation of the dynamics.

7.3 What this settles, and what it does not

Assumption 10.1 asserts that maximally bonded embedded states approach Barlow close packing.

The enumeration supports it in the range computed: Arkus et al. ﬁnd the maximal-contact clusters

= 10

12 to be subsets of close-packed crystals, and the present enumeration reaches those

same conﬁgurations. The assumption is therefore no longer bare; it is veriﬁed at small

and open

in the thermodynamic limit.

One qualiﬁcation is essential and works in the paper’s favor. The small-

maximal-contact

clusters are reported as subsets of hexagonal close packing. At twelve spheres a cluster is far too

small to distinguish FCC from HCP: both are Barlow packings, and small fragments are common to

the two. Contact maximization therefore delivers close packing but does not by itself select FCC.

This is precisely the tie that the chirality-transporting Lift of Section 10 breaks. Assumptions 10.1

and 10.4 are consequently not two hedges on the same step but a decomposition of the geometric

problem into distinct parts: the contact Hamiltonian selects Barlow packing, and the transported

frontier orientation selects the ABC branch within it.

What remains open is the thermodynamic statement. Twelve nodes cannot exhibit an FCC bulk

node, which requires twelve neighbors together with populated second and third shells. Establishing

that the reachable maximizers of

cont

approach Barlow packings as

n → ∞

, and reformulating the

detailed-balance construction of Section 3 with

cont

in place of

hist

, are the two remaining steps

between the stochastic mechanism proved here and the geometric conclusion. Neither is addressed

in this paper.

8 From relational order to an eﬀective metric

The metric is not fundamental in the microscopic state. It is reconstructed from the dominant bond

geometry after an embedding compatible with unit bonds has been selected.

For a node x with bond directions n

(x) ∈ R

, deﬁne the local second-moment tensor

µν

(x) =

z(x)

j=1

jµ

(x)n

jν

(x), g

eﬀ

µν

(x) =

z(x)

µν

(x). (5)

The normalization makes g

eﬀ

= I for an isotropic set of z unit directions.

Lemma 8.1 (FCC reference metric). For the twelve nearest-neighbor directions of the FCC lattice,

µν

= 4δ

µν

, g

eﬀ

µν

= δ

µν

Proof.

The twelve FCC directions are the normalized vectors obtained from permutations of

(

, ±

0). Oﬀ-diagonal contributions cancel by sign symmetry. Each coordinate is nonzero in eight

directions with squared component 1/2, giving diagonal sum 8/2 = 4.

Thus the FCC branch realizes the identity spatial reference metric through a local bond moment,

rather than by stipulation.

9 Spatial Regge ﬂatness of the FCC branch

The current argument is an equal-time spatial statement. In three-dimensional Regge calculus the

hinges are edges [9], with action

(3)

ℓ

This avoids mixing the spatial edge-deﬁcit calculation with a four-dimensional triangular-hinge

action.

Lemma 9.1 (Tetrahedral–octahedral closure). In the regular FCC tetrahedral–octahedral honeycomb,

two regular tetrahedra and two regular octahedra meet around every edge. Their internal dihedral

angles satisfy

= arccos(1/3), θ

= arccos(−1/3) = π − θ

2θ

+ 2θ

= 2π.

Therefore every spatial Regge edge deﬁcit vanishes and S

(3)

= 0.

Proof.

The incidence structure gives two tetrahedra and two octahedra per edge. Since the two

dihedral angles are supplementary, the total angle is exactly 2π, hence δ

= 0 for every edge.

By contrast, ﬁve regular tetrahedra leave the positive deﬁcit

= 2π − 5 arccos(1/3) ≈ 0.1284. (6)

This local identity is exact. It does not require the false general implication that maximal rigidity

always implies ﬂatness.

10 Directed crystallization and FCC selection

The concentration theorem selects conﬁgurations of maximal bond number inside the reachable

state space. To identify their geometry, ﬁrst assume that completed Stitch growth produces ﬂat

equilateral triangular sheets and that Lift joins successive sheets through tetrahedral holes.

Assumption 10.1 (Close-packed sheet classiﬁcation). Every translation-invariant accumulation

point of maximally bonded embedded SSM conﬁgurations is a stack of complete triangular sheets,

with each new sheet occupying one of the two hollow registries of the preceding sheet.

This assumption is stronger than the density result of the Kepler conjecture [

]: it concerns the

structure of the SSM-reachable maximally bonded states. It is veriﬁed for

n ≤

12 by the embedded

enumeration of Section 7, in agreement with the maximal-contact clusters of Arkus et al. [

], and

remains open in the thermodynamic limit. Note that the small-

veriﬁcation delivers close packing

but not the choice between its stacking isomers; that choice is made by the chirality transport

introduced below. Under the assumption, the registry of sheet n can be encoded by

∈ Z

= {0, 1, 2},

corresponding to the conventional A, B, C lateral registries. Close-packed attachment requires

n+1

− s

∈ {+1, −1} (mod 3).

A propagation direction selects the outward side of Lift, but FCC selection requires the crystallization

front to transport its orientation as well. We represent that transported orientation by the stacking

chirality

= s

n+1

− s

(mod 3) ∈ {+1, −1}.

10.1 Operator-level transport of stacking orientation

Let an oriented equilateral frontier face be the ordered triple

f = (x

, x

with local frame

− x

, e

− x

√

, n

= e

× e

The outward geometric Lift is

L(f) = x

+ x

L n

, (7)

the unique point on the selected side satisfying

−x

for

= 0

2. The normal ﬁxes the side

of growth, but Eq.

(7)

alone does not distinguish the two close-packed continuations of a two-layer

stack. That distinction is carried by a discrete frontier chirality.

Deﬁnition 10.2 (Chirality-transporting Lift). An oriented frontier state is a pair (

f, χ

) with

χ ∈ {

, −

}

. A Lift produces the apex

of Eq.

(7)

, an ordered daughter frontier

′

, and

transports the same chirality:

: (f, χ) 7−→ (x

, f

′

, χ). (8)

On completed triangular sheets, the induced registry map is required to be equivariant with the

cyclic registry labels,

(s) = s + χ (mod 3). (9)

An odd permutation of the ordered frontier changes χ and is recorded as a stacking-fault event.

Proposition 10.3 (Operator transport law). For the oriented operator deﬁned by Eqs.

(8)

–

(9)

, the

stacking increment is conserved along every defect-free growth history:

n+1

= χ

= χ.

Proof.

The frontier state includes

, and the transport rule leaves this component unchanged.

Equation

(9)

therefore applies with the same sign at every defect-free Lift. A sign change can

occur only through an explicitly orientation-reversing update, which by deﬁnition belongs to the

stacking-fault set.

The operator statement is deliberately explicit. The geometry of Eq.

(7)

ﬁxes the tetrahedral

apex, while Eq.

(9)

speciﬁes how a coherent crystallization front transports in-plane registry

information. Cyclic face ordering by itself would not select between the two nontrivial rotations

; the conserved sign is additional state carried by the front. No stacking-dependent energy is

introduced, but the microscopic implementation must preserve this state for FCC selection to follow.

Assumption 10.4 (Coherent oriented Lift). The microscopic crystallization dynamics realizes the

operator of Eqs. (8)–(9) away from a stacking-fault set.

Theorem 10.5 (Directed-wave FCC theorem). Under Assumptions 10.1 and 10.4, every defect-free

connected stack generated from two adjacent registries is cubic close packed. More precisely,

= s

+ nχ (mod 3), χ ∈ {+1, −1}.

For

= +1 the sequence is

ABCABC ···

; for

−

1 it is

ACBACB ···

. These are the two

mirror orientations of FCC stacking.

Proof.

Close-packed attachment gives

n+1

modulo three. Proposition 10.3 makes

independent of

. Induction then gives

nχ

modulo three. The two possible signs generate

the two cyclic three-layer sequences, both crystallographically FCC. An HCP sequence

ABAB ···

would require the increment to alternate +1

, −

, . . .

and hence would reverse the transported

orientation at every layer, contradicting Assumption 10.4.

The normal vector chooses forward rather than backward growth. FCC selection additionally

requires the one-bit chirality carried by the advancing front; without that transported state, the

local tetrahedral geometry also permits HCP continuation.

Theorem 10.5 should therefore be read as a minimal-suﬃciency result rather than a derivation

of FCC from geometry alone. Registry-increment conservation is built into the operator of Deﬁni-

tion 10.2, so the induction that follows is not where the content lies. The content is the identiﬁcation

of exactly what must be transported: one conserved binary frontier variable is enough to exclude

HCP, an oriented normal alone is not, and no stacking-dependent energy is required. Whether the

microscopic dynamics conserves that variable, and treats its reversal as a stacking fault, remains an

implementation condition and is the substance of Assumption 10.4.

Corollary 10.6 (Thermodynamic FCC bulk). Let

be a sequence of embedded crystallites with

N nodes. Suppose Assumptions 10.1 and 10.4 hold except on a defect set Γ

, and suppose

|Γ

| = o(N ).

Then the fraction of nodes whose local neighborhoods are not FCC tends to zero. Consequently every

bounded local observable converges to its FCC bulk value. In particular,

⟨K⟩

→ 12, g

(N)

µν

→ δ

µν

and the regular tetrahedral–octahedral spatial edge deﬁcits vanish away from a zero-density defect set.

Proof.

Outside Γ

, the directed-wave FCC theorem ﬁxes the registry sequence and hence the FCC

local neighborhood. The fraction of exceptional nodes is at most

|/N →

0. A bounded local

observable diﬀers from its FCC value only on this vanishing fraction, which proves convergence.

The metric and coordination conclusions follow from the exact FCC bond-moment lemma, and

spatial Regge ﬂatness follows from tetrahedral–octahedral dihedral closure.

Combining the results gives the paper’s central conditional statement: the equilibrium measure

favors maximally bonded complexes, and a coherent chirality-transporting front ﬁxes their close-

packed bulk registry to FCC.

11 Relation to Gravity from Entropy

The GfE theory uses a zero-entropy reference metric with unit eigenvalues. The FCC result supplies

a microscopic candidate for that reference state, with

eﬀ

the local bond second moment of Eq.

(5)

spatial

µν

= lim

coarse grain

eﬀ

µν

= δ

µν

The present derivation should not be read as a proof that the microscopic chain is gradient ﬂow of

the GfE action. Instead, it establishes a matching of stationary macrostates:

supp(π

)

low T

−→ close-packed reference geometry −→ eg.

A full spacetime theorem requires an additional step. One must either derive a discrete temporal

direction and a 3 + 1-dimensional eﬀective action, or work directly on a four-dimensional complex

such as a D4 construction. The three-dimensional edge-deﬁcit lemma proved here is a theorem about

emergent spatial geometry. Lorentzian signature, gravitational propagation, and the continuum

GfE action remain separate problems.

12 Finite-state veriﬁcation program

The assumptions used above are testable on ﬁnite systems. The ﬁnite-state calculations test the

following items:

1. enumerate Ω

) for small N;

2. list every admissible stitch, lift, and reverse move;

3. build the complete proposal matrix Q and transition matrix P ;

4. verify detailed balance entry by entry;

5. test the removable-exposed-node condition;

6. count |Ω

N,m

| and estimate s

∗

;

measure bond number, coordination, stacking order, metric anisotropy, and spatial deﬁcit under

Useful observables include

, E

∥g

eﬀ

− I∥, E

|E|

|δ

A ﬁnite-size crossing or rapid crossover in these observables as a function of

would provide direct

evidence for an ordering transition into the geometric phase.

12.1 Exact enumeration of a history-resolved ﬁnite model

Proposition 12.1 (Exact bond bookkeeping in the ﬁnite model). Consider a history-resolved state

with n nodes obtained from the triangular seed by s stitches and ℓ lifts. Then

n = 3 + s + ℓ, B = 3 + 2s + 3ℓ = 3n − 6 − s.

Consequently

max

(

) = 3

N −

6, attained exactly by all-lift histories at the node cutoﬀ, and the

bond deﬁcit satisﬁes

m = B

max

(N) − B = 3(N − n) + s.

In particular

for states at the cutoﬀ

, while states below the cutoﬀ carry the additional

term 3(N − n).

Proof.

The seed has three bonds. Each stitch adds two bonds and each lift adds three, while either

move adds one node. Substitution of

ℓ

n −

− s

gives

= 3

n −

− s

. Since

s ≥

0 and

n ≤ N

the bond count is maximized precisely at

and

= 0, giving

max

(

) = 3

N −

6. Subtracting

yields m = 3(N − n) + s.

Because Ω

(

) contains every state with at most

nodes, the deﬁcit sectors are graded by

both quantities: the largest deﬁcit at cutoﬀ

is 3(

N −

3), attained by the seed itself, and exceeds

the largest attainable stitch count

N −

3. Restricted to the cutoﬀ layer

, the proposition turns

the deﬁcit into a transparent combinatorial observable: the low-temperature measure suppresses

histories by their number of planar stitch events relative to the all-lift maximal-bond sector. Away

from that layer the suppression also counts missing nodes. This is a theorem of the history-resolved

veriﬁcation model, not a classiﬁcation of the embedded FCC assembly.

The ﬁnite calculation therefore tests detailed balance and concentration using construction bonds

only; its all-lift maximum is not a geometric proxy for a close-packed crystallite. In the embedded

model, the physical bond count includes all unit-distance proximity contacts, and it is that contact

Hamiltonian, together with the crystallization hypothesis, that selects the close-packed bulk.

We implemented the preceding program for a minimal, history-resolved simplicial realization

of the stitch–lift rules. The seed is one oriented triangle. A stitch adds a vertex to a boundary

edge and forms two new bonds and one triangle. A lift adds a vertex to a boundary triangle and

forms three new bonds and one tetrahedron. Reverse moves remove the most recently exposed

nonseed vertex. States are deduplicated up to colored incidence-graph isomorphism, preserving

the seed. This model is deliberately narrower than the full spatial SSM: it tests the Markov and

conﬁgurational-entropy claims without pretending to classify embedded FCC growth.

For cutoﬀs

= 4

, . . . ,

8, the complete reachable state spaces contain 3

124

and 588 states.

In every case the transition graph is connected, every nonseed state has an admissible reverse move,

and the Metropolis–Hastings kernel satisﬁes detailed balance and stationarity to residuals below

−18

. Table 3 gives the exact counts at

= 1

5, and Fig. 2 shows the stationary weight of

the maximally bonded sector across the full range of β

ϵ.

Table 3: Exact ﬁnite-state veriﬁcation for the history-resolved stitch–lift model. The empirical

entropy slope is bs

= max

m>0

−1

log(|Ω

N,m

|/|Ω

N,0

|).

N |Ω

| B

max

|Ω

N,0

| bs

(Ω

N,0

)

4 3 6 1 0.000 0.810

5 9 9 2 0.000 0.781

6 30 12 4 0.405 0.704

7 124 15 16 0.172 0.738

8 588 18 67 0.186 0.739

0 2 4 6 8 10 12 14

Bond deficit m

Number of reachable states

Finite reachable-state deficit counts

N=4

N=5

N=6

N=7

N=8

Figure 1: Exact counts of reachable conﬁgurations by bond deﬁcit. The ﬁnite systems satisfy an

exponential deﬁcit-count bound with modest empirical slopes. This is direct model-speciﬁc evidence

for Assumption 6.1 at the sizes enumerated, not a proof of its thermodynamic-limit form. The

evidence is also thin: the slopes

vanish identically at

= 4

5, so only three of the ﬁve sizes are

informative, and the sequence is not monotone, its largest value occurring at

= 6. No trend in

should be read from these points, and the prefactor a

of Eq. (3) remains uncontrolled.

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.0

0.2

0.4

0.6

0.8

1.0

Stationary probability of maximal bonding

Exact finite-state concentration

N=4

N=5

N=6

N=7

N=8

Figure 2: Exact stationary probability of the maximally bonded sector as a function of

. The

probability approaches unity at low eﬀective temperature for every enumerated cutoﬀ.

The enumeration therefore closes three gaps at ﬁnite size: proposal asymmetry is handled

exactly, irreducibility is checked on the actual reachable graph, and the deﬁcit-sector entropy is

measured rather than assumed. What remains open is uniform control as

N → ∞

and the geometric

classiﬁcation of maximally bonded embedded states.

12.2 Kinetic suppression does not alter equilibrium

To model the informational lift penalty, assign forward lift proposals a relative weight

ρ >

0 while

stitch and reverse proposals retain unit weight, then normalize each proposal row. The Metropolis–

Hastings acceptance factor includes the reverse-to-forward proposal ratio. Therefore Theorem 3.1

applies for every

ρ >

0 and the stationary measure remains Eq.

(2)

; only relaxation times and path

statistics can change.

We veriﬁed this statement exhaustively for

= 1, 0

1, and

−3

on every enumerated state

space. At

= 1

5, the maximal-sector probabilities are identical to machine precision for all

three proposal kernels, while detailed-balance and stationarity residuals remain below 6

−17

(Fig. 3). This supports the distinction used throughout the paper: lift suppression belongs to

kinetics, whereas bond energy ﬁxes equilibrium.

Forward lift proposal weight

0.0

0.2

0.4

0.6

0.8

1.0

(

, 0

) at

= 1.5

Kinetic suppression changes proposals, not equilibrium

N=4

N=5

N=6

N=7

N=8

Figure 3: Changing the forward lift proposal weight over more than an order of magnitude, including

the physical value

−3

, leaves the exact stationary probability of maximal bonding unchanged.

The Metropolis–Hastings correction separates kinetic suppression from equilibrium selection.

13 Boundaries of the result

The construction begins without a prior metric manifold, but not without inputs. It assumes:



relational nodes and bonds;



a minimal seed;



local reversible move rules;



a bond scale ϵ;



a stochastic bath and inverse temperature β

Entropy does not explain why these objects exist. It explains why, given them, the dynamics selects

an ordered geometric phase.

The continuum limit also remains open. A mathematically complete bridge to GfE would require

a Γ-convergence, hydrodynamic-limit, or renormalization argument showing that the coarse-grained

stationary functional approaches the continuum relative-entropy action. The present paper isolates

the ﬁnite-state and geometric statements that must precede such a limit. The FCC theorem is

exact for the explicitly chirality-transporting Lift operator. Cyclic face ordering and an outward

normal do not by themselves ﬁx the sign of the registry increment; the front must carry and preserve

that sign. What remains implementation-speciﬁc is to show that the detailed Stitch–Lift dynamics

realizes this transport rule and treats sign reversal as a stacking fault.

14 Conclusion

The model begins with a relational state space rather than a metric manifold. Its stochastic

assembly need not be gradient ﬂow of a continuum gravitational action; the relevant question is

which macroscopic phase the ﬁnite-state dynamics selects.

For ﬁnite reachable bond complexes, the Metropolis–Hastings assembly is reversible and, under

an exposed-node condition, ergodic. Exact enumeration through

= 8 veriﬁes these properties on

the history-resolved stitch–lift state graph and shows low-temperature concentration toward maximal

bonding. Under an explicit bound on conﬁgurational entropy, its stationary measure concentrates

exponentially on maximally bonded states. This is a statement about the construction-bond count

hist

. A second enumeration, of embedded conﬁgurations for

n ≤

12, shows that

hist

and the

contact count

cont

share their maximizers through

= 9 and separate from

= 10, where the

contact maximizers acquire octahedral cells and reproduce the known maximal contact numbers of

hard-sphere clusters. The construction-bond model is therefore adequate at the sizes it is veriﬁed

on, and inadequate beyond them in a way whose onset and mechanism are now known rather than

assumed. If the maximizers of

cont

approach complete close-packed sheets and the microscopic

front realizes the chirality-transporting Lift operator, the directed-wave theorem forces ABC or

mirror ACB stacking within that model. The resulting FCC bulk reconstructs the identity spatial

metric and has exactly zero spatial Regge edge deﬁcits away from a vanishing-density defect set.

Under these stated conditions, the zero-entropy spatial reference geometry used by GfE can arise as

the equilibrium phase of microscopic dynamics with no prior metric manifold. Two steps remain

between the mechanism proved here and that conclusion: the thermodynamic classiﬁcation of

contact-bond maximizers, and the reformulation of the detailed-balance construction with

cont

place of B

hist

A Exact ﬁnite partition functions

Let x = e

. Exhaustive enumeration gives

(x) =

m≥0

|Ω

N,m

max

(N)−m

(x) = x

+ x

(x) = 2x

+ 2x

+ x

(x) = 4x

+ 6x

+ 5x

+ 8x

+ 2x

+ x

(x) = 16x

+ 19x

+ 22x

+ 25x

+ 22x

+ 5x

+ 8x

+ 2x

+ x

(x) = 67x

+ 79x

+ 89x

+ 117x

+ 95x

+ 74x

+ 25x

+ 22x

+ 5x

+ 8x

+ 2x

+ x

The coeﬃcients of

are the complete deﬁcit-sector counts

Ω

N,m

for

= 4

, . . . ,

8, and they sum

Ω

= 3

124

588 respectively. These polynomials make the ﬁnite concentration curves

exact algebraic quantities rather than Monte Carlo estimates. In particular,

(Ω

N,0

) =

|Ω

N,0

max

(N)

(x)

B Reproducibility identiﬁers

The SHA-256 hashes of the delivered veriﬁcation scripts are

finite_ssm_verification_v3.py

1201ae4442a2d4577306a334c42be3e9fa7574c8ee125f60805bd0baf27e7780

run_finite_verification_v3.py

8c79276a8970dc4d720218c62295aed2dac811809a06fbcc1a3b763b5ba32223

embedded_crossover_v1.py

fc174ad18a986d93a20a972e3bcd63e33ed9d166e6b74471f723bbd63a29a02a

Code availability

The ﬁnite-state veriﬁcation scripts are publicly available in the SSMTheory GitHub

repository:

finite ssm verification v3.py

run finite verification v3.py

, and

embedded crossover v1.py

. The ﬁrst two enumerate reachable simplicial stitch–lift complexes,

deduplicate states by colored incidence-graph isomorphism, build the full Metropolis–Hastings

transition matrix, verify detailed balance and stationarity, and generate Table 3 and Figs. 1–3. The

third performs the embedded enumeration of Section 7 and generates Table 2. In our tests, the

history-resolved calculation through

= 8 completed in about ten seconds, and the embedded

enumeration through

= 11 in under two minutes, on a contemporary laptop using Python,

NumPy, NetworkX, and Matplotlib. The

= 12 level of the embedded enumeration requires

roughly twenty minutes and several gigabytes.

Declarations

Funding. No funding was received for conducting this study.

Conﬂict of interest. The author declares no competing interests.

Data availability. All data supporting this study are contained within the article. The complete

deﬁcit-sector counts

Ω

N,m

for

= 4

, . . . ,

8 are the coeﬃcients of the partition polynomials in

Appendix A, and the reachable-state counts, stationary-measure values, and detailed-balance and

stationarity residuals are reported in Table 3. These quantities are exact enumerations rather than

samples, and are regenerated deterministically, with no random seed, by the veriﬁcation scripts

cited under Code availability. No experimental or third-party data were used.

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