A Master Equation for the Selection-Stitch Model

Entropy-Producing Crystallization Selects a Flat Spatial Vacuum:

Detailed Balance and Local Close Packing in Stitch–Lift Assembly

Raghu Kulkarni

raghu@idrive.com

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

Abstract

The Selection–Stitch Model (SSM) treats spatial geometry as the large-scale phase of a

network of unit bonds. Gravity from Entropy (GfE) writes the gravitational action as a quantum

relative entropy between the spacetime metric and a metric induced by matter and curvature [

We ask whether SSM dynamics can produce a ﬂat spatial vacuum, the conﬁguration at which the

GfE action vanishes. The states are point sets built from a seed triangle by two moves, Stitch and

Lift. The energy is minus the number of unit-distance contacts. A Metropolis–Hastings chain on

these states is reversible and irreducible, and its stationary measure is

(

)

∝ exp

[

ϵB

cont

(

)].

We verify this exactly for all 16 129 states with

N ≤

10 points. At

= 10 and 11, every contact

maximizer is a fragment of a close packing. In the thermodynamic limit, vacancy entropy rules

out a uniform entropy bound when the number of maximizers grows subexponentially. Under

a Peierls-type counting assumption, the density of contact defects falls as

−c β

. The local

content of Hales’ proof of the Fejes T´oth contact conjecture then shows that all but this density

of points sit in FCC or HCP environments. There the eﬀective metric is the identity and space

is ﬂat. Energy does not ﬁx the stacking. The Stitch move is a point inversion, so Stitch growth

stays on one face-centered cubic (FCC) lattice. The growth simulation of a companion paper

conﬁrms this: 97% of stacking triplets are cubic. We do not derive the GfE action.

1 Introduction

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

quantum relative entropy between the metric of spacetime and a metric induced by matter ﬁelds

and curvature [

]. The action vanishes in the vacuum with no matter and no curvature. In the

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

bond tensor is the identity [

]. We ask whether such microscopic dynamics can produce a ﬂat

vacuum geometry without assuming a metric manifold at the start.

One might guess that SSM assembly follows the gradient ﬂow of the GfE action. The microscopic

rules do not support that guess, and we do not need it. Ordering is controlled by a stochastic

transition kernel. A continuum action describes the large-scale theory near an emergent phase. The

bridge is statistical:

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

The main results are:

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

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

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

a ﬁnite-size concentration theorem; a proof that vacancy entropy rules out concentration

on exact maximizers in the thermodynamic limit, when the number of maximizers grows

subexponentially; and a Peierls-type theorem that makes an extensive contact-deﬁcit density

exponentially unlikely;

a bound on the number of points outside close-packed environments in terms of the contact

deﬁcit, from the local content of Hales’ proof of the Fejes T´oth contact conjecture;

exact veriﬁcation of the contact-energy chain through

= 10, contact-deﬁcit counts through

N = 11, and exact veriﬁcation of a simpler history-resolved model through N = 8;

6. an identity eﬀective metric reconstructed from the close-packed phase;

7. an exact lemma that tetrahedral–octahedral cells close around every edge, so space is ﬂat;

a theorem that Stitch growth from a tetrahedron stays on one FCC lattice, with the seed

choosing its orientation;

9. a test of that theorem on the growth simulation of the companion matter paper.

The claim is narrow. Entropy-producing microscopic dynamics selects a phase with a spatial

metric and zero spatial Regge curvature. We assume the microscopic degrees of freedom and rules;

we do not derive them. Table 1 gives the logical status of each claim.

The companion matter paper [

] reaches a close-packed bulk with coordination 12 by another

route: an irreversible growth simulation that starts from the same seed and uses the same Stitch

and Lift moves (Section 2.3 of Ref. [

]). This paper supplies the equilibrium route. There, the

stationary measure of a reversible chain concentrates on maximal bonding. Both routes use the same

Stitch move. Section 10 shows that this move, not the energy, selects FCC among close packings.

Section 14 tests this on the growth simulation.

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

means checked by exhaustive ﬁnite enumeration. “Conditional” means proved under the hypothesis

named.

Claim Status

Metropolis–Hastings reversibility and stationary bond

measure

Exact, all ﬁnite reachable spaces

Equilibrium chain deﬁned directly with

cont

on assemblable

embedded conﬁgurations; reversible and irreducible

Exact (Prop. 2.6); veriﬁed N ≤ 10

Unrestricted insertion–deletion closure is inﬁnite at ﬁxed N Observed at N = 5 (Remark 2.7)

Finite-state irreducibility of the history-resolved model under

the exposed-node condition

Exact; condition exhaustively

veriﬁed for N ≤ 8

Maximum bond count B

max

= 3N − 6 and bond deﬁcit

equals stitch count for states at the node cutoﬀ

Exact; general form

m = 3(N − n) + s

Stationary measure independent of lift proposal suppression

ρ > 0

Exact; veriﬁed for ρ = 1, 0.1, e

−3

in both models

Finite-N concentration under the deﬁcit-entropy bound Exact conditional theorem

Uniform deﬁcit-entropy slope is impossible as

N → ∞

if the

maximal sector is subexponential (vacancy entropy)

Exact conditional theorem

(Prop. 6.3); requires

subexponential maximal-sector

degeneracy (open) and vacancy

assemblability (veriﬁed

numerically)

Defect-density concentration, m/N ≲ δ

∗

∝ e

−cβ

Exact conditional theorem

(Thm. 6.5)

Points outside FCC/HCP patterns ≤ 26(6n − B

cont

) Exact, from the local content of

Hales’ proof, traced in App. C

(Cor. 11.2)

Equilibrium conﬁgurations locally close-packed outside

density O(δ

∗

)

Exact given Assumptions 6.4, 11.3

(Thm. 11.4)

Contact maximizers are close-packed fragments with

octahedra where maximal contact exceeds 3N − 6

Veriﬁed N = 10, 11 (Table 4)

hist

and B

cont

maxima coincide for n ≤ 9 and separate at

n = 10

Veriﬁed n ≤ 12;

history-independently n ≤ 10

(Sec. 7)

Peierls counting bound for assemblable excitations

(Assumption 6.4)

Open

Compact assembly, B

max

(N) ≥ 6N − C

2/3

(Assumption 11.3)

Veriﬁed N ≤ 3 055; open for all N

Maximally bonded embedded SSM states approach Barlow

close packing

Veriﬁed n ≤ 12; locally, given

Assumption 11.3 (Thm. 11.4)

Close-packed bond tensor is the identity metric; spatial edge

deﬁcits vanish (FCC and HCP alike)

Exact geometric lemmas

Stitch is point inversion; stitch closure of a regular

tetrahedron is one FCC lattice

Exact (Thm. 10.4)

First seed Lift selects the global FCC orientation; seed twin

if both seed apexes are occupied

Exact (Prop. 10.6)

Kinetic simulation of Ref. [

]: cubic stacking, Lift as the only

lattice-changing move

Veriﬁed numerically (Sec. 14)

Misoriented material from Lift nucleation has zero density as

N → ∞

Open

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

the local combinatorial data needed to identify allowed stitch and lift moves. The state contains no

background coordinates, metric tensor, or curvature ﬁeld.

The local rules come from the SSM construction. A stitch adds a node along an occupied edge.

A lift adds a node over an occupied triangular face. We specify each move combinatorially ﬁrst and

then construct its geometric realization at unit bond length. Reverse moves delete exposed nodes

and never delete the seed.

Deﬁnition 2.2 (Minimal seed). Let

be a ﬁxed oriented triangle. It is the smallest seed on which

both stitch and lift are deﬁned. The seed is not spacetime. It is a ﬁnite boundary condition that

avoids starting from an empty state.

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

)|, let

Ω

)

be the set of conﬁgurations reachable from

by stitch, lift, and reverse moves with at most

nodes.

The cutoﬀ makes the state space ﬁnite. The thermodynamic question is then a sequence of ﬁnite

problems, followed by the limit N → ∞.

2.1 The embedded state space

Once its face and side are ﬁxed, each move has a unique realization at unit bond length. A Lift

on a triangle (

, x

) with pairwise contacts places one of the two points at distance

from all

three vertices,

+ x

L ˆn, (1)

where

ˆn

is the unit normal of the triangle. A Stitch on the edge

{u, v}

of such a triangle (

u, v, w

)

places the equilateral apex in the plane of the triangle, opposite

, at

− x

(Lemma 10.3).

So a conﬁguration built from the seed ﬁxes a ﬁnite point set in

up to isometry. The coordinates

follow from the relational construction and the unit bond length. The dimension three enters

through the Lift rule, which intersects three spheres.

Deﬁnition 2.4 (Assemblable conﬁguration and contact count). A contact triangle of a ﬁnite point

set

is a triple of points with pairwise distances

. A forward move at

adds a Lift apex

(1)

a Stitch apex

− x

of some contact triangle of

. The new point must lie at distance at

least

from every point of

(overlap exclusion). A point set is assemblable if a ﬁnite sequence of

forward moves builds it from the seed triangle. For a cutoﬀ

, let Ω

emb

be the set of assemblable

point sets with at most

points, identiﬁed up to isometry, including reﬂection. The contact count

cont

(X) = #{{x, y} ⊂ X : |x − y| = L}.

A state of Ω

emb

carries no record of how it was built. Assemblability is a property of the point

set. The forward moves available at

depend only on its contact triangles. In contrast, the

construction-bond count of Section 7 depends on a history.

Deﬁnition 2.5 (Reverse moves). A reverse move at

deletes a point

such that

X \ {y} ∈

Ω

emb

and y is a forward-move apex of X \{y}.

Proposition 2.6 (Reversibility and irreducibility of the embedded move graph). For

X, X

′

∈

Ω

emb

a forward move takes

′

if and only if a reverse move takes

′

. Every

X 

admits a

reverse move. So the move graph on Ω

emb

is connected, and Assumption 4.1 holds with no further

hypothesis.

Proof.

′

X ∪ {y}

with

a forward-move apex of

, then deleting

from

′

gives back the

assemblable set

. The converse holds for the same reason. An assemblable

X 

has a growth

sequence from the seed. Its last move gives a point

such that

X \ {y}

is assemblable and

is a

forward apex of it. Repeated deletion reaches C

Remark 2.7 (Why the state space is restricted to assemblable sets). The restriction is necessary.

Suppose deletion were allowed whenever

is a forward apex of

X \ {y}

, without requiring

X \ {y}

to be assemblable. Then the conﬁgurations connected to the seed would be inﬁnite at ﬁxed

. A

point held by a single contact can be rotated about that contact by insertions and deletions. The

tetrahedral dihedral angle

arccos

3) is an irrational multiple of

, so the reachable rotations form

a dense set. A breadth-ﬁrst search at

= 5 ﬁnds 146 311 distinct conﬁgurations after 26 layers,

and the frontier still grows by a factor of about 2

3 every two layers. Distinct distance values diﬀer

by about 10

−3

, far above the numerical tolerance of 10

−6

, so the growth is not a rounding artifact.

Requiring each state to have some growth history from the seed, without recording which one, keeps

Ω

emb

ﬁnite.

The physical equilibrium model of this paper is the chain on Ω

emb

with

cont

. We keep

the combinatorial conﬁgurations of Deﬁnition 2.1, with the construction-bond count, as a simpler

veriﬁcation model (Section 13).

3 Reversible stochastic dynamics

Let

(

) be the bond count of a conﬁguration. In the physical model it is the contact count

cont

on Ω

emb

. In the veriﬁcation model it is the construction-bond count

(

)

. In this section and

the next two, Ω

denotes whichever state space is in use, and the results require only that

be a

function of the state. Assign the microscopic energy

with bond scale

ϵ >

0. We put kinetic suppression in the proposal kernel, not in the energy. A

suppressed lift can then still lower the energy.

For each ordered pair

C, C

′

∈

Ω

connected by one move, let

(

C, C

′

) be the proposal probability.

Q may treat stitch and lift attempts diﬀerently and may suppress lifts. 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

′

)



, (2)

where

= (

)

−1

. This is the Metropolis–Hastings construction [

]. The transition kernel is

P (C, C

′

) = Q(C, C

′

)A(C, C

′

)

for C = C

′

, and P (C, C) makes each row sum to one.

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

exp



ϵB(C)



, Z

C∈Ω

ϵB(C)

. (3)

Proof.

For adjacent conﬁgurations, Eq.

(2)

is the standard Metropolis–Hastings rule. Direct

substitution gives

(C)Q(C, C

′

)A(C, C

′

) = π

′

)Q(C

′

, C)A(C

′

, C).

For nonadjacent pairs both sides are zero. So detailed balance holds for every pair.

Remark 3.2. With symmetric proposals and a reverse move that removes one bond, the rule

reduces to the local logistic identity

− q

) =

−β

with

. The full kernel is needed when

the numbers of possible insertions and deletions diﬀer.

4 Ergodicity on the reachable complex

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

relevant state space is the set reachable from the seed by the allowed moves, not the set of all

connected graphs.

Assumption 4.1 (Removable exposed node). Every conﬁguration C ∈ Ω

with C = C

contains

at least one exposed nonseed node whose deletion is an allowed reverse stitch or reverse lift and

stays in Ω

The assumption states that construction histories can be reversed. It can be checked on ﬁnite

enumerated state spaces. For the embedded model it holds by Proposition 2.6.

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

(

C, C

)

0 for every

, the

Markov chain on Ω

is irreducible and aperiodic. So

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

. So any two states communicate through

the seed, and the chain is irreducible. A positive hold probability gives each state a self-loop, so

the chain is aperiodic. 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 change in system entropy

∆s

sys

= −k

log π

′

) + k

log π

(C)

and the change in medium entropy

∆s

med

= k

log

P (C, C

′

)

P (C

′

, C)

Detailed balance gives

∆s

sys

+ ∆s

med

= 0

for a pair of equilibrium trajectories. During relaxation from a nonequilibrium distribution

, the

relative entropy

D(ρ

∥π

) =

(C)

cannot increase. So local bond order can grow while the system exports entropy to the bath.

Three roles should be kept apart. Detailed balance and the bond energy

−ϵB

ﬁx the

stationary measure

. Entropy production drives the relaxation toward it. A competition between

bond energy and the conﬁgurational entropy of the deﬁcit sectors decides which macrostate dominates

. In that competition, entropy opposes concentration and energy drives it. The condition

ϵ > s

∗

in Theorem 6.2 below expresses this balance. Entropy production makes the relaxation irreversible

and the equilibrium phase reachable. It does not create the microscopic degrees of freedom, the

energy scale ϵ, or the direction of selection.

GfE asks a similar question at the macroscopic level. Its thermodynamic analysis [

] identiﬁes

the GfE energy density with an emergent dark-energy term. For Friedmann–Robertson–Walker

universes, it shows that the total entropy of GfE cosmologies never decreases, which oﬀers a way to

reconcile local order with global entropy growth. The same reconciliation holds exactly in our ﬁnite

system. Bond order increases while

(

∥π

) decreases and entropy ﬂows to the bath. The ordered

spatial phase does not conﬂict with the second law. It is the stationary endpoint of the second law.

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

∗

. (4)

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

(4)

holds and

ϵ > s

∗

, then for every

M ≥ 1,



max

(N) − B(C) ≥ M



≤

−(β

ϵ−s

∗

1 − e

−(β

ϵ−s

∗

)

. (5)

So at each ﬁnite

the tail falls exponentially in the deﬁcit threshold

. A thermodynamic statement

also needs control of the prefactor a

and of how M scales with N.

Proof. Relative to the maximal sector,

(Ω

N,m

)

(Ω

N,0

)

|Ω

N,m

|Ω

N,0

−β

ϵm

≤ a

−(β

ϵ−s

∗

Summing the geometric series over m ≥ M gives Eq. (5).

Bond energy alone does not give concentration; conﬁgurational entropy must also be controlled.

Theorem 6.2 is exact at every ﬁnite

. At ﬁnite

, Assumption 6.1 holds for suitable constants,

because Ω

is ﬁnite. The thermodynamic question is whether it holds with a slope

∗

that does not

depend on

and a prefactor with

log a

(

). Under a mild hypothesis on the maximal sector,

it does not. The reason is physical: crystals have vacancies.

6.1 A uniform entropy slope is excluded by vacancy entropy

Proposition 6.3 (Vacancy entropy excludes a uniform slope). Suppose that for all suﬃciently large

: (i)

Ω

N,0

o(N)

; and (ii) there are constants

∈

1] and

, C

< ∞

such that for every

k ≤ c

the state space contains at least



⌊c

N⌋



conﬁgurations with deﬁcit at most

2/3

Then there are no

-independent

∗

< ∞

and

with

log a

(

) for which Eq.

(4)

holds for

all large N.

Proof.

Summing Eq.

(4)

over

m ≤ M

gives

m≤M

Ω

N,m

| ≤ a

Ω

N,0

∗

− e

−s

∗

) for

∗

and a stronger bound for

∗

≤

0. Take

⌊δN⌋

with

δ < c

and

2/3

. Hypothesis

(ii) and





≥ (n/k)

give

k log

≤ log a

+ o(N) + s

∗



δN + C

2/3



+ O(1).

Divide by N and let N → ∞. This gives log(c

/δ) ≤ s

∗

, which fails for every δ < c

−s

∗

In the embedded contact model, both hypotheses have concrete content. For (ii), start from

a compact close-packed conﬁguration with

points (Assumption 11.3 of Section 11). Remove

interior points from a set of

interior sites, no two of them adjacent. The sites of the sublattice of

spacing 2

give

→

8. Each removal costs twelve contacts, so the deﬁcit is at most 12

2/3

and

= 12,

. Every such conﬁguration we tested is assemblable (Appendix B). Hypothesis

(i) says the number of maximal-contact conﬁgurations grows subexponentially. It agrees with the

enumeration (3 maximizers at

= 10 and 1 at

= 11) and with the collapse of maximal-contact

degeneracy beyond nine points [

], but it is not proved. If it failed, the maximal-contact sector

would itself carry extensive entropy. We know of no mechanism for this, and the enumeration

shows the opposite trend. The proposition is a conditional theorem, and we state its consequences

conditionally.

Under its hypotheses, the proposition has two consequences. First, a uniform slope would force

the equilibrium vacancy density to zero at every ﬁnite temperature, which is why no such bound

exists. Second,

(Ω

N,0

)

→

0 as

N → ∞

at every ﬁxed

, because the number of ways to place

a single defect grows with

. Exact concentration on maximizers, as seen in Section 13.3, is a

ﬁnite-size property. The thermodynamic statement concerns the density of the contact deﬁcit.

6.2 Concentration of the defect density

Assumption 6.4 (Peierls counting). There are constants

K ≥

min

C >

0 and a sequence

with log b

= o(N) such that for all N and m,

|Ω

N,m

| ≤ b

|Ω

N,0

⌊m/c

min

⌋

j=0



⌈CN⌉



. (6)

The assumption says three things. A conﬁguration with deﬁcit

diﬀers from a maximizer by

at most

m/c

min

localized excitations. Each excitation sits at one of

(

) positions. Each unit of

deﬁcit allows at most

local shapes. This is the counting behind Peierls’ argument [

] and its

extension to general low-temperature phases [

]. It allows the entropy of order

log N

per excitation

that rules out Assumption 6.1.

Theorem 6.5 (Concentration of the defect density). Assume Assumption 6.4 and set

∗

(β

ϵ) = e c

min

C exp



− c

min

(β

ϵ − log K)



. (7)

If β

ϵ > log K + (log 2)/c

min

and δ > δ

∗

(β

ϵ), then there is η > 0 such that for all large N,



max

(N) − B ≥ δN



≤ e

−ηN

So the contact-deﬁcit density is at most

∗

with probability tending to one, and

∗

falls exponentially

in β

ϵ.

Proof.

Since

≥ |

Ω

N,0

ϵB

max

, we have

(Ω

N,m

)

≤ |

Ω

N,m

−β

ϵm

Ω

N,0

. Write

m/N

and

J = ⌊m/c

min

⌋. The binomial sum in Eq. (6) is bounded by

j≤J



⌈CN⌉



≤

(

(eCN/J)

≤ exp



min

log

min



, J ≤ CN,

⌈CN⌉

≤ exp



min

log 2



, J > CN.

Hence

log π

(Ω

N,m

) ≤ log b

− m Γ(u),

Γ(u) = β

ϵ − log K −

min

max



log

min

, log 2



For

u ≥ δ > δ

∗

, Eq.

(7)

and the condition on

give Γ(

)

≥ γ

for a constant

γ >

0. So

(Ω

N,m

)

≤ b

−γδN

for every

m ≥ δN

. There are at most

max

(

) + 1

≤

+ 1 sectors. The

tail probability is therefore at most (6N + 1)b

−γδN

, which is below e

−ηN

for large N.

Theorem 6.2 is the exact ﬁnite-

statement, and the enumerations test it. Theorem 6.5 is the

thermodynamic statement. It concerns the density of defects, not the maximal sector, as it must for

a crystal at ﬁnite temperature. Section 11 turns it into a statement about local geometry.

7 Two levels of bond counting

The concentration theorems apply to whatever bond count enters the energy

−ϵB

. This paper

uses two counts. This section deﬁnes both and compares their maximizers.

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 a construction move joined the pair.

The physical equilibrium model uses

cont

(Section 2.1), because unit-distance contact deﬁnes a

close packing and construction history does not. Section 13.3 reports its exact enumeration. The

history-resolved veriﬁcation model of Section 13 uses

hist

. The two functionals diﬀer, and we can

compute how their maximizers relate.

7.1 Embedded enumeration

Once a face and a side are ﬁxed, stitch and lift each have a unique apex, so a construction history

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

cont

is not

deﬁned on it directly. We place the seed triangle in

and apply the Lift rule

(1)

and the Stitch

rule recursively, with the overlap exclusion of Deﬁnition 2.4 at each step. Each history then gives a

point set

X ⊂ R

, and

cont

is deﬁned on

. We identify states up to isometry, including reﬂection,

by color reﬁnement on the weighted complete graph, followed by minimization over the remaining

color classes.

The two enumerations count diﬀerent objects, so their totals should not be compared directly.

The history-resolved enumeration of Section 13 covers Ω

(

), which contains every state with at

most

nodes. It identiﬁes states by isomorphism of the colored incidence graph of the abstract

complex. The embedded enumeration ﬁxes the node count

, imposes overlap exclusion, and identiﬁes

states by isometry of the point set. The diﬀerence is informative. At

= 8 the combinatorial model

has 464 states with exactly eight nodes, but only 303 of them can be embedded. The pairs at

= 5

7 are 6

3, 21

12, and 94

55. So about a third of the abstract complexes cannot be realized

geometrically under the exclusion rule. This measures 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 13.1 requires. The contact-bond maximum departs from it at

= 10, and the gap then

grows linearly.

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 matter here. First, the two counts agree for

n ≤

9 (Proposition 7.3).

So in that range the history-resolved veriﬁcation of Section 13 tests the right object: the construction-

bond energy selects contact-bond maximizers. Second, from n = 10 the maxima separate, the gap

grows, and the maximizer sets are disjoint. At

= 12 the unique contact maximizer has

hist

= 23,

while the construction maximum is 30. Third, the cell type changes. Every contact maximizer from

= 10 on contains octahedral cells. No construction maximizer contains one at any

. The reason

is Proposition 13.1: the sector Ω

N,0

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

All-tetrahedral complexes are also strained, since a fan of tetrahedra around an edge cannot close

and leaves the deﬁcit

of Eq.

(9)

. The ﬂatness of Lemma 9.1 needs the tetrahedral–octahedral

honeycomb, in which two tetrahedra and two octahedra meet at every edge. So the octahedral cell

is the structure the geometry needs, and the construction-bond sector cannot supply it.

The enumeration behind Table 2 generates moves from the construction triangles of one stored

history per point set. We repeated it on the history-independent space Ω

emb

of Section 2.1, taking

hist

of each point set from its best growth history (the one with fewest stitches). The result is the

same: every

hist

maximizer is a

cont

maximizer for

n ≤

9, and the maximizer sets are disjoint at

= 10. Some counts diﬀer slightly, because contact-triangle moves reach more conﬁgurations. For

example, there are 2 003 nine-point states instead of 1 851, and 32 contact maximizers instead of 31.

7.2 Agreement with the sphere-packing literature

The crossover does not come from the stitch–lift move set. Arkus, Manoharan, and Brenner [

]

found the same transition for clusters of hard spheres with short-range attraction, using graph

enumeration and geometry instead of lattice growth. They report that ground-state degeneracy

grows exponentially for

n ≤

9 and decreases for

n >

9, where packings with more than 3

n −

contacts appear. They identify octahedra and half-octahedra as the cause. Their maximal contact

numbers at

= 10

12 are 3

n −

5, 3

n −

4, and 3

n −

3, the values in Table 2. The enumeration

also reproduces the collapse in degeneracy, from 31 contact maximizers at

= 9 to a unique one

from n = 11 on. Bezdek and Khan [7] review the contact-number problem for small n.

The stitch–lift moves reach the known optima. So the move set of Section 2 is rich enough, and

the separation in Table 2 reﬂects the choice of energy, not a limit of the dynamics.

7.3 What this settles, and what it does not

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

The enumeration supports it where computed. Arkus et al. ﬁnd that the maximal-contact clusters

= 10

12 are subsets of close-packed crystals, and our enumeration reaches the same

conﬁgurations. So the assumption is veriﬁed at small n.

One qualiﬁcation matters. The small maximal-contact clusters are reported as subsets of

hexagonal close packing. A twelve-sphere cluster is far too small to tell FCC from HCP, since

both are Barlow packings and share small fragments. So contact maximization gives close packing

but does not select FCC. Energy cannot break this tie (Remark 10.2). The Stitch move breaks it,

because it is a lattice inversion (Section 10). The geometric problem therefore has two parts: the

contact energy selects Barlow packing, and the move set selects FCC within it.

The thermodynamic statement needs more. Twelve nodes cannot contain an FCC bulk node,

which needs twelve neighbors and ﬁlled second and third shells. Section 13.3 carries out the

detailed-balance construction with

cont

at ﬁnite

. Sections 6 and 11 treat the large-

limit. A

uniform bound on the contact-deﬁcit entropy fails when the maximal sector has subexponential

degeneracy (Proposition 6.3). A Peierls counting assumption replaces it and gives concentration

of the defect density. Local close packing then follows from the proof of the Fejes T´oth contact

conjecture, given that compact clusters are assemblable.

8 From relational order to an eﬀective metric

The microscopic state has no metric. We reconstruct one from the bond geometry, once an embedding

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). (8)

The normalization gives g

eﬀ

= I for any isotropic set of z unit directions.

Lemma 8.1 (FCC eﬀective 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 permutations of (

, ±

0). Oﬀ-diagonal

terms cancel by sign symmetry. Each coordinate is nonzero in eight directions, with squared

component 1/2, so each diagonal entry is 8/2 = 4.

So the FCC phase has a Euclidean, isotropic spatial metric, obtained from a local bond moment

and not put in by hand. The same identity holds for hexagonal close packing and for every Barlow

stacking (Remark 10.2). It needs close packing, but not a particular stacking.

9 Spatial Regge ﬂatness of the FCC branch

This section makes a statement about space at one time. In three-dimensional Regge calculus the

hinges are edges [9], and the action is

(3)

ℓ

We do not use the four-dimensional action, whose hinges are triangles.

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.

Each edge meets two tetrahedra and two octahedra. The two dihedral angles are supplemen-

tary, so the total angle is 2π and δ

= 0 for every edge.

By contrast, ﬁve regular tetrahedra around an edge leave the positive deﬁcit

= 2π − 5 arccos(1/3) ≈ 0.1284. (9)

This identity is exact and local. It does not rely on the claim that maximal rigidity implies ﬂatness,

which is false in general.

10 Stitch inversion and FCC selection

The concentration theorems select conﬁgurations of maximal bond number. Section 7 shows that,

for the contact count, these are close packings. This section asks which close packing the stitch–lift

dynamics produces. Two points should be kept apart. The identity eﬀective metric of Section 8 and

the ﬂatness of Lemma 9.1 hold for every Barlow stacking, FCC and HCP alike (Remark 10.2). So

the geometric conclusions relevant to GfE do not depend on the stacking. The companion matter

construction [

] does need FCC, because its charge assignment uses the Bravais translation group.

We show here that FCC follows from the geometry of the Stitch move.

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 maximally bonded states that SSM can reach. The embedded enumeration of

Section 7 veriﬁes it for

n ≤

12, in agreement with the maximal-contact clusters of Arkus et al. [

Section 11 proves its local form under Assumptions 6.4 and 11.3: equilibrium conﬁgurations are

close-packed except on a set of small density. The global stacking statement remains open. Under

the assumption, the registry of sheet

is a label

∈ Z

for the usual lateral registries

A, B, C

, and

n+1

− s

∈ {

, −

}

(

mod

3). A constant increment gives FCC (

ABC ···

or its mirror

ACB ···

An alternating increment gives HCP (ABAB ···).

Remark 10.2 (The contact Hamiltonian does not distinguish Barlow stackings). In every Barlow

stacking, each bulk node has twelve unit contacts. So

cont

gives FCC and HCP the same energy,

and a stacking fault costs no bond energy. Both ﬁll ﬂat space with regular tetrahedra and octahedra,

so the edge deﬁcits of Lemma 9.1 vanish for both. A direct calculation over the HCP neighbor shell

gives

µν

= 4

µν

, as for FCC. The equilibrium route of Sections 3–7 therefore selects close packing,

coordination

= 12, the identity eﬀective metric, and spatial ﬂatness, but not FCC. For hard

spheres, the free-energy diﬀerence between FCC and HCP is about 10

−3

per particle [

]. In

our discrete model, which has no vibrational entropy, it is zero. The move set decides the stacking,

as we now show.

10.1 Stitch is lattice inversion

We ﬁrst ﬁx the geometric form of the two moves. For an oriented equilateral frontier face

, x

) with unit normal n

, the Lift places

(f) =

+ x

L n

, (10)

the two points at distance

from all three vertices. The Stitch acts on an occupied edge

{u, v}

an occupied triangle (

u, v, w

). It places the equilateral apex in the plane of that triangle, on the

side opposite w.

Lemma 10.3 (Stitch is point inversion). The Stitch apex is

S(u, v; w) = x

+ x

− x

, (11)

the image of

under point inversion through the midpoint of the edge

{u, v}

. Consequently, if

, x

lie in an aﬃne lattice Λ, so does S(u, v; w).

Proof.

Let

= (

)

2. The vector

m − x

is the altitude of the equilateral triangle. It has

length

√

and lies in the plane of the triangle. The apex opposite

at the same altitude is

+ (

m −x

) =

−x

. An integer combination whose coeﬃcients sum to one maps an aﬃne

lattice to itself.

The embedded enumeration of Section 7 uses the same formula. So does the growth simulation

of Ref. [

], whenever the edge belongs to an occupied triangle. The plane of the triangle need not

be a growth sheet. A Stitch across a triangle that contains an interlayer bond places its apex in an

adjacent layer. In the growth simulation, this out-of-plane Stitch starts most new layers (Section 14).

Theorem 10.4 (Stitch closure of a tetrahedron is FCC). Let

, x

}

be a regular

tetrahedron of edge L, and let

= x

+ Z(x

− x

) + Z(x

− x

) + Z(x

− x

Then Λ

is a face-centered cubic lattice of nearest-neighbor distance

, and every node generated

from

by any sequence of Stitch moves lies in Λ

. In particular, the close-packed layers of the

stitch-generated structure along each

⟨

111

⟩

direction of Λ

follow a constant registry increment: no

HCP stacking occurs among stitch-generated nodes.

Proof.

Up to isometry,

= 0 and

1,2,3

√

1). These are the primitive

vectors of the FCC lattice with cubic constant

√

. Their cell volume,

√

4, conﬁrms

that they generate it. By Lemma 10.3 and induction on the number of moves, every stitch-generated

node lies in Λ

. For the registry, let e

= x

− x

and Λ

= Ze

+ Ze

, the triangular lattice of the

face (

, x

), and let

−x

. Every point of Λ

with integers

a, b, k

, so

the layers of Λ

parallel to the face are the cosets

+ Λ

, at heights

2/3 L

. The in-plane

projection of

is the centroid oﬀset (

)

3. Labeling the three lateral registries

A, B, C

by the

cosets Λ

+ r(e

+ e

)/3 with r = 0, 1, 2, layer k occupies registry

≡ s

+ k (mod 3).

The increment

k+1

− s

= 1 is the same for every

. This gives

ABCABC ···

along the normal

toward

, and

ACBACB ···

in the opposite direction. HCP would need

k+2

, but here

k+2

− s

≡

2 (

mod

3). Every face of

spans a close-packed plane of Λ

, and the same argument

applies to each.

So FCC selection needs no extra state variable, such as a stacking chirality carried by the growth

front. The lattice Λ

itself carries the constant registry increment, and the Stitch transports the

lattice by inversion. The Lift is the only move that can leave the lattice.

Lemma 10.5 (Exactly one Lift apex is on the lattice). For every unit triangle of an FCC lattice Λ,

exactly one of the two apexes L

of Eq. (10) lies in Λ.

Proof.

In the tetrahedral–octahedral honeycomb of Λ, each unit triangle is shared by one regular

tetrahedron and one regular octahedron. The apex on the tetrahedron side is the fourth vertex of

the tetrahedron, a lattice point. The apex on the octahedron side lies at height

2/3 L

above the

face centroid. This height equals the distance between opposite faces of the octahedron, so the apex

is the centroid of the opposite face. That point is at distance

√

3 < L

from three lattice points,

so it is not a lattice point.

So a Lift into empty space on the octahedron side starts a new lattice. On a triangle parallel to

the growth sheet this is a stacking fault; on a tilted triangle it is a grain with a diﬀerent orientation.

Lifts are needed for three-dimensional growth, but each octahedral-side Lift can create disorder.

Lift suppression therefore has a clear kinetic role. It sets the rate at which misoriented material

forms, and it leaves the stationary measure unchanged (Section 13).

10.2 The seed selects the global orientation

The seed triangle

has two Lift apexes

(

). Theorem 10.4, applied to each tetrahedron, gives

two lattices Λ

and Λ

−

. They are mirror images under reﬂection through the seed plane, and they

share the seed layer. They are the two FCC orientations, with registry sequences

ABC ···

and

ACB ··· along the seed normal. By Lemma 10.5, L

∈ Λ

\ Λ

−

and L

−

∈ Λ

−

\ Λ

Proposition 10.6 (Seed orientation and the seed twin). The ﬁrst Lift on the seed selects Λ

or Λ

−

Stitch growth then carries the selected lattice through the whole connected stitch-generated region.

Suppose both seed apexes are occupied before the two growth regions join. Then the two sides grow

on mirror lattices, and the registry sequence along the seed normal is

···CBABC ···

. This is a

coherent twin boundary in the seed plane between two FCC domains of opposite orientation. Every

node of a twin plane keeps twelve unit contacts.

Proof.

The ﬁrst two statements follow from Theorem 10.4 and Lemma 10.5. Label the seed layer

. Then Λ

gives

B, C, . . .

above it, and its mirror Λ

−

gives

B, C, . . .

below it. Read along the

normal, the sequence is

···CBABC ···

. At the twin plane each node has six neighbors in its layer

and three in each adjacent layer, as in any Barlow packing.

So a spontaneous binary choice at the seed ﬁxes the global orientation, and inversion carries

it outward. Regions far from the seed keep that orientation, except where later octahedral-side

Lifts have started misoriented grains. Two statements should be kept apart. For stitch growth, the

orientation is inherited from the seed exactly. Whether a region far from the seed lies in a single

domain depends on how large the Lift-nucleated grains are compared with the region. Section 14

measures this.

Corollary 10.7 (Thermodynamic FCC bulk). Let

be a sequence of embedded crystallites with

nodes grown by Stitch and Lift. Let Γ

be the set of nodes within a bounded distance of a stacking

fault, twin plane, or grain boundary. If

(

), the fraction of nodes whose neighborhoods

are not FCC tends to zero. Every bounded local observable then converges to its FCC bulk value.

In particular

⟨K⟩

→

12 and

(N)

µν

→ δ

µν

, and the spatial edge deﬁcits vanish outside a set of zero

density.

Proof.

Outside Γ

, each node lies inside a stitch-generated region. By Theorem 10.4, that region is

a subset of one FCC lattice. The fraction of exceptional nodes is at most

|/N →

0. The metric

and ﬂatness statements follow from Section 8 and Lemma 9.1.

A single seed twin plane contributes

(

2/3

) nodes, so it satisﬁes the hypothesis. The hypothesis

fails if Lift nucleation produces grains of bounded size. The remaining quantitative input is how

grain size scales with N and with the Lift rate.

11 From contact deﬁcit to local close packing

Every assemblable conﬁguration is a packing. Its points are at least

apart, and contacts are pairs

exactly

apart. This section shows that a small contact deﬁcit puts all but a proportionally small

set of points in FCC or HCP environments. The argument uses Hales’ proof of L. Fejes T´oth’s

contact conjecture [

]. We state results with contact distance

; Hales uses contact distance 2.

Lemma 11.1 (Local form of Fejes T´oth’s contact theorem). Let

be a ﬁnite packing and

u ∈ X

point with twelve contacts, each of which also has twelve contacts in

. Then the twelve contacts of

u are arranged in the FCC or the HCP pattern.

Proof.

The proof of Theorem 1 of Ref. [

] establishes this. Appendix C traces it step by step and

lists the hypotheses of each step. Hales’ Lemma 2 uses only the twelve contacts of a point and

an inequality for ﬁnite packings. It shows that every other point of the packing lies at least

from that point, with

= 1

26. Applied at each contact of

, it shows that any two contacts of

either touch or lie at least

apart. This is the deﬁning property of the kissing conﬁgurations

that Hales classiﬁes. The classiﬁcation (Theorem 3 and Lemmas 8–10 of Ref. [

]) uses only the

twelve points around

, and it gives the FCC or HCP pattern. The published theorem assumes that

every point has twelve contacts, and its last step applies the local argument at an arbitrary point.

The hypothesis enters only through Lemma 2 at the contacts of u, so the local form holds.

Corollary 11.2 (Defects are controlled by the contact deﬁcit). In any ﬁnite packing with

points

and

contacts, the number of points whose contacts are not arranged in the FCC or HCP pattern

is at most 26(6n − B).

Proof.

Every degree is at most twelve, the kissing number. A point fails the hypothesis of Lemma 11.1

only if it, or one of its at most twelve contacts, has degree below twelve. There are at most

(12 − deg v) = 13(12n − 2B) = 26(6n − B) such points.

The corollary helps only if the maximal contact count is close to 6

. Bezdek and Reid proved

B <

n−

926

2/3

for every packing of

unit balls [

]. A matching lower bound on the assemblable

state space needs compact close-packed clusters to be assemblable.

Assumption 11.3 (Compact assembly). There is a constant

such that for every

N ≥

3 some

X ∈ Ω

emb

has N points and B

cont

(X) ≥ 6N − C

2/3

We verify the assumption exactly over a wide range (Appendix B). Deciding whether a target

cluster is assemblable reduces to a monotone closure computation (Lemma B.1). FCC balls of up to

3 055 points, all four cuboctahedral clusters up to 309 points, and HCP balls of up to 763 points are

assemblable. When FCC balls are grown shell by shell, every intermediate state with 20

≤ N ≤

3 055

satisﬁes 6

N −B

cont

≤

2/3

. With the Bezdek–Reid bound, this gives 6

N −B

max

(

) = Θ(

2/3

)

over that range. A proof for all

would follow if FCC balls of every radius were shown to be

assemblable. That is a statement about the closure of Theorem 10.4 restricted to a ball.

Theorem 11.4 (Equilibrium conﬁgurations are locally close-packed). Assume Assumption 11.3.

Let X ∈ Ω

emb

have n points and contact deﬁcit m = B

max

(N) − B

cont

(X).

(a)

At most 26(

2/3

) points of

lie outside the FCC and HCP patterns, and

n ≥

N − (m + C

2/3

)/6.

(b) Every maximal-contact conﬁguration is locally close-packed except at O(N

2/3

) points.

(c)

If Assumption 6.4 also holds, and

and

are as in Theorem 6.5, then with

-probability

at least 1

− e

−ηN

the fraction of points outside the FCC and HCP patterns is at most 26(

−1/3

)

− δ/

− C

−1/3

6), and the mean coordination satisﬁes

⟨K⟩ ≥

−

−1/3

)/(1 − δ/6 − C

−1/3

/6).

Proof.

A packing has

cont

≤

, and

max

(

)

≥

N −C

2/3

. So 0

≤

n −B

cont

≤

N −B

cont

N −B

max

m ≤ m

2/3

. Corollary 11.2 gives the ﬁrst claim of (a), and 6(

N −n

)

≤

N −B

cont

gives the second. Part (b) is the case

= 0. For (c), Theorem 6.5 gives

m < δN

with the stated

probability. Divide the bound of (a) by n, and use ⟨K⟩ = 2B

cont

/n = 12 −2(6n −B

cont

)/n.

Corollary 11.5 (Eﬀective metric and ﬂatness at ﬁnite temperature). Assume the hypotheses of

Theorem 11.4(c). At every point with the FCC or HCP pattern, the eﬀective metric of Eq.

(8)

equals

µν

(Remark 10.2). At every edge that meets such a point, the pattern ﬁxes the four common

neighbors of the edge. Their angular gaps around the edge are two tetrahedral and two octahedral

dihedral angles, so the closure of Lemma 9.1 holds. So the fraction of points with

eﬀ



and the

fraction of edges with nonzero spatial deﬁcit are both

(

∗

(

) +

−1/3

). Both vanish as

N → ∞

and then β

ϵ → ∞.

This result is the equilibrium counterpart of the kinetic corollary in Section 10. In the equilibrium

route, temperature ﬁxes the defect density. In the kinetic route, the defects are set by the size of

Lift-nucleated grains. Lemma 11.1 allows both FCC and HCP patterns, as Remark 10.2 requires.

Energy selects close packing. The Stitch move selects the stacking.

12 Relation to Gravity from Entropy

What GfE is. In GfE the gravitational action is the quantum relative entropy between two

metrics [

]. One is the spacetime metric

˜g

. The other is a metric

induced by the matter ﬁelds and

the curvature. Both are topological metrics, that is, direct sums of metrics on 0-, 1-, and 2-forms.

Matter is described by Dirac–K¨ahler ﬁelds. The eigenvalues of a rank-2 tensor

are deﬁned as

those of

−1

. So every metric has all eigenvalues equal to one with respect to itself, and zero

entropy, Tr ˜g ln ˜g

−1

= 0. The Lagrangian then reduces to the cross-entropy

L = Tr ˜g ln

−1

= −Tr



G˜g

−1



G = ˜g + α

M − β

where

is built from the matter ﬁelds and

collects the Ricci scalar, Ricci tensor, and Riemann

tensor. The couplings

and

are constants of GfE;

is unrelated to the inverse temperature

At low coupling, the action reduces to the Einstein–Hilbert action coupled to matter.

What SSM does not supply. By the GfE deﬁnition of eigenvalues, every metric, ﬂat or curved,

has unit eigenvalues and zero entropy. So GfE needs no special reference metric, and the identity

bond tensor of Section 8 does not supply a missing input. The statement

eﬀ

µν

says that the

emergent spatial metric is Euclidean and isotropic in coordinates adapted to the lattice.

Where SSM and GfE meet. In vacuum and at zero curvature,

= 0 and

= 0, so

˜g

and

= 0. The spatial phase selected here has zero Regge edge deﬁcits (Lemma 9.1 and Corollary 11.5).

ϵ → ∞

, its defect density vanishes (Theorem 6.5). So it is the spatial counterpart of the

conﬁguration at which the GfE Lagrangian vanishes. This matches one conﬁguration, not two

actions. Spatial ﬂatness is necessary but not suﬃcient for the four-dimensional condition

= 0. We

do not show that the microscopic chain is a gradient ﬂow of the GfE action. At ﬁnite temperature,

the selected conﬁgurations carry defects at density

(

∗

), where ﬂatness fails. Reading those defects

as curvature or matter in

G requires the coarse-graining discussed next.

What a derivation of GfE from SSM would require. Four ingredients are missing. (i) Time

and Lorentzian signature. The Markov chain has no physical time, while GfE is formulated on

Lorentzian spacetime. (ii) Metric dynamics. The long-wavelength theory of a crystal is elasticity,

with phonons and a preferred frame. It is not a diﬀeomorphism-invariant theory of the metric.

The closest developed framework is the world-crystal approach, in which curvature comes from the

density of disclinations and torsion from the density of dislocations [

]. The Regge deﬁcit used

here measures the disclination content of the tetrahedral foam, so this route is natural. (iii) Matter.

is built from Dirac–K¨ahler ﬁelds, while matter in SSM is carried by trapped lattice defects [

(iv) The entropy. This paper uses the classical conﬁgurational entropy of bond complexes. The GfE

action is a quantum relative entropy between metric operators.

Two concrete steps toward (iv) are available. First, under

the probability of a coarse-grained

bond-tensor ﬁeld should obey a large-deviation principle whose rate functional is a relative entropy.

Comparing that functional with

Tr ˜g ln

−1

is a well-posed calculation. Proving the large-deviation

principle is a separate problem. Second, GfE has a discrete formulation on cell complexes, in which

the metric matrix of a higher-order network is the dynamical variable [

]. The conﬁgurations of

this paper are tetrahedral–octahedral cell complexes. That formulation may allow a comparison

without a continuum limit.

13 Finite-state veriﬁcation program

The assumptions above can be tested on ﬁnite systems. The ﬁnite-state program has seven steps:

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 a sharp crossover in these observables as a function of

would be direct

evidence of an ordering transition into the geometric phase. Exact enumeration carries out steps

1–6. Step 7 needs systems with a bulk interior, which exact enumeration cannot reach. Section 14

measures stacking and coordination in grown clusters instead.

13.1 Exact enumeration of a history-resolved ﬁnite model

Proposition 13.1 (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, and each move

adds one node. Substituting

ℓ

n −

− s

gives

= 3

n −

− s

. Since

s ≥

0 and

n ≤ N

, the bond

count is largest at

and

= 0, so

max

(

) = 3

N −

6. Subtracting gives

= 3(

N −n

) +

Ω

(

) contains every state with at most

nodes, so the deﬁcit depends on both

and

. The largest deﬁcit at cutoﬀ

is 3(

N −

3), reached by the seed itself. It exceeds the largest

possible stitch count,

N −

3. On the cutoﬀ layer

, the deﬁcit is simply the number of

stitches: the low-temperature measure penalizes each history by its number of stitches, relative to

the all-lift sector. Below the cutoﬀ, the penalty also counts missing nodes. This is a result about

the history-resolved veriﬁcation model. It does not classify embedded FCC growth.

So this calculation tests detailed balance and concentration with construction bonds only. Its

all-lift maximum is not a stand-in for a close-packed crystallite. The embedded model counts all

unit-distance contacts. Section 13.3 shows that its maximizers are close-packed fragments.

We carried out the program for a minimal, history-resolved simplicial version of the stitch–lift

rules. The seed is one oriented triangle. A stitch adds a vertex to a boundary edge, with two new

bonds and one triangle. A lift adds a vertex to a boundary triangle, with three new bonds and one

tetrahedron. Reverse moves remove the most recently exposed nonseed vertex. We merge states

that are isomorphic as colored incidence graphs, keeping the seed ﬁxed. This model is narrower than

the full spatial SSM. It tests the Markov-chain and entropy claims, not embedded FCC growth.

For cutoﬀs

= 4

, . . . ,

8, the reachable state spaces contain 3, 9, 30, 124, and 588 states. In

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

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

−18

. Table 3 gives the exact counts at

= 1

5. Figure 1 shows the deﬁcit counts, and Fig. 2

shows the stationary weight of the maximal sector as a function 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

construction-bond deficit

= 4

= 5

= 6

= 7

= 8

Figure 1: History-resolved model: exact number of reachable conﬁgurations

Ω

N,m

for each

construction-bond deﬁcit

. The counts obey an exponential bound with modest slopes. This

supports Assumption 6.1 at these sizes but does not prove it for large

. The evidence is thin:

= 0 at

= 4

5, so only three sizes are informative, and

is not monotone (its largest value is

= 6). No trend in

should be read from these points, and the prefactor

of Eq.

(4)

is not

controlled.

0 1 2 3 4 5 6

0.0

0.2

0.4

0.6

0.8

1.0

(

, 0

)

= 4

= 5

= 6

= 7

= 8

Figure 2: History-resolved model: exact stationary weight

(Ω

N,0

) of the maximal sector as a

function of

. It approaches one at low temperature for every cutoﬀ. The dotted line marks

ϵ = 1.5, the value used in Table 3.

So at ﬁnite size the enumeration settles three points. It handles proposal asymmetry exactly, it

checks irreducibility on the actual reachable graph, and it measures the deﬁcit entropy. Section 13.3

repeats the program for the embedded contact model, whose maximizers are the ones that matter

for geometry.

13.2 Kinetic suppression does not alter equilibrium

To model lift suppression, give forward lift proposals a relative weight

ρ >

0, keep unit weight for

stitch and reverse proposals, and normalize each row. The Metropolis–Hastings acceptance factor

contains the ratio of reverse to forward proposals. So Theorem 3.1 holds for every

ρ >

0, and the

stationary measure is still Eq. (3). Only relaxation times and path statistics can change.

We checked this for

= 1, 0

1, and

−3

on every enumerated state space. At

= 1

the maximal-sector probabilities agree to machine precision for all three proposal kernels. The

detailed-balance and stationarity residuals stay below 6

−17

(Fig. 3). This supports a distinction

used throughout the paper: lift suppression belongs to kinetics, and bond energy ﬁxes equilibrium.

Kinetics still matters. An octahedral-side Lift is the only move that leaves the lattice (Lemma 10.5),

so the Lift rate sets how much misoriented material an irreversible growth history contains. Section 14

measures this.

Lift proposal weight

0.0

0.2

0.4

0.6

0.8

1.0

(

, 0

) at

= 1.5

= 4

= 5

= 6

= 7

= 8

Figure 3: History-resolved model: exact stationary weight of the maximal sector at

= 1

5 for

lift proposal weights

= 1, 0

1, and

−3

(dashed line; the value used in Ref. [

]). The weight does

not depend on ρ. The curves for N = 7 and N = 8 diﬀer by 0.001 and overlap.

13.3 Exact enumeration of the embedded contact model

We now carry out the program for the physical model. This is the chain on the assemblable state

space Ω

emb

of Section 2.1, with energy

−ϵB

cont

. We enumerate states by forward moves from

the seed and identify them up to isometry with the canonical form of Section 7. We then ﬁnd every

reverse move by deleting each point and testing the two conditions of the reverse-move deﬁnition.

The proposal gives weight one to each Stitch choice (contact triangle and edge), weight

to each

Lift choice (contact triangle and side), weight one to each reverse move, and weight one to a hold

move that makes the chain aperiodic. We build the full Metropolis–Hastings matrix for every cutoﬀ

N ≤ 10 and count the contact-deﬁcit sectors |Ω

N,m

| for N ≤ 11.

For every

N ≤

10, the reverse moves pair exactly with the forward moves, as Proposition 2.6

requires. The transition graph is a single strongly connected class. Detailed-balance residuals

are below 10

−18

, and stationarity residuals are below 2

−16

. The stationary weight of the

maximal-contact sector agrees to machine precision for

= 1, 0

1, and

−3

. The canonical form is

exact for every state through

= 10. At

= 11, 9 of 96 632 keys fall back to an invariants-only

form, which could merge at most a few distinct states.

Table 4: Exact enumeration of the embedded chain with contact energy.

Ω

emb

counts assemblable

conﬁgurations with at most

points.

is the empirical entropy slope deﬁned in Table 3, here

for contact deﬁcits.

1/2

and

0.9

are the values of

at which the stationary weight of the

maximal-contact sector reaches 0

5 and 0

9. “Close-packed” counts maximizers that are subsets of

a Barlow packing, tested with every contact triangle as a basal plane. “With octahedra” counts

maximizers that contain at least one regular octahedron; the number of octahedra per maximizer is

in parentheses. We verify the Metropolis–Hastings matrix through

= 10; the

= 11 row uses

exact counts only.

N |Ω

emb

| B

max

maximizers bs

1/2

0.9

close-packed with octahedra

6 18 12 1 1.39 1.60 3.61 0 0

7 74 15 4 0.92 1.42 3.20 1 1 (1)

8 393 18 10 1.41 1.79 3.67 4 4 (1)

9 2 396 21 32 1.61 2.00 3.87 10 11 (1)

10 16 129 25 3 3.72 3.85 5.93 3 (1 FCC, 2 HCP) 3 (1–2)

11 112 761 29 1 3.16 3.59 5.24 1 (HCP) 1 (2)

0 2 4 6 8 10

0.0

0.2

0.4

0.6

0.8

1.0

(

, 0

)

= 6

= 7

= 8

= 9

= 10

= 11

Figure 4: Embedded contact model: exact stationary weight

(Ω

N,0

) of the maximal-contact

sector as a function of

. Dashed:

= 6–9. Solid:

= 10

11, past the crossover. There the

concentration moves to larger β

ϵ, and the N = 11 curve is the steepest.

Table 4 and Fig. 4 show four things.

The energy and the geometry use the same count. Through

= 9 the contact maximum

is 3

N −

6, and the maximizers are mixed. Most are polytetrahedral and not close-packed, as in the

exponential ground-state degeneracy of small hard-sphere clusters [

]. At

= 10 and 11, where the

contact maximum ﬁrst exceeds 3

N −

6, every maximizer is a close-packing fragment with octahedral

cells. These are the known maximal-contact clusters: three at

= 10 with 3

N −

5 contacts, and

a unique HCP fragment at

= 11 with 3

N −

4. So at low temperature the stationary measure

concentrates on close-packed conﬁgurations, and the chain

∝ e

ϵB

cont

−→ max B

cont

−→ close packing

holds at both sizes in the enumeration where maximal contact exceeds 3N − 6.

The contact-deﬁcit entropy is larger. At the crossover, the number of maximizers drops

from 32 to 3, while the sector one contact below stays large. So

jumps from 1

61 to 3

72, and

the stationary weight of the maximizers at

= 1

5 falls to 0

03. Concentration still occurs, but

it needs

ϵ ≈

4 for a majority and

≈

6 for 90%. The construction-bond model of Section 13

understates this cost, because its maximal sector never loses degeneracy.

The slope is a small-system quantity. From

= 10 to 11,

falls from 3

72 to 3

16 and

1/2

from 3

85 to 3

59. The concentration curve gets steeper: its maximum slope is 0

41 at

= 11,

against 0

29 at

= 10 and 0

32–0

38 for

N ≤

9. These trends should not be extrapolated. Under

the hypotheses of Proposition 6.3,

cannot stay bounded once clusters have an interior. The

number of ways to place a single defect grows with

, so the slope should grow like

log N

, and

the stationary weight of the exact maximizers must eventually fall. The thermodynamic statement

is the concentration of the defect density (Theorem 6.5). These sizes are too small to contain an

interior point with ﬁlled outer shells, so they cannot test it.

Energy selects close packing, not stacking. The unique

= 11 maximizer is an HCP

fragment, as for hard-sphere clusters. This agrees with Remark 10.2: the contact count selects close

packing and does not favor FCC in the bulk. The Stitch move selects FCC (Section 10).

The enumeration measures the ﬁnite-size content of Assumption 6.1 for the contact count, whose

maximizers deﬁne the geometry. The thermodynamic input is Assumption 6.4.

Kinetic realization: the growth simulation of the matter paper

Theorem 10.4, Lemma 10.5, and Proposition 10.6 make three predictions for any growth process

built from these moves. Stitch-generated material stacks cubically. Material leaves the seed lattice

only through a Lift. The seed plane is a preferred place for a twin. We test these predictions on the

irreversible growth simulation of Ref. [

] (Section 2.3 there). It grows a cluster from a triangular seed

by Stitch and Lift, with Lift probability

lift

, overlap exclusion 0

, and proximity bonding within

. We use the reference implementation without changes. We record each move by subclassing,

without altering the move. The Stitch places the apex by Eq.

(11)

whenever the edge belongs to an

occupied triangle. For the rare edges that do not (68 of about 3

moves at

lift

= 0

05), it falls

back to an in-plane construction that is not an inversion. The Lift picks one of the two apexes of

Eq. (10) by a fair coin. So the simulation carries no stacking or chirality information.

Measurements. The seed lies in the plane

= 0, so the two seed lattices Λ

are known exactly.

We classify each node as on Λ

, on Λ

−

, or on neither. The lattice with more nodes is the seed

lattice of that run. We measure stacking separately from this classiﬁcation. Every node at height

2/3 L

goes in layer

. Each layer with at least ten nodes gets its majority lateral registry. Every

run of three consecutive layers with distinct adjacent registries is cubic (

k−1



k+1

) or hexagonal

(

k−1

k+1

). All results use

= 1000, with seeds 0–29 at

lift

= 0

05 and seeds 0–14 at

lift

= 0

and 0.2. Table 5 summarizes the results.

Table 5: Stacking and lattice statistics of the growth simulation of Ref. [

] at

= 1000. “Seed

lattice” is the fraction of nodes on the lattice selected at the seed (mean

sample standard deviation

over runs). “First exit” is the move that created the ﬁrst node oﬀ the seed lattice. In every case,

that node’s three parent nodes were on the seed lattice.

lift

runs cubic : hex cubic frac. seed lattice ⟨K⟩ ﬁrst exit Lift : other

0.01 15 140 : 3 0.979 0.89 ±0.10 8.72 ±0.35 12 : 1 (2 runs none)

0.05 30 310 : 11 0.966 0.84 ±0.10 8.90 ±0.26 29 : 1

0.20 15 156 : 5 0.969 0.61 ±0.16 8.06 ±0.39 15 : 0

Cubic stacking without a stacking rule. Across all lift rates, 606 of 625 stacking triplets

(97%) are cubic, even though the Lift side is random. At

lift

= 0

05, 384 of 434 new layers were

started by a Stitch across a triangle with an interlayer bond, and only 50 by a Lift. So Eq.

(11)

, not

a coin, ﬁxes the registry of most layers. This is the mechanism of Theorem 10.4 at work, and it

explains the mostly

ABC

stacking reported in Ref. [

]. The stacking is mostly cubic, not entirely.

The remaining hexagonal triplets are the stacking faults and twins discussed below.

Lift is the only exit from the lattice. In 56 of the 58 runs that contain an oﬀ-lattice node,

the ﬁrst such node came from a Lift whose three parents were on the seed lattice, as Lemma 10.5

requires. In the other two runs, it came from the non-inversion fallback Stitch described above. No

inversion Stitch ever left the lattice. In two runs at

lift

= 0

01, every node lies on the seed lattice.

The fraction of misoriented material rises steeply with the Lift rate, from about 11% at

lift

= 0

to about 39% at 0

2. The fraction of hexagonal triplets among horizontal layers stays near 3%. So

most Lift-nucleated material forms tilted grains, not horizontal stacking faults. This is the kinetic

counterpart of the equilibrium result of Section 13. The Lift rate does not change the stationary

measure, but it does set the defect content of an irreversible growth history.

lift

−3

a kinetic optimum? Ref. [

] identiﬁes

−3

as the Lift amplitude consistent with

saturation at

= 12. The stationary measure does not depend on the Lift rate, so any special

value would have to be kinetic. We tested this with a sweep of 16 values of

lift

from 0

003 to

2, including

−3

, with 40 seeds each at

= 1000. We ﬁxed three measures before running the

sweep: the mean coordination

⟨K⟩

; the fraction of bulk-interior nodes (at least 42 nodes within

, the criterion of Ref. [

]) with

= 12; and the fraction of nodes on the seed lattice (Fig. 5).

The mean coordination has a broad maximum. A quadratic ﬁt in

log P

lift

around the largest value,

bootstrapped over seeds, puts the peak at

lift

≈

026 (95% interval 0

014–0

050). Every value

from 0

015 to 0

08 lies within about two standard errors of the maximum. The other two measures

have no interior maximum. Both fall steadily as the Lift rate increases, from 0

65 and 0

94 at

lift

= 0

003 to 0

43 and 0

61 at 0

2. The second growth implementation distributed with Ref. [

]

(

ssm bulk analysis.py

) shows the same pattern. Its mean coordination peaks near 0

025, and

its bulk

= 12 fraction falls from 0

77 at

lift

= 0

003 to 0

58 at 0

1. So

−3

lies inside a broad

plateau of mean coordination, but it is not an optimum of any of the three measures. Lifts trade

dimensionality against order. Too few give sheet-like clusters with low mean coordination. Each

additional Lift can nucleate misoriented material and lower bulk crystallinity. Neither this kinetic

analysis nor the equilibrium analysis selects a unique Lift rate.

lift

8.0

8.2

8.4

8.6

8.8

9.0

mean coordination

lift

0.40

0.45

0.50

0.55

0.60

0.65

bulk-interior

= 12 fraction

lift

0.60

0.65

0.70

0.75

0.80

0.85

0.90

0.95

fraction on seed lattice

Figure 5: Lift-probability sweep of the kinetic simulation of Ref. [

] at

= 1000 (16 values, 40

seeds each; error bars are standard errors). Left: mean coordination, with a broad maximum near

lift

≈

026. Center: fraction of bulk-interior nodes (at least 42 nodes within 2

) with

= 12.

Right: fraction of nodes on the seed lattice. The dashed line marks e

−3

The seed plane. Proposition 10.6 (Section 10.2) predicts a twin in the seed plane whenever both

seed apexes are occupied before the two growth regions join. The seed plane is the middle layer of a

hexagonal triplet in 7 of 60 runs (11

7%). The other layers give 12 hexagonal triplets out of 565

1%). The diﬀerence is signiﬁcant (Fisher exact test, odds ratio 6

p ≈

001). The seed is the

one place where both global orientations often form. Away from it, the orientation chosen at the

seed is inherited, and only later Lift nucleation changes it.

Table 6: Fraction of bulk-interior nodes (at least 42 nodes within 2

, the criterion of Ref. [

]) lying

on the seed lattice, by distance

from the seed centroid. Shells with fewer than 100 bulk nodes are

omitted.

lift

r < 2L 2L ≤ r < 4L 4L ≤ r < 6L 6L ≤ r < 8L

0.01 0.900 0.906 0.913 0.977

0.05 0.894 0.895 0.874 0.870

0.20 0.849 0.786 0.759 –

Far from the seed. Whether a region far from the seed lies in a single domain depends on two

competing eﬀects: inheritance from the seed and Lift nucleation along the way. Table 6 shows that

both regimes occur. At

lift

= 0

01 the bulk becomes more coherent with distance from the seed. At

05 coherence is nearly constant. At 0

2 it decreases. Clusters with

= 1000 extend only about

from the seed, so these trends show the start of the far-ﬁeld regime, not its limit. A region far

from the seed has a single orientation only if the Lift-nucleated grains are larger than the region.

This is the quantitative content of the hypothesis

(

) in the thermodynamic corollary of

Section 10. Larger simulations can test it directly.

15 Boundaries of the result

The construction starts without a metric manifold, but it does have inputs. It assumes:



relational nodes and bonds;



a minimal seed;



local reversible move rules and their Euclidean realization at unit bond length (the Lift brings

in dimension three);



a bond scale ϵ;



a stochastic bath and inverse temperature β

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

an ordered geometric phase.

The continuum limit is open. A bridge to GfE needs a coarse-graining argument (for example

large deviations, Γ-convergence, or renormalization) that connects the stationary measure to the

relative-entropy action. It also needs the ingredients listed in Section 12: time and Lorentzian

signature, metric dynamics beyond elasticity, and matter ﬁelds. This paper supplies the ﬁnite-state

and geometric results that must come before such a limit.

Several inputs remain open. On the kinetic side, the FCC selection of Theorem 10.4 is exact

for stitch growth and needs no extra state on the growth front. What remains is the density of

Lift-nucleated misoriented material, and in particular whether grain size grows with

so that

(

). The growth data of Section 14 stop at

= 1000 and do not settle this. On the

equilibrium side, vacancy entropy rules out a uniform entropy slope when the maximal sector has

subexponential degeneracy (Proposition 6.3). That hypothesis agrees with the enumeration but is

not proved. Two further inputs remain. One is the Peierls counting bound of Assumption 6.4, a

standard form of low-temperature control not yet proved for this state space. The other is compact

assembly (Assumption 11.3), veriﬁed exactly up to 3 055 points. The local close-packing result

also relies on reading Hales’ proof as a local statement (Lemma 11.1). His argument supports this

reading, but his theorem does not state it. Appendix C traces the reading step by step.

16 Conclusion

The model starts from a relational state space, not a metric manifold. Its stochastic assembly need

not follow the gradient ﬂow of a continuum gravitational action. The question is which large-scale

phase the ﬁnite-state dynamics selects.

The physical states are assemblable point sets, which carry no history, and the energy is the

contact count that deﬁnes close packing. On every ﬁnite state space, the Metropolis–Hastings chain

is reversible and irreducible with no further assumption. Restricting to assemblable sets keeps

the state space ﬁnite. Exact enumeration through

= 10 veriﬁes detailed balance, stationarity,

irreducibility, and independence from lift suppression. Exact counts through

= 11 measure the

contact-deﬁcit entropy. At

= 10 and 11, the maximal-contact conﬁgurations are close-packing

fragments with octahedral cells. So the energy that concentrates is the same count that deﬁnes the

geometry. The simpler construction-bond count shares its maximizers with the contact count only

through n = 9, and it understates the entropy that concentration must overcome.

In the thermodynamic limit, if the maximal sector has subexponential degeneracy, vacancy

entropy rules out a uniform entropy bound. Under a Peierls counting bound, the contact-deﬁcit

density falls below

∗

∝ e

−cβ

. The local content of Hales’ proof of the Fejes T´oth contact conjecture

then shows that equilibrium conﬁgurations are close-packed, with identity eﬀective metric and ﬂat

spatial cells, except on a set of density O(δ

∗

). This holds for every Barlow stacking.

The Stitch move ﬁxes the stacking. It is a point inversion, so Stitch growth from a tetrahedron

stays on one FCC lattice. The ﬁrst Lift on the seed selects the global orientation, and later

octahedral-side Lifts are the only source of misoriented material. The growth simulation of the

companion paper conﬁrms this: 97% of stacking triplets are cubic, and in 56 of 58 runs the ﬁrst

node to leave the seed lattice comes from a Lift. The equilibrium route of this paper and the kinetic

route of Ref. [

] reach the same close-packed phase with coordination 12, through the same Stitch

move.

Under these conditions, a ﬂat, isotropic spatial vacuum arises as the equilibrium phase of

microscopic dynamics with no prior metric manifold. It is the spatial counterpart of the conﬁguration

at which the GfE action vanishes. Three inputs remain open: the Peierls counting bound, compact

assembly at all sizes (veriﬁed to 3 055 points), and, on the kinetic side, how grain size scales with

system size. Deriving the GfE action itself would further require time and Lorentzian signature,

metric dynamics beyond elasticity, matter ﬁelds, and a coarse-graining of conﬁgurational entropy

into a relative entropy of metrics.

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

These polynomials belong to the history-resolved veriﬁcation model. Their coeﬃcients are the

deﬁcit-sector counts

Ω

N,m

for

= 4

, . . . ,

8, and they sum to

Ω

= 3, 9, 30, 124, and 588.

They make the ﬁnite concentration curves exact algebraic quantities, not Monte Carlo estimates.

The contact-deﬁcit counts of the embedded model for

= 4

, . . . ,

11 are in the repository ﬁle

embedded deficit counts.json. For both models,

(Ω

N,0

) =

|Ω

N,0

max

(N)

(x)

B Assemblability tests

Lemma B.1 (Assemblability is a monotone closure). Let

be a ﬁnite packing and

S ⊆ T

a contact

triangle. Let

(

) be the smallest subset of

that contains

and contains every forward-move

apex of its contact triangles that lies in

. Then

is assemblable if and only if

(

) =

for

some contact triangle S of T , and cl

is monotone in S.

Proof.

Forward moves only add points, and adding a point never removes a contact triangle. Overlap

exclusion never blocks a point of

, because

is a packing. Any growth sequence that builds

uses only points of

. Any maximal growth sequence inside

reaches

(

), whatever the order of

moves. A larger S can only give a larger closure.

The lemma makes assemblability an exact computation that does not depend on move order.

By monotonicity, once a cluster is assembled, any larger target that contains it can be reached from

it, and every intermediate state is assemblable. Table 7 lists the tests. Two negative controls show

that the test can fail. An isolated regular octahedron is not assemblable: its closure stops at the

seed triangle, because building the octahedron needs an extra tetrahedral apex (Section 2.1). A ball

with one detached point is not assemblable either.

Table 7: Exact assemblability tests by monotone closure. Vacancy conﬁgurations are drawn at

random from interior sites (distance below 3.5L from the center) of the FCC ball of radius 5L.

Target points assemblable

FCC cuboctahedra, 1–4 shells 13, 55, 147, 309 all

FCC balls, radius 1.5, 2.5, 3.5, 5, 6, 7, 8 L 19 to 3 055 all

HCP balls, radius 1.5, 2.5, 3.5, 5 L 19 to 763 all

FCC ball with 2–59 pairwise non-adjacent interior vacancies

60 conﬁgurations 60/60

FCC ball with a compact interior vacancy cluster of 1–10

sites

60 conﬁgurations 60/60

Isolated octahedron; ball with a detached point 6; 20 neither

Assemblability belongs to a conﬁguration, not to each of its triangles. In one vacancy conﬁgura-

tion, the ﬁrst seed triangle we tried was trapped. Its tetrahedral-side Lift apex and its three Stitch

apexes are lattice sites, no two of them adjacent, and all four had been removed. So the closure

could not leave the triangle. The same conﬁguration assembles completely from 9 of 10 other seed

triangles.

We assembled the FCC ball of radius 8

shell by shell. Every intermediate state with 20

≤ N ≤

3 055 satisﬁes 6

N −B

cont

≤

2/3

, and the complete ball has 6

N −B

cont

= 8

2/3

. This gives

the constant C

of Assumption 11.3 over that range.

C Derivation of Lemma 11.1 from Hales’ argument

This appendix traces the proof of Theorem 1 of Ref. [

] step by step and records what each step

assumes about the packing (Table 8). Hales uses balls of radius one, so contacts are at distance 2.

All statements below use his units; multiply by

2 to convert to ours. His truncation parameter is

= 1.26, and his weight function is ℓ

(h) = (h

− h)/(h

− 1).

Fix a ﬁnite packing

and a point

u ∈ X

with twelve contacts

, . . . , u

. Suppose each

also

has twelve contacts in X. Translate so that u = 0, and let V

= {u

, . . . , u

Table 8: The steps of Hales’ proof of Theorem 1 [

] and what each uses. “Local” means that the

step involves only u, its twelve contacts, and at most one further point.

Step in Ref. [13] Content What it uses

Lemma 1 For a ﬁnite packing V with 2 ≤ ∥v∥ ≤ 2h

for

all v ∈ V :

v∈V

ℓ

(∥v∥/2) ≤ 12. Proved in

Ref. [14].

Only the ﬁnite packing V .

Applies to every ﬁnite subset of

Lemma 2 If a point has twelve contacts, every other

point of the packing is at distance 2 or at least

from it.

Proof applies Lemma 1 to the

twelve contacts plus one further

point. Local.

Deﬁnition 1 (class

Twelve points on S

(2) whose pairwise

distances are 2 or at least 2h

A property of the twelve points

only.

Lemma 3 A conﬁguration in V can be deformed within

V, preserving its contact graph, so that the

graph with edges of length below

√

8 is a

biconnected fan.

Only the twelve points.

Theorem 2,

Lemmas 4–7

Solid-angle estimates on the faces of the

contact fan; node degrees at most four.

Only the twelve points.

Lemma 5 uses

ℓ

= 12,

which holds because all twelve

points have norm 2.

Theorem 3

The contact hypermap of any

V ∈ V

has tame

contact.

Lemmas 3–7.

Lemma 8 Computer classiﬁcation: eight tame-contact

hypermaps up to isomorphism and reversal.

Combinatorics only.

Lemma 9 Six of the eight are not realizable (one

geometric argument, ﬁve linear programs); the

remaining two are the FCC and HCP contact

hypermaps.

Only the twelve points.

Lemma 10 A conﬁguration in V with the FCC or HCP

contact hypermap is congruent to the FCC or

HCP kissing arrangement.

Only the twelve points.

Proof of Thm. 1 Apply the preceding at an arbitrary point of

the packing.

Conjunction of the local

statement over all points.

Where the hypotheses enter. Lemmas 3–10 hold for every conﬁguration in

, so it is enough

to show

∈ V

. The ﬁrst two conditions of Deﬁnition 1 ask for twelve points at distance 2 from

. They hold because

has twelve contacts. The third condition asks that every pair

, u

be at

distance 2 or at least 2

. Ref. [

] states Lemma 2 for a packing in which every point has twelve

contacts, but its proof uses only the twelve contacts of the point where it is applied. Apply it at

which has twelve contacts by hypothesis. It shows that

is either a contact of

or at least 2

away from it. This is the only place the hypothesis on the neighbors of

is used. Nowhere else does

the argument refer to points of X outside V

What the lemma does not claim. If some neighbor of

has fewer than twelve contacts, two

neighbors of

may be at a distance between 2 and 2

. Then

/∈ V

, the classiﬁcation does not

apply, and we draw no conclusion about

. Corollary 11.2 counts such points as defects. The lemma

inherits the computer-assisted parts of Ref. [

]: Lemma 1 (via Ref. [

]) and Lemmas 8–9. We

claim nothing beyond this reading of the published argument.

D Reproducibility identiﬁers

The SHA-256 hashes of the veriﬁcation and analysis scripts are

finite_ssm_verification_v3.py

1201ae4442a2d4577306a334c42be3e9fa7574c8ee125f60805bd0baf27e7780

run_finite_verification_v3.py

8c79276a8970dc4d720218c62295aed2dac811809a06fbcc1a3b763b5ba32223

embedded_crossover_v1.py

fc174ad18a986d93a20a972e3bcd63e33ed9d166e6b74471f723bbd63a29a02a

embedded_contact_mh.py

07bb4a7262fa8d67a878fde34eb6bd2feed9af6fb8e0fa30f5fe31c9ccc98c36

contact_mh_verify.py

0833af5301ba3cc1df8f7335e7f72d06d7ae9cf1ccb5cf2aaab76bf611c47151

barlow.py

406b1ff0dffa8742d94d5619c491a64a8322935381f636a62c4f7c62db75d995

layer11.py

60283e034921365f15f868e12720a7ae7c9ddfcfea1d9e1191001e57727545a3

depth_probe.py

674b4e3d4eef1085db804566d3dd18475f0dbee8942e44a366414dc53dd67628

assemble.py

92f2d3f1d4ea483efd12f9747881b492e362133a687873c14e59837750f7a1ba

vacancy_tests.py

004d3fe6cc5ddc0688383fa2a388aaa88632b19bfbe26d97ca201520a88a214c

compact_path.py

1a05f245607db3b694f3a9cd875d464c4b252c262a3414bf39542a0cf353a914

plift_sweep.py

9e00526a135f4e5305f3789ad305ce3e1963605b34df4eeef410223cb1bf3ba5

summarize.py

098553fe3e8e81a497c7c56ed84781b73fbfaac3d6c1509f794c85f69394eb3c

variant_sweep.py

00147c91191fc8cd06916b8311a02eb412c210f80cc0bd77551593419014b7fb

make_figures.py

2f3f2a6bba21d1acf30b16b41534d99efd283c23c83af738f7db1463325d22bb

stitch_inversion_analysis.py

ce594de3d661385f7bcd79749a8ed993eb934c884f2e1aaec341bfb7139319e9

ssm_sim.py (matter-paper reference implementation, unmodified)

32a55190f04ffe73d5cb69af54109dfbd232f52479d5cad4e50f72145782f38f

Code availability

All scripts are in the SSMTheory GitHub repository. Each name below links to its ﬁle. Appendix D

lists their SHA-256 hashes. The scripts need Python, NumPy, SciPy, NetworkX, and Matplotlib.



History-resolved model (Section 13, Table 3):

finite_ssm_verification_v3.py

and

run_

finite_verification_v3.py

. They enumerate the states, build the Metropolis–Hastings

matrix, and check detailed balance and stationarity. Runtime through

= 8 is about ten

seconds.



Construction versus contact bonds (Table 2):

embedded_crossover_v1.py

. Runtime through

n = 11 is under two minutes; n = 12 takes about twenty minutes and several gigabytes.



Embedded contact model (Sections 2.1 and 13.3, Table 4):

embedded_contact_mh.py

builds

the state space and moves through

= 10 (about two minutes).

contact_mh_verify.py

builds the transition matrices and uses

barlow.py

to classify maximizers.

layer11.py

extends

the counts to n = 11 (about ﬁve minutes in four chunks).



Inﬁnite closure (Remark 2.7): depth_probe.py.



Assemblability (Appendix B):

assemble.py

(closure, clusters, negative controls),

vacancy_

tests.py, and compact_path.py (the constant C



Growth simulation (Section 14):

stitch_inversion_analysis.py

runs the unmodiﬁed refer-

ence simulation

ssm_sim.py

of Ref. [

] and records moves by subclassing, without changing

any rule. summarize.py computes Tables 5 and 6 from its output.



Lift-probability sweep (Fig. 5):

plift_sweep.py

, and

variant_sweep.py

for the second im-

plementation (ssm bulk analysis.py of Ref. [3]).



Figures: make_figures.py regenerates every ﬁgure from the stored data.

All growth runs use ﬁxed random seeds.

Declarations

Funding. No funding was received for conducting this study.

Conﬂict of interest. The author declares no competing interests.

Data availability. The deﬁcit counts of the history-resolved model are the coeﬃcients of the

partition polynomials in Appendix A. The contact-deﬁcit counts of the embedded model for

= 4

, . . . ,

11, the per-run records of the growth simulations, and the Lift-probability sweep are

in the repository as machine-readable ﬁles. The enumerations are exact and deterministic. The

growth simulations use ﬁxed random seeds, so every number can be regenerated with the scripts

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

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