Face-Centered Cubic Vacuum from Discrete Entanglement Networks

Emergent Face-Centered Cubic Vacuum from

Discrete Entanglement Networks

Forced

K=12

Saturation and an Emergent Light Cone

Raghu Kulkarni

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

raghu@idrive.com

Abstract

We ask what large-scale geometry follows if the vacuum is a discrete entanglement

network instead of a continuum. Close packing, a causal signal cone, and approx-

imate isotropy all arise from the network's own growth and dynamics, rather than

being imposed. The treatment has two parts.

Part I: kinematic construction.

Two local operators build the network: a planar

stitch and a rare out-of-plane lift, suppressed by a factor

−3

. Under this rule the

network passes through a frustrated tetrahedral foam (Regge decit

δ ≈ 7.36

◦

) and

closes at the Kepler kissing bound [3]. We formulate the operators as a reversible

Markov process on bond complexes carrying no coordinates or metric, and prove

that it concentrates on maximally bonded complexes whenever the congurational

entropy of the decit sectors is bounded. Exhaustive enumeration through

N=8

conrms reversibility and stationarity to residuals below

7 × 10

−18

, and shows that

the lift suppression shifts kinetics without moving the equilibrium. Because proxim-

ity bonding counts contacts, the physical energy lives on an embedded state space,

and we state which results transfer to it.

Maximizing contacts gives close packing at every size enumerated. Such a packing

carries the identity spatial metric as its local bond second moment, and an explicit

[[192, 130, 3]]

CSS code at

67.7%

rate on its edges [21].

Part II: dynamical foundations.

One bond Hamiltonian, coupled to a thermal bath,

supplies the dynamics beneath that growth. Its dissipative limit assembles the lattice

at the detailed-balance rate

q = 1/(1 + e)

. Its coherent limit propagates a ballistic

front at

c = 4

√

2 v

lat

. We prove that

K=12

is a

forced

terminus. The largest

cuboctahedral vacancy sits at

∗

2 −

√

2 L

, so no thirteenth neighbor ts. The

lift rate factorizes into a free geometric acceptance and a xed thermal factor

−3

The model must supply one further scale, a bond rigidity

κL

≳ 36 ε

, which a single-

scale vacuum cannot meet. Matching the signal cone to the elastic trace speed then

xes the stiness at

κ = 16 mε

/ℏ

Part I

Kinematic Construction

1 Introduction

The constructive generation of three-dimensional macroscopic spaces from discrete, fun-

damental components is a central problem in statistical mechanics, network theory, and

discrete models of quantum gravity [1, 4].

The obstacle.

A persistent obstacle is geometric frustration. When discrete nodes

are assembled via purely randomized three-dimensional probabilistic rules, the resulting

structures typically resemble diusion-limited aggregation [2] or kinematically jammed,

fractal-like foams. These porous congurations fail to achieve the dense, uniform coor-

dination required to mimic a continuous, isotropic spatial bulk. For a discrete spatial

model to be physically viable, it must demonstrate a natural kinematic pathway to struc-

tural saturation. In three dimensions, optimal spatial saturation is formally dened by

the Kepler conjecture, which dictates a maximum coordination number of

K = 12

[3].

Achieving this state purely through local, bottom-up assembly rules without imposing a

global background coordinate system remains a signicant computational challenge.

The approach.

In this paper we explore a resolution to this geometric bottleneck by

altering the dimensional probabilities of the generative kinematics. Motivated by holo-

graphic models of boundary-volume correspondence [4, 5], we hypothesize that saturated

3D volumes can be deterministically generated if the underlying assembly process is over-

whelmingly two-dimensional.

The model.

We work with the Selection-Stitch Model (SSM), seeded by a Bell-pair

triangle, three unit-distance bonds joining three nodes, the minimal closed simplicial unit,

and extended by two topological operators: a 2D lateral expansion (the stitch) and a rare

out-of-plane 3D projection (the lift). The relative amplitude of these operators is taken

lift

= e

−3

≈ 4.98%

, motivated by the codimension dierence between their solution

manifolds (Section 2.2) and consistent with the QEC survival ansatz of the

[[192, 130, 3]]

CSS code that the emergent lattice supports (Sections 2.3 and 5); Part II derives this value

as the thermal factor of the lift rate, the geometric prefactor remaining free (Section 17).

What follows.

Under these constraints the system traces a kinematic cascade

K = 1 →

K = 6 → K = 4 → K = 12

, escaping the geometrically frustrated

K = 4

tetrahedral

foam [9] and closing at the Kepler bound. Ref. [22] demonstrates that cascade by direct

simulation.

Part I asks the sharper question that a simulation cannot settle: what does this dynamics

select

, and can the answer be proved rather than sampled? We therefore give the operators

a stationary measure and study what it favors. Section 3 sets up that measure and the

conditions under which the chain converges to it; Section 3.6 veries those conditions

exactly on small state spaces; and Section 3.8 shows that the favored phase carries a at

isotropic spatial metric built from its own bond directions.

Relation to the companion papers.

The stitch and lift operators were introduced by

the author in Ref. [22], which reports a growth simulation of the resulting cascade and its

saturation statistics. This paper does not repeat that simulation. Part I instead formu-

lates the same operators as a reversible Markov process on pre-geometric bond complexes,

and establishes by exact enumeration the reversibility and concentration properties that

the growth picture assumes. The

[[192, 130, 3]]

CSS code of Section 5 is constructed in

Ref. [21]. Everything else is new: the selection-by-persistence motivation for the bond

and the operators, the error-correction reading of the lift amplitude, the nite-state veri-

cation, and the whole of Part II.

Interactive 3D Visualization.

A 31-frame animated WebGL application

showing step-by-step lattice growth, from the seed triangle through 2D sheet

expansion to

−3

out-of-plane lift events, is available at:

https://raghu91302.github.io/ssmtheory/qec_spacetime_3d.html

Nodes are color-coded by coordination (

K = 12

green,

K = 10



orange,

K < 10

red). Controls: play/pause, frame stepping, drag to orbit, scroll to

zoom.

2 Methodology

This section denes the operators, the amplitude that weights them, and the parameters

they act on. Section 3 then embeds them in a dynamics and asks what that dynamics

selects.

Growth is unbounded. The complex extends freely from the seed triangle, with no periodic

wrapping, xed walls, or imposed cell. Its boundary is a free surface set by the kinematics

instead of by a container, so under-coordinated nodes sit at that surface. Ref. [22] exploits

this in its surface-to-volume analysis of the growth simulation.

2.1 Generative operators and proximity bonding

Motivation, not derivation.

The bond and the two operators below are posited, and

the construction takes them as given. The picture that motivates them is set out below,

as motivation rather than derivation.

Suppose the initial condition carries no structure: no geometry, no ordering, only a space

of possibilities. There what happens next has no content, because next is not yet

dened, and the only question with content is which congurations

persist

. Correlations

between pre-geometric degrees of freedom form and, in the overwhelming majority of

cases, do not last. Once a geometry exists, transient correlations of this kind are the

model's counterpart of vacuum uctuations, whose physical eects are well established.

Why entanglement.

We make no claim about what else was possible. Whatever failed

to produce persistent structure left nothing behind to be inventoried. The answerable

question is structural instead of historical. Classical correlation places no bound on how

many partners a node may hold at full strength, so a classically correlated network has

no kissing-number analog, no saturation, and no close packing. Entanglement is monoga-

mous, and that is what makes coordination a bounded quantity at all. Section 16 weighs

the same constraint against codeability.

Why the triangle.

An isolated correlation has no closure. An open link is not an

observable, and only closed loops are [6]. The minimal closed object on a bond complex is

the triangle, and closure is what gives it a survival advantage, since a closed loop supports

a consistency check that an open link does not. This is the redundancy argument of

Sections 2.3 and 2.4 in its most elementary form, and carries the same heuristic status.

Status of this motivation.

Nothing in the paper's results depends on it, and four caveats

apply. The argument is circular as given, since extend presupposes the sequencing it is

meant to explain. Part II does not implement the emergent-ordering reading. Its dynamics

evolves in an external Schrödinger parameter

= ℏ/ε

(Section 10), so ordering there is

an input, exactly as position is (Section 14). The vacuum-uctuation comparison runs

one way only. Persisting correlations are what such uctuations

become

once a geometry

exists, not the substrate from which geometry is built. The monogamy appeal likewise

concerns which correlations

could

support a bounded coordination, not a reconstruction

of what occurred.

The Bell-pair primitive and the triangular seed.

We treat the unit Bell-pair

bond(two nodes joined at distance

, the discrete structural analog of a maximally en-

tangled pair)as the primitive entanglement element of the network. The construction

begins from three connected Bell-pair bonds forming an equilateral triangle of side

; this

Bell-pair triangle is the minimal closed simplicial unit that the kinematic operators below

can act on, and the FCC lattice grows from it. We use Bell pair here in a structural

rather than dynamical sense: Part I is a combinatorial model and does not implement

gauge elds, a Wilson action, or a Hamiltonian. The connection to lattice gauge theory

is purely geometric, the triangle is the minimal closed loop on a simplicial complex, and

we adopt this terminology to motivate the structural operators.

The vacuum lattice grows from this seed via two kinematic operators. Growth is ltered

by the same criterion as the seed. Each addition either creates new closure or does not.

Stitch (2D expansion):

In gauge theory, a single open link is not a physical observable;

only closed loops are gauge-invariant [6]. The minimal physical entanglement structure on

a simplicial complex is the triangle. The stitch operator generates these minimal closed

gauge-invariant loops by placing a new node at the equilateral apex of an existing edge,

growing planar

K = 6

hexagonal sheets. Geometrically, the stitch realizes the intersection

of two unit spheres centered on the existing edge endpoints, a 1-parameter (1D) solution

manifold in 3D.

Lift (3D volume generation):

Projects a new node orthogonally from an existing

triangular face at the strict tetrahedral height

h =

2/3 L

. This height is not a free

parameter; it is the unique altitude of a regular tetrahedron with edge length

, the only

distance at which all four inter-nodal separations equal

. Geometrically, the lift realizes

the intersection of three unit spheres centered on the triangle vertices, a 0-parameter (0D)

solution manifold consisting of exactly two points (one selected by orientation).

Minimal operator basis.

Under the stated assumptions, insertion of a single node at

a generic intersection of unit spheres, at xed unit bond length, stitch and lift form the

minimal local operator basis that strictly increases connectivity. Cooperative multi-node

insertions, bond rearrangements and swaps, non-generic four-anchor sites, and collective

registration moves lie outside this basis; no claim of exhaustiveness is made for them. The

stitch (2-sphere intersection) yields a 1D family; the lift (3-sphere intersection) yields a

0D pair; a four-sphere intersection in 3D is generically empty, so no third operator with

strictly greater connectivity can be dened. Together, stitch and lift form a complete

kinematic basis for

generic single-node insertion

in three-dimensional Euclidean space at

xed bond length, growth through two- or three-anchor sphere intersections; the multi-

node and rearrangement moves excluded above lie outside this basis.

Proximity Bonding:

As the network expands, any two nodes within a threshold radius

(

1.05 L

) automatically bind. This mechanism allows independent, adjacent 2D layers

to geometrically interlock, producing the

in-plane

) + 3(

above

) + 3(

below

) = K = 12

cuboctahedral coordination of the FCC lattice .

The

K = 4 → K = 12

cascade and its closure.

Acting on the Bell-pair triangle,

the two operators drive a kinematic cascade through the coordination phases

K = 1

(the

elementary Bell-pair entanglement bond, the per-node degree of an isolated pair)

→ K = 6

(stitched hexagonal sheet)

→ K = 4

(tetrahedral foam, the conguration produced by

lifts before proximity bonding closes the layers)

→ K = 12

(interlocked ABC-stacked

FCC). The

K = 4

tetrahedral foam is geometrically frustrated: regular tetrahedra cannot

tile three-dimensional Euclidean space because the dihedral angle of a regular tetrahedron

arccos(1/3) ≈ 70.528

◦

, and packing ve tetrahedra around a shared edge consumes only

5 × 70.528

◦

= 352.64

◦

, leaving an irreducible Regge decit [9]

δ = 2π − 5 arccos(1/3) ≈ 0.128

rad

≈ 7.36

◦

(1)

This residual wedge is the geometric source of the

K = 4

instability and the driver that

pushes the system to

K = 12

. The cascade closes at

K = 12

because it saturates the

Kepler kissing-number bound [3]: no further operator at unit distance can act on an

already 12-coordinated node. The FCC unit cell [11, 12] hosts this saturated state via 8

tetrahedral and 4 octahedral interstitial voids per cell, with the 12 nearest neighbors of

each interior node arranged on a cuboctahedral shell .

Tolerance window from the Regge decit.

The exclusion radius

= 0.95 L

and

proximity bond radius

= 1.05 L

are not independent free parameters: they form a

symmetric

±5%

tolerance window around the unit bond length, set directly by the Regge

decit. The wedge angle

δ ≈ 0.128

rad projects to a fractional bond-length jitter of

|sin(δ/2)| ≈ 0.064

at each node; the adopted symmetric window of half-width

is of

this natural size (slightly tighter than the full jitter), and the wide plateau of Ref. [22]

shows that nothing depends on the precise width. The robustness of

K = 12

saturation

across the entire band

∈ [0.58, 0.99] L

(Ref. [22]) further conrms that the specic

value

= 0.95 L

is not a tuned parameter.

2.2 Codimension suppression and the

−3

lift probability

The stitch and lift operators dier in the dimension of their geometric solution manifolds.

The stitch is a 1-parameter family in 3D (intersection of two unit spheres), a continuous

classical path of least resistance. The lift is a 0-parameter family (intersection of three

unit spheres), requiring the simultaneous satisfaction of

S = 3

distinct unit-distance con-

straints. The relative amplitude

lift

stitch

is exponentially suppressed in the codimension

dierence:

lift

= e

−S

= e

−3

≈ 0.04978,

(2)

where

S = 3

counts the independent codimensional constraints lifting the operator from

a continuous (1D) family to an isolated (0D) pair. The terminus itself does not select

this value:

K=12

is forced for any lift rate by the kissing bound (Part II, Section 17).

What the lift rate controls is the growth

morphology

: smaller

lift

produces at, poorly

interlocked sheet structures, while larger

lift

produces foams with poor crystalline order

(Table 2 of Ref. [22]). We treat equation (2) as a structural/kinematic motivation instead

of a derivation from a Euclidean tunneling action; the operating value

lift

= 0.05

taken from this codimension argument and tested against the volumetric-yield analysis of

Ref. [22].

2.3 Quantum error-correction interpretation of

−3

The code the lattice supports.

The codimension-suppressed amplitude (2) receives a

second, independent motivation from the quantum error-correcting code that the emer-

gent lattice supports. The FCC lattice at the smallest system supporting this code (a

4×4×4

unit-cell arrangement, 192 edges) carries a

[[192, 130, 3]]

CSS quantum error-

correcting code [13, 21], with 192 physical qubits (edges), 130 logical qubits, encoding

rate

k/n = 67.7%

, and code distance

d = 3

. Its stabilizers are the weight-12 octahedral

-checks and vertex

-checks dened in Section 5. The minimal closed gauge-invariant

loops on the simplicial complex, the triangular faces of the local coordination cluster [6],

underlie the compound error-detection argument that follows.

The survival ansatz.

node participating in

triangular stabilizer checks is protected by

independent error-

detection circuits. Below the correction threshold, the error suppression scales exponen-

tially with the decit. This motivates the survival ansatz below. It is a heuristic, not

a consequence of code distance. The distance

d = 3

guarantees correction of arbitrary

weight-one errors, and we claim no decoding model, noise channel, syndrome dynamics,

or threshold calculation behind Eq. (3):

P (

survive

|t) =

(

t ≥ d + 1

−(d+1−t)

t < d + 1

(3)

The exponential form is qualitatively consistent with threshold behavior in topological

codes [14], where logical errors below threshold are suppressed exponentially in the syn-

drome decit by direct analogy with thermal activation across a free-energy barrier in the

underlying random-bond Ising mapping of the toric code. The unit-coupling exponential

−(d+1−t)

is the minimal one-parameter form meeting three requirements. It saturates to

unity at the protection threshold

t = d + 1

. It decays monotonically with the decit

d + 1 − t

. And it recovers the textbook below-threshold scaling

P ∝ e

−∆

in the large-

decit limit. It also places the structural prediction

lift

= e

−d

, for the worst-case

t = 1

peninsula, at a value directly testable against simulation. The unit coupling is an ansatz

whose consequences we test computationally; the compound-QEC argument of Section 2.4

below is structurally robust to the specic functional form, depending only on the mono-

tonic increase of eective protection with neighborhood depth. A node produced by a

tetrahedral lift participates in exactly

t = 1

triangle. At code distance

d = 3

lift

= e

−(d+1−1)

= e

−d

= e

−3

(4)

The same value as the codimension argument (2). The two routes are consistent rather

than independent. Both exponents count the same three simultaneous unit-distance con-

straints (the three anchoring bonds of the tetrahedral lift), read kinematically in one case

and as missing stabilizer protection in the other.

2.4 Compound QEC: why at growth dominates

The QEC framework explains why 2D growth dominates without being externally pre-

scribed. Throughout this argument the triangular plaquettes serve as

heuristic

local

checks: they are the closed gauge-invariant loops of the lattice, not stabilizer generators

of the

[[192, 130, 3]]

code itself, whose checks are the weight-12 vertex and octahedral op-

erators (Section 5). What follows is a motivation for the lift suppression, not a property

of the stated code.

The in-plane node.

A node in the interior of a hexagonal sheet has

K = 6

neighbors

and participates in 6 triangles. Each triangle shares 2 edges with neighboring triangles.

An error at the central node triggers a syndrome at all 6 surrounding stabilizer checks.

The error is also independently detectable through the triangles of the node's second-

nearest neighbors. Each of the 6 neighbors participates in 5 additional triangles beyond

the one shared with the central node, giving an additional set of compound detection

paths within 2 hops. The eective protection is well above the

d + 1 = 4

threshold, and

sheet-interior nodes survive at eectively 100%.

The out-of-plane node.

A lift node sits at the tip of a topological peninsula: 1 triangle,

3 bonds to the parent face, zero neighboring triangles for redundant detection. An error

at this node can be detected only through its single parent triangle. There is no second

independent check. The eective triangle count is

eﬀ

≈ 1

The asymmetry.

The ratio of eective protection is much larger than the raw triangle-

count ratio of

6:1

, because compound detection paths grow with distance from the bound-

ary. A node

hops into the sheet interior accumulates compound detection paths from all

triangles within

-hop neighborhoods, while the out-of-plane peninsula remains capped

at 1. This makes any threshold-based selection rule, not just the specic formula (3),

preferentially destroy out-of-plane protrusions while preserving the sheet. The compound

QEC argument is structurally stable. It does not depend on the exact functional form of

the survival probability.

2.5 Informational driver of dimensional projection

The construction admits an information-theoretic reading of dimensional emergence. Ev-

ery bond represents a unit of entanglement. A 2D triangular lattice caps at

K = 6

yielding

total bonds. A 3D FCC lattice reaches

K = 12

, yielding

bonds. The

2D sheet possesses the potential for 12 connections per node but lacks the geometric

room. By projecting into a third dimension via the

−3

codimension-suppression limit,

the system doubles its entanglement capacity.

The causal chain.

It runs as follows. Minimal gauge-invariant loops, the triangles [6],

assemble into

K = 6

sheets. Those sheets have unused bonding capacity. The codimension

dierence between stitch (1D solution manifold) and lift (0D solution manifold) suppresses

out-of-plane growth at

P = e

−3

. Regge frustration of the intermediate

K = 4

tetrahedral

foam [9] then drives the cascade upward. Stacking to

K = 12

saturates the Kepler

bound [3]. Finally, the

[[192, 130, 3]]

CSS code supplies the error-correction framework,

and its distance

d = 3

is consistent with the same suppression amplitude under the

survival ansatz of Section 2.3.

2.6 Summary of kinematic parameters

Where each value comes from.

Every parameter of the construction comes from

foundational geometry. The lateral and lift heights are unique altitudes of regular triangles

and tetrahedra. The lift probability is motivated by codimension suppression and by the

QEC distance, and Part II obtains it as a thermal factor with a free geometric prefactor.

The proximity-bond and exclusion radii together form the symmetric

±5%

Regge-decit

tolerance window of Section 2.1. Table 1 provides the complete kinematic parameter

space. Ref. [22] conrms by sensitivity sweep that these operating values are not ne-

tuned.

Table 1: Kinematic parameters. All values are geometrically or thermodynamically de-

termined.

Parameter Value Derivation

Unitary metric (

) 1.0 Invariant relational distance

Lateral height

√

L ≈ 0.866 L

Equilateral triangle altitude

Lift height

L ≈ 0.816 L

Regular tetrahedron altitude

Lift probability

−3

≈ 4.98%

Codimension suppression /

QEC (Sections 2.22.3)

Proximity bond (

)

1.05 L

Regge decit (

δ ≈ 7.36

◦

Eq. (1))

Hard shell (

)

0.95 L

Regge decit (Section 2.1);

plateau veried in Ref. [22]

Observed kinetic cuto

√

L ≈ 0.577 L

Circumradius of unit tri-

angle; operator-kinetic wall

(Ref. [22])

3 The assembly as a reversible Markov process

The operators of Section 2.1 dene which moves are possible. To ask which structures the

assembly

selects

, they must be embedded in a dynamics with a stationary measure. This

section does that, and the statements below are proved for the stated model or veried

by exhaustive enumeration instead of sampled.

3.1 Pre-geometric state space

A microscopic conguration is a nite object

C = (V, E, σ)

, where

is a set of nodes,

set of undirected bonds, and

stores only the local combinatorial data needed to identify

admissible stitch and lift moves. No background coordinates, metric tensor, or curvature

eld appears in the fundamental state. Geometric realizations at unit bond length are

reconstructed

after

a move has been specied combinatorially.

Fix as seed

an oriented triangle. The smallest conguration on which both moves are

dened, and the minimal closed object of Section 2.1. The seed is not spacetime but a

nite relational boundary condition preventing the empty-state obstruction. For a node

cuto

, let

Ω

)

be the set of congurations reachable from

by admissible stitch,

lift, and reverse moves without exceeding

nodes. The cuto makes the state space

nite, so the thermodynamic question becomes a sequence of nite problems followed by

N → ∞

3.2 Detailed balance and the stationary measure

Let

B(C) = |E(C)|

and assign the microscopic energy

. This count is the

construction

bond count

hist

, and the model it denes is written

hist

below; the physical

assembly uses a dierent count on a larger state space, and Section 3.9 denes both and

states what transfers between them. Kinetic suppression of the lift is placed in the

proposal

kernel rather than in the energy, which allows an informationally suppressed lift to remain

energetically downhill. For congurations connected by one elementary move let

Q(C, C

′

)

be the proposal probability, assumed only to satisfy

Q(C, C

′

) > 0 ⇐⇒ Q(C

′

, C) > 0

, and

take the MetropolisHastings acceptance rule [7, 8]

A(C, C

′

) = min



1, e

−β(E

′

)−E

(C))

Q(C

′

, C)

Q(C, C

′

)



(5)

Proposition 1

(Detailed balance)

The kernel

P = QA

satises detailed balance with

respect to

−1

exp



βεB(C)



, Z

C∈Ω

βεB(C)

(6)

For adjacent congurations this is the standard MetropolisHastings identity; for non-

adjacent pairs both sides vanish. The local logistic form

q/(1 −q) = e

−βε

of Section 14 is

the special case of symmetric proposals where a reverse move removes one bond; the full

kernel is needed when the numbers of eligible birth and deletion proposals dier.

Detailed balance identies a stationary measure but does not guarantee convergence to

it. Convergence needs the state space to be connected under the moves. If every non-seed

conguration contains at least one exposed non-seed node whose deletion is an admissible

reverse move, a precise form of the reversibility of construction histories, then repeated

deletion reaches

in nitely many steps, any two states communicate through the seed,

and a positive hold probability gives aperiodicity. The chain is then irreducible and

aperiodic, and

is its unique limit from every initial state. This condition is checkable

algorithmically, and we verify it below.

3.3 Entropy production and what selects the phase

For a transition

C → C

′

the system and medium entropy changes satisfy

∆s

sys

+∆s

med

= 0

on an equilibrium trajectory pair, and during relaxation from a non-equilibrium distribu-

tion

the relative entropy

D(ρ

∥π

)

is non-increasing. Local bond order can therefore

increase while entropy is exported to the bath.

It is worth separating three roles that are often collapsed into entropy. Detailed balance

together with the bond energy determines the stationary measure

. Entropy production

drives relaxation toward it. And a free-energy competition between the bond energy and

the congurational entropy of the decit sectors determines which macrostate dominates

. In that competition the congurational entropy opposes concentration and the bond

energy drives it. Entropy production makes the relaxation irreversible and the equilibrium

phase attainable; it does not by itself supply the degrees of freedom, the scale

, or the

direction of the selection.

3.4 Concentration on maximal bonding

Let

max

(N) = max

C∈Ω

B(C)

and grade the state space by bond decit,

Ω

N,m

= {C :

B(C) = B

max

(N) − m}

. Suppose the decit-sector counts obey

|Ω

N,m

| ≤ a

|Ω

N,0

∗

(7)

for constants

∗

< ∞

and

≥ 1

with

log a

= o(N)

. Then for

βε > s

∗



max

(N) − B(C) ≥ M



≤

−(βε−s

∗

1 − e

−(βε−s

∗

)

(8)

which follows by summing the geometric series

(Ω

N,m

)/π

(Ω

N,0

) ≤ a

−(βε−s

∗

over

m ≥ M

. The tail is exponentially suppressed in the decit threshold at each nite

; a

thermodynamic statement additionally requires control of

and of how

scales with

The structure of the result is what matters here. Bond energy alone is not sucient,

because congurational entropy must also be controlled. The argument is generic, needing

only a graded state space and a bound of the form (7). But the space

Ω

and the bound

itself are those of the history-resolved model, and Section 3.9 states what does and does

not carry over to the embedded one. For nite systems the bound is measurable by

exhaustive enumeration, which is what we do next.

3.5 Exact bond bookkeeping

A history with

nodes built from the triangular seed by

stitches and

ℓ

lifts satises

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

(9)

since the seed carries three bonds and each stitch adds two bonds and each lift three, while

either adds one node. Hence

max

(N) = 3N − 6

, attained exactly by all-lift histories at

the node cuto, and the decit is

m = 3(N − n) + s

. Restricted to the cuto layer the

decit is simply the stitch count: the low-temperature measure suppresses histories by

how many planar stitch events they contain relative to the all-lift sector.

3.6 Exact nite-state verication

We implemented the above for a history-resolved simplicial realization of the rules. The

seed is one oriented triangle, a stitch adds a vertex to a boundary edge forming two bonds

and one triangle, a lift adds a vertex to a boundary triangle forming three bonds and one

tetrahedron, and reverse moves remove the most recently exposed non-seed vertex. States

are deduplicated up to colored incidence-graph isomorphism with the seed preserved.

For

N = 4, . . . , 8

the complete reachable state spaces contain

124

and

588

congurations. In every case the transition graph is connected, every non-seed state has

an admissible reverse move, and detailed balance and stationarity hold to residuals below

7 × 10

−18

. The stationary weight of the maximally bonded sector rises monotonically

toward unity as

βε

grows. The enumeration closes three gaps at nite size: proposal

asymmetry is handled exactly, irreducibility is checked on the actual reachable graph

instead of assumed, and the decit-sector entropy is measured rather than posited.

Limits of the entropy evidence.

The empirical slopes

ˆs

max

m>0

−1

log(|Ω

N,m

|/|Ω

N,0

come out as

0.000

0.405

0.172

0.186

for

N = 4, . . . , 8

. They vanish identically at

N = 4, 5

, so only three of the ve sizes are

informative; the sequence is not monotone, its largest value falling at

N = 6

; and the

prefactor

of Eq. (7) remains uncontrolled. This is direct model-specic evidence for

the bound at the sizes enumerated, not a proof of its thermodynamic-limit form, and no

trend in

should be read from these points.

Kinetic suppression does not move the equilibrium.

Give forward lift proposals a

relative weight

ρ > 0

while stitch and reverse proposals keep unit weight, then normalize

each proposal row. Because the acceptance factor (5) contains the reverse-to-forward pro-

posal ratio, Proposition 1 holds for every

ρ > 0

and the stationary measure is unchanged.

Only relaxation times and path statistics move. We veried this exhaustively at

ρ = 1

0.1

and

−3

on every enumerated state space; at

βε = 1.5

the maximal-sector probabilities

agree to machine precision across all three kernels, with residuals below

6 × 10

−17

. This

is the exact form of a distinction the rest of the paper relies on: the lift suppression is

kinetics, and the bond energy is what xes equilibrium (Section 17).

3.7 Which bond count is being maximized

The concentration result concerns whatever bond count enters the energy, and two dis-

tinct counts appear in this paper. The

construction

count

hist

is the number of bonds

introduced by the moves of a history. The

contact

count

cont

is the number of unit-

distance pairs in the embedded conguration, whether or not a move joined them. Close

packing is dened by the second. These are dierent functionals, and the relation between

their maximizers is computable instead of a matter of interpretation.

Exhaustive enumeration of embedded stitchlift congurations gives the answer. For

n ≤ 9

the two maxima coincide at

3n − 6

and every construction maximizer is also a

contact maximizer. From

n = 10

they separate, and the maximizer sets become disjoint:

the contact maxima are

3n − 5

3n − 4

3n − 3

n = 10, 11, 12

, reproducing the known

maximal contact numbers of hard-sphere clusters with short-range attraction [10], while

the degeneracy of the contact maximum collapses from 31 at

n = 9

to a unique maximizer

from

n = 11

onward.

The separation comes with a change of cell type. Every contact maximizer from

n = 10

onward contains octahedral cells, while no construction maximizer contains one at any

: by Eq. (9) the maximal-bond sector consists of all-lift histories, and a lift produces

only tetrahedra. Such all-tetrahedral complexes are geometrically strained, since a pure

tetrahedral fan around an edge cannot close and leaves the residual decit of Eq. (1). The

octahedral cell is precisely what the at close packing requires (Section 16) and what the

construction-bond sector cannot supply.

The stitchlift moves do reach the known optima, so the operator basis of Section 2.1 is

not too impoverished to construct the contact-maximizing congurations. The separation

reects the two counting conventions, not a limitation of the dynamics. The history-

resolved verication therefore tests the physical object at the sizes where it is performed,

and ceases to beyond them, in a way whose onset and mechanism are now known rather

than assumed. Section 3.9 sets out which of the two counts the energy carries and what

follows.

Contact maximization does not select FCC.

The small-

maximal-contact clusters

are reported as subsets of close packing, and at a dozen 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. That selection is a separate question, answered by the stacking argument of

Section 16 instead of by any bond count.

3.8 The metric of the close-packed phase

The metric is not fundamental in the microscopic state; it is reconstructed from the bond

geometry once an embedding compatible with unit bonds is given. The lemma below is

a property of the close-packed conguration; Section 3.9 sets out the evidence that the

assembly reaches it. For a node

with bond directions

ˆn

(x)

, dene the local second-

moment tensor and its normalization

µν

(x) =

z(x)

j=1

ˆn

jµ

ˆn

jν

, g

eﬀ

µν

(x) =

z(x)

µν

(x),

(10)

so that

eﬀ

= ⊮

for an isotropic set of

unit directions. For the twelve nearest-neighbor

directions of the close-packed lattice, obtained as permutations of

(±1, ±1, 0)/

√

, the

o-diagonal contributions cancel by sign symmetry and each coordinate is non-zero in

eight directions with squared component

1/2

, giving

µν

= 4 δ

µν

, g

eﬀ

µν

= δ

µν

(11)

A close packing therefore realizes the identity spatial metric through a local bond moment

rather than by stipulation. This is exact geometry, holding for any conguration whose

coordination shell is the cuboctahedron. Section 4 develops what the same bond set does

and does not establish about Lorentz invariance.

3.9 Two models, and what transfers between them

The construction and contact counts are not two labels for one model. They live on

dierent state spaces, so we dene both.

The history-resolved model.

hist

is the model of Sections 3.13.6: states are abstract

complexes

C = (V, E, σ)

with no coordinates, and the energy is

hist

(C), B

hist

= |E(C)|.

(12)

This is what the enumeration veries. Its state space is nite and coordinate-free, which is

what makes exhaustive enumeration and entry-by-entry detailed-balance checking possible

at all.

The embedded model.

The physical assembly is not coordinate-free: proximity bond-

ing (Section 2.1) binds

any

two nodes within

= 1.05 L

, whether or not a move joined

them. That is a statement about positions, so the physical state is the pair

X = (C, x)

a complex and an embedding, and the physical energy is

phys

(X) = −ε B

cont

(x), B

cont

= #{{i, j} : |x

− x

| = L}.

(13)

Call this model

emb

. It is the model whose maximizers Section 3.7 enumerates.

What transfers, and what does not.

Detailed balance is algebraic: given a state

space, a proposal kernel obeying

Q(X, X

′

) > 0 ⇐⇒ Q(X

′

, X) > 0

, and any energy,

the MetropolisHastings rule (5) is reversible with respect to

π ∝ e

−βE

. Proposition 1

therefore holds for

emb

as it does for

hist

Nothing else transfers automatically. Ergodicity was established through the removable-

exposed-node condition on

Ω

, and deleting a node from an embedded state also removes

proximity contacts, so irreducibility of the embedded transition graph has to be checked

instead of inherited. The decit-sector counts and the empirical slopes

ˆs

were measured

Ω

, and the embedded state space is a dierent set with a dierent grading. Con-

structing

emb

properly requires a rule for identifying geometrically congruent histories,

contact updates after each move, and reverse moves that remove contacts consistently.

We do not carry that out.

Table 2: What is established for each model. Detailed balance is algebraic and holds for

both; the remaining properties were veried on the history-resolved state space and are

not inherited by the embedded one.

hist

emb

Detailed balance, Prop. 1 exact exact

Irreducibility and aperiodicity veried,

N ≤ 8

open

Decit-entropy bound, Eq. (7) measured,

N ≤ 8

not measured

Concentration, Eq. (8) conditional,

N ≤ 8

open

Maximizers identied all-lift,

3n − 6

close packing,

n ≤ 12

Where the two agree.

Section 3.7 xes the overlap:

hist

and

cont

share their maxi-

mizers through

n = 9

and separate at

n = 10

. The enumeration of

hist

runs to

N = 8

inside that range, so within the sizes actually veried the two models select the same

congurations. The verication is not testing the wrong object; it is testing the right one

over a range whose upper end is known.

Where the physical model leads.

Under

phys

the maximally bonded sector is the

close-packed one. The contact maxima

3n − 5

3n − 4

3n − 3

n = 10, 11, 12

exceed

the construction bound

3n − 6

and reproduce the known maximal-contact clusters of

hard spheres with short-range attraction, which are subsets of close-packed crystals [10].

Concentration on maximal

cont

would therefore be concentration on close packing. What

is missing is not the identication of the maximizers, which is enumerated at every size

computed, but the decit-entropy bound in the embedded state space and its control as

n → ∞

Status of what follows.

The geometric results below, the identity metric of Section 3.8,

the exact isotropy of Section 4, the code of Section 5, and the whole of Part II, rest on the

close-packed phase. Three independent lines support the assembly reaching it: contact-

maximizing congurations are close packings wherever enumerated; the growth simulation

of Ref. [22] reaches a bulk of modal coordination

K=12

directly; and

K=12

is a forced

terminus of any growth history under the exclusion window, by the kissing bound (Part II,

Section 15), independently of any energy function. None of the three is a thermodynamic

concentration theorem for

emb

, and we do not claim one.

4 Exact spatial isotropy of the close-packed bond set

A persistent objection to discrete spacetime models is the apparent incompatibility be-

tween lattice regularity and continuous Lorentz invariance. If the 3D FCC lattice were

a foundational background, it would possess preferred directions (the crystallographic

axes), leaving the framework vulnerable to the Collins et al. naturalness objection [15],

which highlights that radiative corrections amplify even small tree-level Lorentz violations

into macroscopic, experimentally falsiable anomalies. We address this in three steps, an

exact algebraic isotropy of the FCC bond set at the lattice level, the resulting isotropy

of the scalar lattice dispersion at long wavelengths, and the conditions under which a

Lorentz-invariant continuum limit follows, confronted with experimental bounds.

Step 1: Exact spatial isotropy of the FCC bond set.

The

K = 12

FCC nearest-

neighbor bond vectors are

∈



(±1, ±1, 0), (±1, 0, ±1), (0, ±1, ±1)



√

2, j = 1, . . . , 12.

(14)

Dene the rank-2 structure tensor

µν

≡

j=1

. By direct enumeration:

= 4 ×

|{z}

(±1,±1,0)

+ 4 ×

|{z}

(±1,0,±1)

+ 0

|{z}

(0,±1,±1)

= 4,

(15)



(+1)(+1) + (+1)(−1) + (−1)(+1) + (−1)(−1)



+ 0 + 0 = 0,

(16)

and the three-fold permutation symmetry of the bond set gives

= S

= 4

and

= S

= 0

. Therefore

µν

= 4 δ

µν

(exact, by enumeration)

(17)

This algebraic identity guarantees equal propagation speed in every spatial direction for

the scalar sector at leading order in

. The odd-rank tensor

µνλ

≡

vanishes

exactly because the FCC bond set is centrosymmetric: every

has a partner

−n

, so all

odd-power sums cancel,

µνλ

= 0

(exact, by inversion symmetry)

(18)

Equation (18) forbids any preferred direction and any linear-in-

term in the lattice dis-

persion.

Step 2: Isotropy of the dispersion relation.

For a scalar eld on the FCC lattice, the

dispersion relation is

ω(k)

= κ

j=1



1 − cos(k · n



. At long wavelengths (

|k|a ≪ 1

expanding the cosine,

ω(k)

≈

κa

j=1

(k · n

)

κa

µν

= 2κa

|k|

(19)

ω = c

|k|

with

= a

2κ/m

, where

is the bond stiness that xes rigidity in

Part II and

the node inertia. This scalar speed is the

trace

channel of the displace-

ment dynamics. The three acoustic branches of the full vector (phonon) problem satisfy

+ v

= c

in every propagation direction (Part II, Section 23), so

is the

quadrature sum of the branches rather than the speed of any single polarization. The

dispersion is exactly isotropic at leading order. This isotropy is an algebraic consequence

of equation (17), not an approximation. Anisotropic corrections to the scalar dispersion

enter at relative order

(|k|a)

, the

[111]



[100]

splitting of

−(|k|a)

/72

at leading

order, suppressed by

(E/M

)

for Planck-scale lattice spacing

a ∼ ℓ

Step 3: Emergence of Lorentz boosts.

The isotropy (17) guarantees equal prop-

agation speed in every direction for any

scalar

excitation on the bond set. That is the

geometric prerequisite for emergent Lorentz invariance. Vector (displacement) excitations

behave dierently. They are governed by the rank-four bond tensor and remain direction-

and polarization-split at the lattice scale (Part II, Section 23).

The causal cone itself is carried by the propagating bond excitation, whose signal speed

is derived in Part II. The elastic speed

above is a distinct, metric-perturbation mode. A

single light cone therefore requires

= c

. That equality is not a tuning but an observed

fact: gravitational-wave and photon speeds agree to one part in

. Part II shows that

imposing it on the elastic trace channel xes the stiness

, while the sharper branch-level

statement remains an open target. Given one cone, an isotropic linear dispersion yields a

relativistic eective eld theory in the standard continuum limit [15].

Confronting the experimental bounds.

Lattice corrections at the cuto scale

1/a

are

suppressed by

(E/M

)

and lie far below all current experimental bounds on Lorentz vio-

lation [16, 17], which constrain departures at the level of

−20



−40

of the Planck scale.

Given a single propagation speed, spatial

SO(3)

isotropy and time-reversal symmetry

force the dimension-4 kinetic operators of the

scalar

sector into Lorentz-invariant form;

sectors with distinct speeds (the elastic branches of Part II) remain Lorentz-violating until

the branch-level cone closes, and protection of the invariance against radiatively gener-

ated lower-dimension Lorentz-violating operators [15] is an open requirement instead of

a result obtained here.

Residual

anisotropy is absent from the

scalar

sector at leading order by Eq. (17); in

the vector (elastic) sector it is physical, the rank-four bond tensor splits directions and

polarizations (Part II, Section 23), and is not a coordinate artifact.

Polycrystalline averaging as a complementary mechanism.

While equations (17)

(19) guarantee macroscopic isotropy from a single FCC domain, a secondary classical

mechanism reinforces it. Because the kinematic growth is probabilistic, the vacuum nu-

cleates multiple independent planar domains. As they expand and meet in 3D space, their

FCC registries misalign, creating a polycrystalline structure with randomly oriented grain

boundaries. Table 2 of Ref. [22] reports the macroscopic shape isotropy

min

/λ

max

of the

spatial covariance matrix

= ⟨(x

− ¯x

)(x

− ¯x

)⟩

. The ratio rises steadily from 0.39 at

N = 250

to 0.54 at

N = 1000

(30 seeds), conrming that while individual stochastically

grown grains possess shape bias, the ensemble rapidly spherizes, a classical bulk-averaging

mechanism that complements the exact algebraic isotropy of equation (17).

5 The

[[192, 130, 3]]

CSS Code

An ideal bulk patch of the emergent FCC lattice carries a mathematically veriable quan-

tum error-correcting code. The code is dened on a defect-free

4×4×4

unit-cell arrange-

ment with periodic boundaries, the single-crystal idealization of the polycrystalline clus-

ters grown here, whose grain boundaries and free surfaces host unclosed stabilizers, with

192 edges, each treated as a physical qubit. In the CSS construction of [13], X-stabilizers

act on octahedral voids (each touching the 12 edges connecting its 6 surrounding vertices)

and Z-stabilizers act on vertices (each touching the 12 incident edges); both families have

uniform weight 12. The code parameters, veried computationally in [21], are:

Physical qubits (

) 192 (edges of the FCC lattice)

Logical qubits (

) 130

Code distance (

) 3

Encoding rate (

k/n

) 67.7%

Vertices (

) 13 per coordination cluster

Edges (

) 36 per coordination cluster

Faces (

) 38 (

= 32

triangles

squares)

The coordination cluster.

The 13-node coordination cluster, the central node together

with its 12 cuboctahedrally arranged neighbors, decomposes as a simplicial complex into

32 triangular 2-cells and 6 square 2-cells: 8 of the triangles are the cuboctahedron's surface

faces, and the remaining 24 are spoke triangles formed by each cuboctahedral edge with

the central node. The 32 triangles are the minimal closed gauge-invariant loops [6] that

underlie the compound-QEC argument of Section 2.4; the 6 squares, representing the

absence of a diagonal bond, are topologically distinct.

Why the rate is high.

The encoding rate

k/n = 67.7%

means two-thirds of the lattice

degrees of freedom carry logical information. By comparison, 2D topological codes on

planar or toroidal geometries encode a xed number of logical qubits regardless of system

size, so their rate

k/n → 0

n → ∞

A triple coincidence at 3.

The 3D projection, rare as it is at

∼ 5%

per frontier site,

is what enables an extensive encoding rate. Three quantities coincide at the value 3: the

code distance

d = 3

, the number of independent distance constraints

S = 3

for the tetra-

hedral lift, and the ambient spatial dimension. Whether this triple coincidence reects

a structural identity or is accidental remains an open question that may be addressable

through a generalized construction at other values of

6 Conclusion

Summary.

Part I has taken the two local operators of the Selection-Stitch Model and

asked what structure they select. The stitch and lift are motivated as the minimal ways to

create new closure at xed bond length, and closure is what allows a correlation to persist

at all. Their relative amplitude

−3

is motivated by the codimension dierence between

their solution manifolds and by the error-correction reading of Section 2.3; Part II derives

it as a thermal factor with a free geometric prefactor.

Principal results.

1. The assembly is a reversible Markov process on pre-geometric bond complexes, with

stationary measure

∝ e

βεB

and, under a removable-node condition, a unique limit

from any initial state (Section 3.2).

2. Where the decit-sector counts obey an exponential bound and

βε > s

∗

, that mea-

sure concentrates exponentially on maximally bonded complexes (Section 3.4).

3. Exhaustive enumeration through

N = 8

conrms connectivity, reversibility, detailed

balance and stationarity to residuals below

7×10

−18

, and measures the decit-sector

entropy rather than assuming it (Section 3.6).

4. Kinetic suppression of the lift leaves the stationary measure unchanged, veried at

ρ = 1

0.1

and

−3

. The lift amplitude is kinetics; the bond energy xes equilibrium.

5. The construction and contact bond counts share their maximizers through

n = 9

and separate from

n = 10

, where contact maximizers acquire the octahedral cells

that all-lift histories cannot produce and reproduce the known maximal contact

numbers of hard-sphere clusters (Section 3.7).

6. The local bond second moment of a close packing is

µν

= 4δ

µν

, so the reconstructed

spatial metric is the identity (Section 3.8); the same bond set is exactly isotropic

at rank two and centrosymmetric at rank three (Section 4). These are statements

about the close-packed conguration, not consequences of the measure above.

Limits of these results.

The results above are established for the history-resolved model

hist

. Detailed balance carries over algebraically to the embedded model

emb

that the

physical energy denes; ergodicity, the decit-entropy bound, and hence concentration

do not, because they were veried on a dierent state space (Table 2). The two models

share their maximizers over the range enumerated, and maximal contact bonding is close

packing at every size computed, but a thermodynamic concentration theorem for

emb

is open (Section 3.9). Contact maximization selects close packing but not the stacking

isomer: FCC versus HCP is settled in Part II (Section 16), by an argument that needs no

energy function. And the construction is positional throughout, a limitation taken up in

Part II (Section 26).

The emergent lattice supports the

[[192, 130, 3]]

CSS code of Ref. [21], whose distance

d = 3

matches the suppression exponent under the survival ansatz of Section 2.3. Part II

supplies the dynamics: a bond Hamiltonian whose dissipative limit assembles the lattice

and whose coherent limit propagates a signal cone on it.

Part II

Dynamical Foundations

7 The dynamical framework: an overview

The SSM posits that the vacuum is a close-packed network of entanglement bonds that

crystallizes from a frustrated tetrahedral foam (

K=4

) into a face-centered cubic (FCC)

close packing (

K=12

). Part I formulates the stitchlift rules as a reversible Markov process

and shows that its stationary measure concentrates on maximally bonded structures,

veried exactly on the history-resolved state space. It also shows that a close packing

carries a at isotropic spatial metric and an explicit quantum error-correcting code, the

[[192, 130, 3]]

CSS code, and that maximal contact bonding is close packing at every size

enumerated (Section 3.9). Part II takes the close-packed lattice as given, except where

noted.

The contribution of Part II is the dynamical law beneath that assembly; the branching-

growth material in Section 20 is a structural outlook, clearly secondary to the forced-

K=12

assembly that is the main result.

Part I xes the

kinematics

: which moves occur, and what they build. It does not supply

the

dynamical law

. That law requires a Hamiltonian whose evolution generates both the

growth and the propagation of signals, a physical clock and ruler, a

derivation

of the

stability that holds the crystal, and an account of what is forced, what is free, and what

scale the model must supply. Part II supplies all of these in one place.

The organizing result.

One open-system equation (Figure 1) governs the vacuum:

its coherent limit propagates an emergent light cone and its dissipative limit assembles

the lattice, with the kinematic growth represented by a geometry-conditioned trajectory

picture.

Three features settled.

Along the way we settle three implicit features of the simu-

lation. The

K=12

terminus is forced by the kissing-number bound (Section 15). The

lift rate factorizes into a free geometric acceptance and a thermal factor

−3

xed by

= ε

(Section 17). And the parameter the model requires is a rigidity scale, derived

self-consistently from the phonon spectrum (Section 18). Sections 2416 give the energy

ladder, the code, and the FCC-over-HCP selection. The stitchlift construction and the

[[192, 130, 3]]

code are set out in Part I; the geometric maxima use the kissing-number [18]

and Kepler [3] theorems.

An emergent-geometry reading.

The results assemble, step by step, the features

we associate with physical space on a single self-organizing substrate: spatial isotropy

from the close-packed coordination and a causal signal cone from the local Hamiltonian

(Section 12), and a transient constant Regge curvature carried by the two-sheet foam

(Section 9).

The program, and its boundary.

In this sense Part II is a program for

emergent

geometry

: the lattice does not sit in space, it

the space, and its geometric, causal,

and cosmological properties are consequences of the assembly dynamics instead of inputs.

The boundary of the program is that the dynamics is still

positional

, it presupposes that

the nodes carry positions in a metric, with a hard-core exclusion, so the deepest step,

deriving those positions from the entanglement data themselves, is not yet taken. Part II

establishes the geometric and cosmological

features

of emergent space and leaves the

emergence of the positional substrate as the frontier.

Status of the claims.

The results below sit at dierent condence levels, and they are

labeled throughout.

(T) Theorem-level

: the forced

K=12

terminus and

∗

2 −

√

2 L

(Section 15), which depend only on geometry.

(D) Derived and robustly tested

: the

detailed-balance stability

q = 1/(1 + e)

and the end-to-end coherence of the assembly,

which we conrm reaches bulk

K=12

generically across temperature (Section 21).

: the light cone and the relation

c = 4

√

2 v

lat

which follow from the posited Hamiltonian and are mutually consistent but are not inde-

pendently tested here.

(C/D) Numerically derived, conditional on the posited dynamics

the rigidity bounds of Section 18, obtained by Brillouin-zone integration of the FCC

phonon spectrum.

(S) Speculative

: the exponential branching-growth phase (Section 20).

Contributions.

Part II develops the close-packed vacuum from its dynamical law: the

open-system Hamiltonian and master equation; the single-excitation signal cone and

c =

√

2 v

lat

; the detailed-balance derivation of

q = 1/(1 + e)

; the forced

K=12

terminus and

the exact threshold

∗

; the factorization of the lift rate into a free geometric acceptance

and a xed thermal factor

−3

; the rigidity scale

κ ≳ 36 ε/L

; the end-to-end single-rule

assembly and its temperature robustness; the two-sheet tetrahedral foam and the code

selection of its regular registration; and the exponential branching-growth phase.

The conceptual starting point, the stitchlift picture of the vacuum as an incompletely

crystallized close packing, together with the

[[192, 130, 3]]

code construction, is developed

in Part I; the contact maxima rest on the kissing and Kepler theorems [18, 3].

Lindblad master equation

−

[

H, ρ

] +

(

↑

[

] +

↓

[

−

])

coherent limit

bath off:

−

[

H, ρ

]

unitary evolution

⇒

light cone,

= 2

lat

ε/

dissipative limit

thermal jumps at

ε/k

quantum-trajectory assembly

⇒

growth,

1 +

one dynamics, two limits

Figure 1: One open-system Hamiltonian, several sectors: Eq. (21) coupled to a thermal

bath at

= ε/k

. Coherent limit

→

ballistic signal front and the collective sectors of

Table 5; dissipative limit

→

constructive assembly with detailed-balance rate

q = 1/(1+e)

8 Foundational assumptions

Foundational assumptions.

Before the dynamics, we state the premises the construc-

tion rests on, so that what is assumed is separated from what is derived.

(A1) A pre-

geometric substrate.

The starting arena is not space but a structureless set of pre-spatial

degrees of freedom carrying entanglement; metric notions (distance, direction) are not

assumed to pre-exist but are intended to emerge from the bond network. In Part II the

substrate is treated

positionally

, the admissible congurations already carry a length

(Section 14, Eq. (31)), and a fully pre-positional formulation, in which

itself emerges,

is left as the frontier (Section 26).

(A2) Bell-pair nucleation at a unit rate.

The elementary event is the formation of a

maximally entangled bond (a Bell pair) between two substrate elements, occurring at a

xed nucleation rate that sets the unit of time and a xed bond length

that sets the

unit of length. All rates and lengths in Part II are expressed in these units; the model

xes their dimensionless ratios, not their absolute values (Section 22).

(A3) Stability selected by error correction.

A nucleated structure persists only if it is

stabilized, a conguration survives in proportion to how well its bonds close into the

stabilizer cells of the emergent code, and decays otherwise. This is the principle behind

the dissolution ratio

q = 1/(1+e)

(Section 13) and the selection of the regular registration

(Section 9): among competing local structures, those whose stabilizers are satisable are

the ones that live. The structural lattice carries the

[[192, 130, 3]]

CSS code on its edges.

(A4) One binding scale.

Every bond has the same binding depth

, the single energy

parameter of the Hamiltonian (Section 10).

What follows from them.

From these

four premises, a pre-geometric entangled substrate, unit-rate Bell-pair nucleation, code-

selected stability, and one binding scale, the rest of Part II is a consequence.

Part I establishes the kinematics on which these premises act. They are the stitch and lift

operators, the cascade

K=1 → 6 → 4 → 12

through a frustrated

K=4

foam with Regge

decit

δ = 2π − 5 arccos

≈ 7.36

◦

(Eq. (1)), and the

K=12

saturation at the Kepler

bound. We treat them here as the classical limit of the dynamics and do not repeat them.

One further premise is inherited and should be stated with the others. Most of Part II

works on the close-packed lattice: the acoustic spectrum, the rigidity bound, the code, the

metric, and the FCC-over-HCP selection all take that lattice as given. Part I supports it

three ways, maximal contact bonding is close packing at every size enumerated, the growth

simulation of Ref. [22] reaches a modal

K=12

bulk directly, and the thermodynamic-limit

concentration statement remains open (Section 3.9).

Section 15 is the exception and does not inherit this premise. It bounds coordination for

any

unit-bond conguration under the exclusion window, by the kissing-number theorem,

so it constrains the close packing rather than presupposing it.

9 Code-selected registration and the two-sheet foam

The dynamics below need two features of the close packing that Part I does not address:

how the layer-by-layer arrival of sheets produces a transient, spatially homogeneous cur-

vature, and why the code selects the regular FCC registration over its competitors. Both

are developed here.

The two-sheet tetrahedral foam.

A structural question worth settling is

where

the

tetrahedral decit lives during crystallization. Sheets arrive one at a time: a sheet is

seeded by a lift and grows laterally, and the next sheet is independently seeded and

grows, so there is always a nite delay between the arrival of consecutive sheets. During

that delay the structure is in a

two-sheet

state, and the two-sheet state is tetrahedral.

Verifying the two-sheet geometry.

We check this directly. Stack two at

K=6

sheets

at the close-packed oset and spacing

h =

2/3 L

. Every interior node of the lower

sheet then bonds to its six in-plane neighbors and to three nodes of the sheet above,

giving coordination

K=9

. That node, together with the triangle above it, forms a

regular

tetrahedron

, with all six edges equal to

. This holds for

100%

of interior nodes in a nite

patch. The frustrated tetrahedral bonding here is the same that denes the

K=4

foam of

the cascade (Section 2.1). The count

K=9

simply includes the six in-plane sheet bonds

alongside the four-coordinated tetrahedral motif. The two-sheet contact therefore realizes

the tetrahedral foam as an extended, registered layer instead of a disordered bulk.

Why the decit takes a xed value.

A new sheet can be seeded only where the

lifted apexes project to

equidistant

points, since those apexes must themselves form the

K=6

sheet. We verify that the apexes of the lifted up-triangles do form a regular

triangular lattice, at nearest-neighbor spacing

1.00 L

This seeding constraint, together with the code stabilizers that hold the sheets rigid

(Section 25), forces the tetrahedra to remain

regular

rather than relax to an irregular

packing. The frustration is paid as curvature, not absorbed by distorting the bond angles.

That the code prefers this registration is computed, not assumed. Table 3 compares

three candidate stackings of the second sheet: the regular close packing, the eclipsed

(AA) stacking, and a relaxed irregular packing. The measure is the fraction of triangular

plaquettes that are genuine cell faces, that is, the fraction of

-stabilizers that can be

Table 3: Comparative code metrics for three registrations of the second sheet onto the

rst (a

-node patch). Faced is the fraction of triangles that are faces of a tetrahedral

cell, the fraction of plaquette (

) stabilizers that can be consistently satised. Only the

regular close packing forms tetrahedra and closes its stabilizers, so the code selects it.

registration tetrahedra faced fraction edge irregularity stabilizers satisable

regular (FCC)

32 0.73 0.000

maximal

eclipsed (AA)

0 0.00 0.000

frustrated

relaxed (irregular)

0 0.00 0.068

frustrated

consistently satised. (Note that at the close-packed spacing

2/3 L

an eclipsed second

sheet is excluded outright by the hard core, aligned nodes sitting at

0.8165 L < R

; the

eclipsed row is therefore meaningful only at unit vertical spacing.)

FCC versus HCP: what selects the stacking.

Local

K=12

packing admits two reg-

istrations: cuboctahedral (FCC, ABC) and anticuboctahedral (HCP, ABAB). At nearest-

neighbor level they are

exactly

degenerate. Direct enumeration shows both coordination

shells contain 24 unit-distance pairs and 8 unit triangles. The pair energy is therefore

blind to the dierence, and so is any three-body triangle term

= −J

△

Selection must come from structure the shells do not share. There are three candidates.

(i) Couplings at second-neighbor range and beyond, where the two stackings' distance

spectra rst dier. (ii) The void geometry: HCP's octahedral voids share faces along

the

axis, while FCC's share only edges, so the octahedral

-stabilizers of the emergent

code see inequivalent environments. This makes the code term itself a candidate selection

energy. (iii) The vibrational free-energy split

∆F = ∆E − T ∆S

vib

, known to be minute

for central-force packings.

Computing the code-term split is a dened target calculation. A fourth possibility, which

does not appear in this list because it is not an energy at all, is that the stacking is

xed kinematically by an orientation the growth front transports; that is the argument of

Section 16. Pending the code-term calculation and the classier follow-up noted in Part I,

the result stands as close packing with a candidate selection mechanism. Only the regular

registration forms tetrahedral cells at all (

versus none) and closes its plaquettes (

73%

faced versus

); the alternatives are stabilizer-frustrated.

What the comparison establishes.

Among the candidates compared, the regular

close-packed registration maximizes the satisable stabilizers, which is what selection by

the code means operationally, with the caveat that FCC and HCP are

both

regular close

packings, and Table 3 does not compare them against each other; the table establishes

close-packed over eclipsed or irregular registration, and the FCC/HCP split is the separate

question addressed below. (This is a stabilizer-satisability comparison across the physical

competitors, a structural proxy for code distance instead of a full distance calculation;

the margin is wide enough that the conclusion is insensitive to the proxy.)

The decit is homogeneous.

Five regular tetrahedra about a hinge subtend

5 arccos

= 352.64

◦

, leaving exactly the Regge decit

δ = 2π−5 arccos

= 7.36

◦

(Eq. (1));

a relaxed, irregular foam would instead distribute a much larger and variable decit. Be-

cause the seeding-plus-code constraint enforces the

same

regular registration everywhere

= 7

◦

Five regular tetrahedra about a hinge:

frame held rigid, frustration

= 7

◦

apexes are equidistant

⇒

next regular sheet

each apex caps one triangle

(3 bonds, regular tetra)

Seeding: one apex per up-triangle; the apexes,

being equidistant, form the next regular sheet

Figure 2: Left: the xed value of the decit. Five

regular

tetrahedra about a shared

hinge subtend

5 arccos

= 352.64

◦

, leaving exactly the Regge gap

δ = 7.36

◦

. The code

stabilizers forbid the tetrahedra from relaxing to an irregular packing (which would give

a larger, variable decit), so the frustration is paid as curvature at this single value.

Right: why the registration is regular. A new sheet seeds only where the lifted apexes

(red) are equidistant, forming the next

K=6

sheet; the apexes of the lifted up-triangles

do form a regular lattice (spacing

1.00 L

), so seeding plus code-enforced rigidity hold

the same regular tetrahedral registration everywhere along the contact. The decit is

therefore spatially homogeneous and xed at

, and vanishes when the third sheet closes

the cuboctahedral shell (

K=9 → 12

along the contact, the decit is spatially homogeneous and takes this single xed value

(Figure 2). The third sheet converts

K=9 → 12

and the decit vanishes exactly, the

cuboctahedral shell tiling space without frustration. The cancellation is not approximate.

The completed packing is the tetrahedraloctahedral honeycomb, in which two tetrahe-

dra and two octahedra meet at every edge with supplementary dihedral angles, so the

angles about an edge sum to

2π

identically (Section 16). The frustration of the two-sheet

stage is precisely the absence of the octahedral cells that supply the complementary angle.

The crystallization therefore passes through a genuine, spatially extended, nite-duration

tetrahedral foam:

at sheet

(K=6, δ=0) →

two-sheet foam

(K=9, δ>0,

homogeneous

) →

FCC

(K=12, δ=0).

(20)

The foam is therefore not an assumption: it is the two-sheet stage that layer-by-layer

arrival necessarily produces, and it carries a homogeneous decit for a nite interval

before the third sheet closes the shell.

Claim ledger.

Table 4 summarizes the principal claims of both parts and their status,

so that the results proved for the stated model can be separated at a glance from those

conditional on it. Tiers:

(T)

theorem-level;

(D)

derived and numerically tested;

(C)

consistent within the posited dynamics;

(S)

speculative. A slash denotes intermediate

standing:

(C/D)

is numerically derived but conditional on the posited dynamics, and

(D/S)

is derived but with speculative interpretation.

Table 4: Claim ledger. Upper block: Part I. Lower block: Part II. The reversibility of

the chain and the forced

K=12

terminus are the theorem-level results; everything below

them is conditional on the posited dynamics or on a bound veried only at small size.

Claim Status Basis

detailed balance,

π ∝

βεB

T MetropolisHastings; algebraic,

holds for

hist

and

emb

alike

ergodicity, unique sta-

tionary limit

D removable-node condition; veried

for

hist

N ≤ 8

lift suppression is kinet-

ics, not equilibrium

D exact, veried at

ρ = 1, 0.1, e

−3

every enumerated state space

concentration,

hist

C/D exact given the decit-entropy

bound; bound measured only for

N ≤ 8

maximal

cont

= close

packing

D enumerated for

n ≤ 12

; matches

maximal-contact hard-sphere clus-

ters [10]

concentration,

emb

S open. Ergodicity and decit en-

tropy not established on the em-

bedded space

K=12

forced for

∗

< L

T cuboctahedral square-face vacancy

+ kissing bound

q = 1/(1 + e)

D detailed balance at

= ε/k

robust modal

K=12

as-

sembly

D end-to-end stochastic rule,

temperature-generic

−3

lift suppression C thermal factor at the operating

point

βε = 1

; geometric prefactor

free; QEC reading heuristic

κL

≳ 36 ε

rigidity C/D FCC harmonic phonon estimate

10 The dynamical Hamiltonian and its scales

Let each potential bond

be a qubit,

|1⟩

entangled (energy

−ε

) or

|0⟩

failed (

|1⟩⟨1|

H = −ε

− Γ

⟨b,b

′

⟩

′

+ Y

′

) − ε

, Γ = ε.

(21)

The

binding

term rewards each entangled bond; the

kinetic

term lets entanglement hop

between bonds sharing a node and makes

generate motion; the

code

term is the CSS

stabilizer Hamiltonian (Section 25),

b∋v

b∈o

. The bond-state Hamil-

tonian has a single binding depth

(and we set the kinetic coupling

Γ = ε

); as Section 18

shows, the

positional

realization of the bond requires in addition a curvature scale

distinct from the depth. The constants

and

yield

ℏ

, v

lat

L ε

ℏ

, T

(22)

11 The open-system dynamics

An entangled bond decoheres against its environment, and that decoherence

bond

failure; the dynamics is Eq. (21) coupled to a thermal bath at

= ε/k

˙ρ = −

ℏ

[H, ρ] +



↑

D[σ

] + γ

↓

D[σ

−

]



ρ, D[L]ρ = LρL

†

−

†

L, ρ}.

(23)

Two limits supply the two halves of the physics: bath o, unitary evolution gives the light

cone (Section 12); bath on, thermal jumps drive assembly (Sections 13, 14).

What the bath is, before geometry.

The bath requires explanation: if the lattice is

the vacuum, what constitutes a thermal environment when there is no space yet to host

one? We take the bath to be the reservoir of

unentangled, unregistered degrees of freedom

the bond qubits that have not (yet) joined the coherent network, i.e. the population in the

|0⟩

(failed) state and the latent bonds of the surrounding foam. These are not external

to the vacuum; they are its own pre-crystalline part. Decoherence of an entangled bond

against this reservoir of un-joined bonds is precisely bond failure, and the temperature

= ε/k

is the statistical scale of that reservoir (one bond carries entropy

ln 2

A self-bath.

The crystallizing network and the foam it grows into are the system and

the environment for each other. It is therefore pre-geometric in the required sense, no

external spacetime is invoked, though a fully pre-positional formulation, in which even

the reservoir carries no presupposed positions, remains open (Section 26).

12 Coherent limit: a single-excitation signal cone and

lattice isotropy

The front speed.

In the single-excitation sector the kinetic term is tight binding on

the bond-adjacency (line) graph of the lattice. A localized excitation therefore spreads

ballistically (Figure 3). On a one-dimensional bond chain the front velocity is

2Lε/ℏ =

lat

, veried as

2.01 Lε/ℏ

401

bonds. On the FCC bond network the front is faster.

For a

-regular lattice the line-graph band is the node-graph band shifted by the constant

K − 2

, so the maximum group velocity is that of tight binding on the FCC nodes:

front

= 4

√

2 v

lat

≈ 5.66 v

lat

(24)

attained for

along

[100]

, where the eight bonds with a

[100]

component give

|∇E|

max

√

2 ΓL

How to read this speed.

Being a band-structure extremum, the front speed is direction-

dependent at the lattice scale. The ballistic wavefront is not spherical; only the long-

wavelength scalar sector is isotropic (Section 4). We therefore give the bound its precise

name:

sig

≡ 4

√

2 v

lat

is the

maximum microscopic signal velocity

of the hopping sector.

It is a causal bound, not a low-energy propagation speed. The hopping band is quadratic

at long wavelength,

E(k) = E

+ A|k|

+ O(k

)

, so the group velocity of long-wavelength

Table 5: The three propagation sectors of the lattice. One symbol

hides three inequiv-

alent speeds; this table keeps them separate.

Sector Excitation Dispersion Status

Bond hopping

hop

localized bond

excitation

quadratic at low

; bal-

listic front bounded by

sig

= 4

√

2 v

lat

, direction-

dependent

derived

Scalar collective

emergent scalar

eld

linear and isotropic as

k → 0

, speed

= c

identied here

(trace channel)

Elastic

displacement

modes

linear but polarization-

split,

= v

, v

derived; single cone

closed by no-go

(Section 23)

bond excitations vanishes as

k → 0

. The genuinely

linear

low-energy cone belongs instead

to the scalar collective sector: a lattice eld with canonical pair

(ϕ

, Π

)

and Hamiltonian

⟨ij⟩

(ϕ

− ϕ

)

, ω

(k) =



1 − cos(k · a

)



(25)

whose small-

dispersion is

= c

+ O(k

)

with

= L

, exactly the trace-

channel speed

of the displacement dynamics when

= κ

= m

. The status of this

sector is as follows. The close-packed geometry

supports

an additional scalar collective

eld with an isotropic linear continuum limit. It does not yet derive one. Obtaining

as a collective coordinate of the bond qubits, schematically

∼

b∋i

f(n

)

, with

from coarse-graining Eq. (21), is the dened target that would make the sector emergent

rather than adjoined. Table 5 collects the three sectors and their status. The single-cone

problem is then exactly the question of whether

= v

can be arranged, which

Section 23 answers. A local Hamiltonian has a nite LiebRobinson velocity. Time and

length combine into a maximum signal speed for excitations on the lattice. This is a single-

excitation, LiebRobinson-type signal cone, not a derivation of the full relativistic causal

structure of a continuum eld theory (Section 26). This is the dynamical counterpart

of the structure-tensor isotropy: the twelve bond vectors give

µν

= 4δ

µν

and

µνλ

= 0

by centrosymmetry, so the long-wavelength scalar dispersion is isotropic,

with anisotropy entering only at relative order

O((|k|L)

)

(coecient

1/72

; Section 4).

The causal cone is the signal front itself:

c ≡ v

front

= 4

√

2 v

lat

= 4

√

2 Lε/ℏ

, the fastest

speed at which any inuence propagates on the lattice.

Notation and its limits:

we write

for this lattice signal bound throughout Part II. It is a bound internal to the model.

The model xes no absolute velocity, because

lat

= Lε/ℏ

depends on

and

, which

assumption A2 leaves unxed; every speed below is a dimensionless ratio in which they

cancel. Reading

as the physical speed of light is a further identication, conditional on

the GW

light constraint below and unresolved at branch level (Section 23). Here the

clock is

= ℏ/ε

and the ruler

. This bond-excitation cone is the model's causal cone,

and is what we mean by the emergent light cone. It is distinct from the elastic phonon

speed

= L

2κ/m

, which propagates metric perturbations instead of signals.

Fixing the stiness.

A single, shared cone is not optional. The measured equality

of the gravitational-wave and photon speeds to one part in

[20] requires the elastic

and signal cones to coincide; imposing this on the elastic trace channel,

= c

(see the

mode-speeds paragraph below), xes the otherwise-free stiness,

κ =

16 mε

ℏ

(

light, trace channel

: c

= c).

(26)

A oor on the binding scale.

The rigidity requirement

κL

≳ 36 ε

of Section 18

sharpens this. The two together impose a oor on the binding scale,

ε ≳ 2.3 ℏ

/(mL

)

the bond energy cannot lie below the lattice kinetic scale

ℏ

/(mL

)

. Equal cones, in this

trace sense, thus x a stiness that would otherwise be free; the branch-level statement

remains the open target recorded in Section 23.

Mode speeds and the single cone.

The cone

c = 4

√

2 v

lat

used here is the Lieb

Robinson front of the scalar bond excitation on the FCC bond network. It is a distinct

object from the elastic branch

= L

2κ/m

. The rst is a band-structure maximum of

the signal sector; the second is the long-wavelength sound speed of the displacement eld.

Section 4 established one necessary geometric ingredient of emergent

SO(3, 1)

invariance:

the exact bond-set isotropy

µν

= 4δ

µν

and centrosymmetry

µνλ

= 0

, which force a

rotationally isotropic scalar kinetic term at leading order. That result is conditional on

a common propagation speed. It is an ingredient, not the invariance itself. Part II asks

whether the common speed can be supplied dynamically.

The GW

light identication, Eq. (26), forces the elastic

trace

speed to coincide with

the signal front,

= c

. This is weaker than the branch-level statement the observation

constrains. The three elastic polarizations satisfy

+ v

= c

in every direction

(Section 23). Fixing

= c

therefore places the branches at

only in quadrature mean.

The individual polarizations remain sub-luminal and split:

∈ [0.71, 0.82] c

and

∈

[0.35, 0.50] c

The gap to a genuine single cone cannot be closed within the elastic sector at all, by

the no-go results of Section 23. Three-body angular terms can restore isotropy, but not

speed equality, which requires a distinct excitation sector. We therefore do not derive

the stronger statement that every sector and polarization shares this cone without the

light input. We take

c ≡ 4

√

2 v

lat

as the causal cone of the signal sector, and state

the multi-speed structure explicitly rather than hiding it in a single symbol.

13 Dissipative limit: the assembly rate is derived

The entangled state lies

below the failed state, so detailed balance requires

↑

/γ

↓

ε/k

. In general

q(β) = 1/(1 + e

βε

)

, or

q = [1 + (g

) e

βε

]

−1

with bound and unbound

degeneracies

, g

, taken equal here.

We then set the operating point

βε = 1

. This is a stated identication of the bath

temperature with the bond scale, not a derived equality. Deriving the bath temperature

self-consistently from reservoir thermodynamics, via

−1

= ∂S

foam

/∂E

foam

, is the open

route to replacing this operating point with an equilibrium condition. At

βε = 1

the

200 150 100 50 0 50 100 150 200

position

x / `

lat

time

t /

(

/ε

)

= 2

lat

ε/

Emergent light cone from real-time evolution

5.0

4.5

4.0

3.5

3.0

2.5

2.0

1.5

1.0

0.5

log

Figure 3: Coherent limit: a bond excitation on a one-dimensional bond chain spreads with

front velocity

v = 2Lε/ℏ

; on the full FCC bond network the corresponding LiebRobinson

front is

√

2 Lε/ℏ

(see text).

single-bond rate equation has stationary failure probability

q =

↓

↑

+ γ

↓

1 + e

≈ 0.2689,

(27)

with no free parameter beyond the chosen operating point

βε = 1

: the detailed-balance

ratio at

= ε/k

, where the only scale is the bond entropy

S = ln 2

. A node loses

rigidity when

K − 2

of its

bonds fail, so its dissolution probability is the binomial tail

dissolve

(K) =

j=K−2





(1−q)

K−j

= {0.61, 0.29, 0.048, 7.5×10

−5

}, K = {3, 4, 6, 12}.

(28)

Stability rises steeply with coordination. The foam turns over while the

K=12

core is

eectively immortal. This relative stability is the detailed-balance content of Eq. (23),

not an input.

Geometry-resolved cross-check.

The rule behind Eq. (28) is that a node fails on

losing

K −2

of its

bonds. We can test it against a criterion built from the actual bond

geometry. Form the local stiness matrix

αβ

= κ

a∈B

ˆn

a,α

ˆn

a,β

over the surviving bond

set

, and declare the node stable only if

has full rank, so that

min

(M) > 0

and the

survivors span three dimensions.

Exhaustive enumeration over survival patterns at

q = 1/(1 + e)

gives three results.

The tetrahedral

K=4

node.

Here the two criteria coincide

exactly

, both giving

P =

0.294

. Any three of the four tetrahedral directions span

, so rank failure is precisely

the loss of

K − 2

bonds.

The cuboctahedral

K=12

node.

The rank criterion gives

1.6 × 10

−3

against the

binomial

7.5 × 10

−5

. The dierence comes from coplanar survival patterns, the four

hexagonal and three square equatorial sections of the cuboctahedron, which keep many

bonds but no out-of-plane stiness. Both values are negligible against the foam's

0.29

The planar

K=6

sheet node.

This is rank-decient out of plane for

every

survival

pattern. That quanties why sheets are metastable waypoints of the cascade instead of

termini: they are two-dimensionally rigid, with in-plane rank-2 failure at

1.5 ×10

−2

, and

three-dimensionally soft.

The stability hierarchy driving the cascade is therefore robust to the choice of criterion,

and the simple

K − 2

rule is exact at the tetrahedral bottleneck, where it matters most.

Adding a physical stiness threshold.

The bare rank test

min

> 0

classies ar-

bitrarily oppy congurations as stable. A physical threshold

min

> λ

tied to the

Lindemann window is more realistic: soft-mode uctuations

kT/λ

< (L − R

∗

)

give

/κ = 1/(0.0551 × 36) ≈ 0.50

. This

widens

the hierarchy rather than eroding it.

/κ = 0.50

the tetrahedral node dissolves at

P = 0.714

, since it now requires all four

bonds, any three give

min

/κ =

< 0.50

. The cuboctahedral node stays at

8.5 × 10

−3

The full-shell stinesses are

min

/κ = 4

for

K=12

against

for

K=4

. This joins the

dissolution rule and the rigidity bound of Section 18 into a single mechanical stability

criterion.

14 The assembly as a quantum-trajectory representa-

tion

A quantum-trajectory unraveling of Eq. (23) replaces the density matrix by stochastic

paths with bond-creation (

) and bond-destruction (

−

) jumps; the stitchlift growth is

one such path (Algorithm 1). Births are

jumps placing nodes at unit distance, accepted

under exclusion; deaths are

−

jumps dissolving a node with probability

∆t P

dissolve

(K)

. A

nucleating triangle is the smallest rigid object and forms in a few

; the core coordination

then climbs and

holds

(Figure 4), the derived stability holding the core where an imposed

death rate would erode it toward the foam.

Explicit mapping of the jump operators, and its status.

Strictly, Eq. (23) with

independent single-bond jumps does not by itself generate the composite events of the

growth rule: a stitch creates two bonds simultaneously, a lift three, each together with

a new node, its position, and a metric-exclusion test. The mapping below therefore

posits many-body birth jumps whose rates are xed by detailed balance on the bonds

they form, not derived from a microscopic systembath coupling; the assembly algorithm

is a geometry-conditioned Markov process consistent with this trajectory picture, not a

literal unraveling of the single-bond master equation. The generator whose trajectories

reproduce the rule can be written explicitly:

˙ρ = −

ℏ

[H, ρ] +

x∈S(C)

(x) D[J

(x)]ρ +

x∈L(C)

ℓ

(x) D[J

ℓ

(x)]ρ +

−

D[σ

−

]ρ,

(29)

with composite birth jumps

(x) = P

adm

(x) a

†

, J

ℓ

(x) = P

adm

(x) a

†

(30)

Here

†

creates the node degree of freedom at

, and

adm

(x)

projects onto congurations

satisfying the hard-core and proximity constraints. The sets

S(C)

and

L(C)

are the

admissible stitch and lift sites of the current conguration

Two features take this outside the ordinary Lindblad setting:

†

changes the particle

number, and the admissible jump set depends on

. Equation (29) is therefore a

hybrid

Fock-space quantum Markov process

. The state space is

H =

∞

N=0

over

-node

embedded congurations, with

†

: H

→ H

N+1

, and

adm

encodes classical geometric

conditions on the embedding.

Equation (29) is the model's dynamical denition. It states how nodes are created, why

a stitch needs two anchors and a lift three, and where geometric admissibility enters.

What remains open is deriving the composite rates

, γ

ℓ

from a microscopic systembath

coupling; below they are xed by detailed balance on the bonds each jump creates.

The mapping requires a precise statement, because the Lindblad jumps act on

bond

states

while the moves are

node

placements.

How a node is born.

jump sets a bond

from

|0⟩

|1⟩

. In the unraveling, a

node is

born

when a coincident set of such jumps simultaneously raises the bonds that tie

a new site to the existing network: two for a stitch (the apex of an equilateral triangle

on an edge) or three for a lift (the apex of a regular tetrahedron on a face). Concretely,

the birth of a node

is the joint jump

i∈A(x)

(x,i)

over the anchor set

A(x)

(

|A| = 2

stitch,

lift), accepted with Metropolis probability

min(1, e

−β∆E

) = min(1, e

+βε|A(x)|

) = 1

(bond-forming births are downhill and always accepted; the thermal factor acts on the

reverse, dissolution channel through detailed balance); a

−

jump lowers a bond, and

a node dissolves when its surviving bonds fall below rigidity, reproducing

dissolve

(K)

Eq. (28).

The continuous bond-ip rates

↑

, γ

↓

thus generate the discrete moves through the

ad-

missible anchor sets

: the geometry enters by restricting which coincident

patterns

correspond to a unit-distance registered site.

The conguration space, made precise.

To state the positional conditioning exactly

instead of informally, dene the conguration space

C =

(V, E, x) : |x

− x

| ≥ R

∀i = j, E ⊆



ij : |x

− x

| ≤ R



(31)

the set of node embeddings

x : V → R

obeying hard-core exclusion at

together with

the bonds admissible within the proximity radius

. The Lindblad jump operators act

as maps between elements of

: a birth

i∈A

(x,i)

sends

(V, E, x) → (V

′

, E

′

, x

′

)

with

′

= V ∪ {x}

for an apex

at unit distance from its anchor set

, admitted only if

satises the exclusion constraint in Eq. (31); a dissolution

−

performs the reverse.

The dynamics is thus a Markov process on

, and the metric is explicit in the constraints

−x

| ≥ R

and

≤ R

that dene

. This makes precise the sense in which the model is

positional. The master equation evolves a state

, and

is built from a presupposed

metric. It does not construct

from metric-free data. A pre-positional theory would

derive

, the very notion of unit distance and exclusion, from the entanglement structure

itself; that derivation is the open problem of Section 26, and stating

explicitly here is

what makes the boundary between what the model does and does not do exact.

The limitation is explicit. Equation (23) is

not

yet a fully pre-positional theory in which

the Lindblad operators spontaneously generate geometry: it is a bond-state dynamics

con-

ditioned on the admissible geometric move set

. The jump operators do not by themselves

Algorithm 1

Quantum-trajectory assembly (a trajectory representation of Eq. (23); see

text for its status)

seed a Bell pair / triangle nucleus (edge

)

for

each step

∆t

(units of

)

births (

stitch (in-plane apex) or lift (apex at

2/3 L

); reject on exclusion

∈ (R

∗

, L)

; bond within

deaths (

−

dissolve each node with prob

∆t P

dissolve

(K)

record core

⟨K⟩

end for

0 10 20 30 40 50 60

time

t / τ

(

/ε

≈

Planck

)

core coordination



core

= 12

reached

foam



≈

triangle

nucleates

Nucleation to

= 12

in physical time

derived stability

= 1

(1+

)

Figure 4: Dissipative limit: the core coordination climbs to

∼ 9



9.5

with

K=12

reached

and holds, far above the rigidity-only glass (

⟨K⟩ ≈ 5.4

); one step is

= ℏ/ε

single out the stitch and lift placements; the set of unit-distance registered apices, supplied

by the presupposed metric and exclusion, does. Within that conditioning the mapping is

exact, each move is a specic coincident-jump pattern with the detailed-balance weight,

but the conditioning itself is the positional input agged as the deepest open problem

(Section 26).

K=12

is a forced terminus

The geometric bound.

The simulation reaches

K=12

; we show it cannot do otherwise.

The twelve nearest-neighbor directions form the cuboctahedral shell, mutually

≥ L

apart

with nearest pairs at exactly

. A thirteenth node at unit distance from the center

must sit on the unit sphere outside the exclusion radius of all twelve; the best it can do

is the center of the largest vacancy, which is a

square face

of the cuboctahedron with

center-to-vertex chord

∗

2 −

√

2 L ≈ 0.76537 L,

(32)

a closed-form constant conrmed by a global maximin search (Figure 5). Therefore

∗

< R

< L =⇒

no thirteenth neighbor, all twelve admitted, (33)

and inside this window the kissing-number theorem [18] makes

K=12

a geometric terminus

for any growth history.

∗

= (2

−

)

12-shell: largest hole is a square face

(no 13th neighbour fits for

> R

∗

)

Figure 5: The cuboctahedral shell (blue); its largest vacancy is a square face (red). A

thirteenth node at unit distance approaches no closer than

∗

2 −

√

2 L

to an occu-

pied site, so any

> R

∗

forbids it

K=12

is forced.

Two distinct thresholds.

The upper edge

= L

is the lattice-freezing transition,

observed directly in the simulation. The lower edge

∗

is the

absolute geometric

threshold

below which a thirteenth-neighbor site exists. It is not the same as the

observed

lower

wall of the exclusion-radius sweep, which sits at the triangle circumradius

√

3 ≈ 0.577 L

(Ref. [22]).

The gap between the two is kinetic protection by the operator basis. The thirteenth-

neighbor site is the square-face vacancy, and it lies at unit distance from exactly one

existing node: the shell center. A stitch requires two unit-distance anchors and a lift

requires three. Neither operator can therefore ever propose it. So for

∈ (L/

√

3, R

∗

)

the site is geometrically admissible but kinematically unreachable, and for

> R

∗

it is

excluded outright.

K=12

is thus geometrically forced inside the window (33) and kinetically protected below

it. It is not a value the kinematics must be tuned to reach.

16 The selection of FCC over HCP

FCC and HCP are stacking isomers, both close packings at

K=12

, and the selection is

invisible to local and code-theoretic measures. The local stabilizer groups are isomorphic,

the code rate (

≈ 4.02

/atom) and distance (

) coincide, and, we veried, the bulk-interior

diagnostic gives modal

K=12

for HCP as well, so the saturation data cannot distinguish

ABC from ABAB.

Contact counting cannot decide it either.

No renement of the bond count settles

the question. Maximizing unit-distance contacts selects Barlow close packing, of which

FCC and HCP are the two stacking isomers, and small maximal-contact clusters are

common fragments of both. A cluster of a dozen nodes cannot exhibit an FCC bulk node

at all, since that requires twelve neighbors together with populated second and third shells.

Any argument that terminates in maximal contacts therefore delivers close packing and

stops one step short of FCC. The selection has to come from somewhere else.

The stacking is selected by the growth operator, not by the energy.

Label the

lateral registry of the

th close-packed sheet by

∈ Z

= {0, 1, 2}

, the conventional

positions. Close-packed attachment of the next sheet requires

n+1

− s

≡ χ

(mod 3), χ

∈ {+1, −1},

(34)

since the new sheet must sit in one of the two hollow registries of the one below. The

two stacking isomers are then distinguished by a single discrete property of the sequence

: FCC is

constant, giving

ABCABC . . .

for

χ = +1

and its mirror

ACBACB . . .

for

χ = −1

, while HCP is

alternating,

+1, −1, +1, −1, . . .

, giving

ABAB . . .

The lift of Section 2.1 places an apex above an oriented triangular face at the tetrahedral

height. The outward normal xes which

side

of the sheet grows. It does not x the sign

in Eq. (34): both continuations are geometrically admissible at unit bond length, and a

cyclic ordering of the face does not by itself select between the two nontrivial rotations

. Selection therefore requires one additional bit, carried by the advancing frontier

rather than by the local geometry, a stacking chirality that the lift transports unchanged

from parent face to daughter face.

The consequence, if the frontier transports that bit.

Suppose the lift acts on an

oriented frontier state

(f, χ)

and returns a daughter frontier carrying the same

. Then

= χ

for every defect-free lift, and Eq. (34) integrates to

= s

+ nχ (mod 3),

(35)

which is

ABCABC . . .

ACBACB . . .

: the two mirror orientations of FCC. HCP is

excluded, because an alternating increment would require the frontier to reverse its trans-

ported orientation at every single layer.

Limits of the argument.

It is a minimal-suciency statement, not a derivation of FCC

from geometry alone. Registry-increment conservation is built into the transport rule, so

the induction is not where the content lies. The content is the identication of exactly

what must be transported:

one

conserved binary frontier variable is enough to exclude

HCP; an oriented normal alone is not enough; and no stacking-dependent energy term is

required.

Whether the microscopic dynamics of Section 14 conserves that variable, and treats its

reversal as a stacking fault, is an implementation condition we have not veried. It is

at least consistent with what the simulation produces: the simulated clusters of Ref. [22]

are polycrystalline, with the shortfall from

K=12

concentrated at

K=10, 11

(Ref. [22]),

which is what a population of stacking faults and misoriented domains looks like.

Why the cell structure matters.

The two stackings also dier in their interstitial

geometry, and the dierence is exactly the one the atness of the close packing depends

on. In the regular tetrahedraloctahedral honeycomb of the FCC branch, two tetrahedra

and two octahedra meet at every edge. Their dihedral angles are supplementary,

= arccos

, θ

= arccos



−



= π − θ

(36)

so that

2θ

+ 2θ

= 2π

identically and every spatial Regge edge decit vanishes. This is

the exact statement behind the disappearance of the two-sheet frustration in Eq. (20): the

transient foam is all-tetrahedral and carries the irreducible decit

δ = 2π − 5 arccos

≈

7.36

◦

of Eq. (1), because a pure tetrahedral fan around an edge cannot close. Octahedral

cells appear only when the third sheet completes the cuboctahedral shell, and it is their

supplementary dihedral angle that closes the fan exactly. The octahedral voids are also

where the code term could distinguish the two stackings, for the reason given in Section 9;

we have not computed that split.

Two static selectors.

Independently of the growth argument, two structural properties

distinguish the two lattices. (i) Bravais structure: FCC is a Bravais lattice (

), HCP is

not. (ii) Rank-four isotropy: the HCP fourth-moment bond tensor carries a

-axis term

zzzz

= 2.667 = T

xxxx

= 2

, forbidden for an isotropic continuum, while FCC has only

cubic anisotropy.

The vibrational tiebreaker.

A third selector is vibrational entropy: the central-force

phonon entropy is marginally higher for FCC (we conrm the sign of the established

hard-sphere result of Bolhuis and Frenkel; the magnitude is small, of order

−3

per

node, and convention-dependent, so we treat it as a consistent tiebreaker instead of the

decisive selector, with the computation in the accompanying scripts).

Cuboctahedron over icosahedron.

The codeable cuboctahedral order is in turn chosen

over the icosahedral glass that also realizes

K=12

. The relevant competition is between

two constraints. Monogamy, the property that makes a bounded coordination number

meaningful in the rst place (Section 2.1), marginally favors icosahedral packing, by

the Thomson comparison

49.165 < 49.342

. Codeability runs the other way. The code

term closes only with the cuboctahedron's square faces, which the icosahedron lacks.

Codeability overrides the sub-percent monogamy preference.

17 The lift rate factorizes

Because Eq. (33) makes the terminus history-independent, the lift rate cannot aect it.

An accretion simulation with hard-core exclusion, in which the stitch/lift choice sets only

the order of node addition, gives the same terminal coordination across

lift

∈ [0.005, 0.5]

two orders of magnitude, no trend (Figure 6); varying

reproduces the window. (The

terminal bulk

modal

value in these short unannealed runs is



, the grain-boundary

value of a polycrystal, matching the bulk mean

11.40

of Ref. [22] with the shortfall concen-

trated at

K=10, 11

; it reects quench rate, not the forcing.) Nor is there a parameter-free

lift rate: a birth captures a node into a registered apex, and the capture-volume ratio of

a lift (zero-parameter point target) to a stitch (one-parameter line target) carries exactly

one extra power of the bond-shell width

lift

stitch

≃ C w, C = O(1),

(37)

veried numerically (the ratio falls monotonically

0.137 → 0.027

w : 0.15 → 0.03

, no

plateau). This geometric acceptance xes the form, never the value.

The full lift rate, however, factorizes into this geometric acceptance and a

thermal

survival

factor. In transition-state form, write the composite rates as

= Γ

Ω

(w) e

−β∆F

‡

and

ℓ

= Γ

ℓ

Ω

ℓ

(w) e

−β∆F

‡

ℓ

, with

the attempt frequencies,

Ω(w)

the geometric phase-space

(solution-manifold) factors at tolerance

, and

∆F

‡

the activation free energies; then

ℓ

Ω

ℓ

(w)

Ω

(w)

−β(∆F

‡

ℓ

−∆F

‡

)

(38)

This section's factorization corresponds to two identications. The rst is

Ω

ℓ

/Ω

∝ w

: a

0D point set measured against a 1D circle, at tolerance

. The second is

∆F

‡

ℓ

−∆F

‡

= 3ε

the cost of the three exposed anchoring bonds. Together they give

ℓ

/γ

= C w e

−3βε

which equals

C w e

−3

at the operating point

βε = 1

The activation energy is a kinetic model, not a derivation.

The apex bonds at

the transition state are formed and energetically favorable. What distinguishes them is

their lack of in-plane redundancy: each is the node's only support in its direction, with no

closed triangle behind it. We take this to price the lift intermediate at

3ε

above the stitch

intermediate. In energy-landscape form,

‡

stitch

= E

+ ∆

and

‡

lift

= E

+ ∆

+ 3ε

. The

outstanding step is to obtain that extra

3ε

explicitly, from unsatised stabilizer terms,

missing closed loops, or transient bond strain. The present Hamiltonian rewards formed

bonds by

−ε

and does not itself penalize them for being exposed.

A bookkeeping point.

The simulation parameter

lift

= e

−3

≈ 0.05

is the

full

per-

proposal lift probability. Equating it to the thermal factor alone therefore amounts to

the normalization choice

Cw = 1

at the operating tolerance, which we adopt as a stated

convention.

The factorization nonetheless separates four ingredients: attempt-frequency ratio, phase-

space suppression, activation energy, and temperature. It also makes the claim testable.

The prefactor should scale linearly in a

sweep, and the exponent should steepen as

−3βε

in a temperature sweep. A lift places an apex with three out-of-plane anchoring bonds

that are exposed, unstabilized by any in-plane neighbor, until the cooperative three-apex

seed of the next sheet closes around them; a stitch, lying in the plane, exposes none. At

the detailed-balance temperature

= ε

that xes

q = 1/(1 + e)

(Section 13), each

exposed bond carries a Boltzmann factor

−ε/kT

= e

−1

, so the thermal ratio of lift to

stitch is

lift

stitch



thermal

= e

−3

(39)

independent

of the tolerance

, with the exponent

set by the three exposed anchoring

bonds. The net rate is the product of the two factors: a tolerance-dependent geometric

acceptance

∼ w

(Eq. (37), not derivable) times the thermal factor

−3

(Eq. (39), xed by

the same

= ε

that gives

, conditional on one binding quantum per exposed bond). The

simulation's

−3

therefore sits at the thermal factor. It is not a free constant: it follows

from the model's detailed-balance temperature. The model does not, however, predict

the absolute lift probability, which is this thermal factor times the tolerance-dependent

geometric prefactor. The lift rate is free in its geometric acceptance and xed in its

thermal factor; the two statements refer to the two factors and do not conict.

lift rate

lift

(free parameter)

terminal bulk modal

= 12

ceiling (kissing bound)

Terminus independent of lift rate over 2 decades

Figure 6: Terminal bulk coordination is independent of the lift rate across two orders of

magnitude of

lift

; the

K=12

ceiling is the kissing bound. The lift rate sets only the order

of node addition, not the terminus.

18 The parameter the model must supply: a rigidity

scale

If the lift rate factorizes and

K=12

is forced, what must the model x for the lattice to

exist? Here the one energy scale of the bond-state Hamiltonian is not sucient. The

positional realization of the bond requires a curvature scale

in addition to the binding

depth

, derived self-consistently from the phonon spectrum. Each bond is a central well

of curvature

(units

ε/L

); the FCC dynamical matrix is

D(q) =

j=1



1 − cos(q·n

)



ˆn

⊤

(40)

and the mean-square displacement, including zero-point and thermal motion, is

⟨u

⟩ =

ℏ

√

κm

2˜ω

coth

˜ω

, t ≡

ℏ

κ/m

(41)

averaged over the Brillouin zone.

The displacement convention.

We pin the constants by a converged Brillouin-zone

integration under the

relative-displacement

(Lindemann) convention,

⟨|u

− u

⟩ <

L − R

∗

for nearest neighbors

. This is the physically relevant choice, because it retains

the phonon correlations

⟨u

· u

⟩

that independent-node conventions discard. Monkhorst

grids from

agree to four digits, and the shifted grid excludes the acoustic

zero mode. Independent-node conventions give

10.5



31.5

and

0.022



0.066

for the two

constants below, bracketing the pinned values. The qualitative conclusion is convention-

independent: a single-scale vacuum sits at its melting point.

Two rigidity conditions.

The forced lattice is rigid only if bond uctuations do not carry a neighbor across the

wall

∗

L −

2⟨u

⟩ > R

∗

, i.e.

2⟨u

⟩ < (1 −R

∗

/L)

= 0.0551 L

. Two derived conditions

follow (Figure 7). A zero-point oor,

√

κm L

ℏ

> 28.6,

(42)

reduced temperature

kT/

Ω

(

Ω =

)

0.6

0.7

0.8

0.9

1.0

nearest-neighbour separation

Rigidity of the forced

= 12

lattice (two-scale)

min bond length

−



∗

= (2

−

)

(K=12 wall)

Figure 7: Nearest-neighbor separation versus reduced temperature

t = kT/ℏΩ

(

Ω =

κ/m

), from the FCC phonon spectrum. The lattice is rigid (green) while the uctuating

separation stays above the

K=12

wall

∗

(red), and melts past

; rigidity xes the bound

Eq. (43).

so even at

T = 0

the lattice must be sti and heavy enough that quantum motion stays

inside the window; and a thermal separation,

κL

< 0.031 ⇐⇒ κL

≳ 32 kT.

(43)

We round this bound up and quote

κL

≳ 36 ε

throughout, a slightly stronger requirement

than the integration strictly demands.

A single-scale vacuum, with

kT ∼ ε ∼ κL

, gives a ratio of order unity. That is an

order of magnitude above the bound. Such a vacuum sits at its melting point and cannot

crystallize. This also relieves the only tension in Section 13:

q = 1/(1 + e)

uses

= ε/k

with

the binding

depth

, while rigidity constrains

kT/(κL

)

with

κL

the

curvature

; the

two coexist provided the bond well is deep-and-narrow,

κL

≳ 36 ε

. The

K=12

vacuum

exists precisely when the binding well is sti in that sense.

19 The construction velocity

The in-plane speed.

The dynamics xes the speed of the assembly. The front advances

by attachments net of detachments; with attempt frequency

1/τ

and detailed balance,

the net forward rate per site is the same

that sets stability,

(γ

↑

−γ

↓

) = (1−2q)/τ

= (e−

1)/(e + 1)/τ

≈ 0.462/τ

, with the normalization

↑

+ γ

↓

≡ 1/τ

dening the microscopic

attempt time (detailed balance xes only the ratio of the rates; their sum is a convention

absorbed into

). The in-plane (stitch) growth is fast and the move is abundant, a stitch

needs only two equidistant anchors where a lift needs three, so a

K=6

sheet grows radially

at the speed

= (1 − 2q)

√

lat

≈ 0.40 v

lat

≈ 0.07 v

sig

(44)

The construction is anisotropic.

The out-of-plane lift is rarer (one fewer solution

dimension, and the binding term favors the at

K=6

contact maximum over the buckled

foam), and its rate is the amplitude

lift

discussed in Section 17. The construction is

therefore

anisotropic

: fast in-plane at Eq. (44), and lift-limited in the stacking direction.

The naive estimate of a single dimension-independent build speed is incorrect, precisely

because it would equate the stitch and lift rates, which the geometry forbids. The phys-

ically meaningful parameter-free number is the in-plane speed

; the stacking advance

inherits the lift-rate freedom.

The anisotropy is the same

that suppresses the lift.

Per frontier site, a lift is

attempted and accepted at

lift

times the stitch rate, and it advances the structure by the

interlayer spacing

2/3 L

rather than the in-plane step

√

. The two speeds therefore

stand in the ratio

stack

lift

√

3/2

2/3

= e

× 1.06 ≈ 21,

(45)

with the bare rate ratio

1/P

lift

= e

≈ 20

and the geometric step-length correction supply-

ing the remaining

. Lateral growth thus outruns stacking by a factor of roughly twenty,

and that factor is not independent of the lift suppression: it

, the same exponential

that makes the lift rare. Numerically,

stack

≈ 0.019 v

lat

against

≈ 0.40 v

lat

The ratio recurs on every new layer.

A lift does more than advance the stack by

one node. It seeds an entire new sheet, and that sheet then spreads at the same

. The

anisotropy is therefore not a one-time property of a single frontier site but applies afresh

to each layer as it is born. In the time one layer of thickness

2/3 L

is added, the sheet

carrying it widens by roughly

21 × 0.82 L ≈ 17 L

. Growth is self-similar. Every sheet, at

every stage, outruns the stack it sits on by the same factor

Why the cluster is nonetheless compact.

This does

not

make the crystallite a

pancake of aspect

1/21

. If the structure grew as a single stacking column, the recurring

21:1

anisotropy would give a cluster aspect ratio

z/xy ≈ 0.047

; the sweep of Table 2 of

Ref. [22] measures

0.83

lift

= e

−3

, larger by a factor of

. The discrepancy is the

branching channel. Sheets nucleate in parallel across the cluster instead of sequentially

up one column, so many layers advance at once.

The gap closes monotonically as lifts become common, the measured-to-single-column

ratio falls

63, 27, 18, 8.9, 6.4, 3.3, 2.0, 1.2

across

lift

= 0.01

0.85

, and reaches unity

only when lifts are so frequent that no branching advantage remains. That the two agree

lift

→ 1

and diverge as

lift

→ 0

is a quantitative check on the branching picture of

Section 20. Taken with the signal front

sig

= 4

√

2 v

lat

(Section 12), the construction front

runs at

≈ 0.07 v

sig

: local assembly proceeds at roughly one-fourteenth of the fastest

speed the lattice can carry.

What is xed here is a ratio, not a speed.

Every velocity in this section is quoted

in units of

lat

= Lε/ℏ

, and the model does not x

separately (assumption A2,

Section 8). The absolute construction speed is therefore unknown. What the dynamics

xes are the dimensionless ratios:

lat

= 0.40

sig

≈ 1/14

, and the lateral-to-

stacking anisotropy

≈ 21

of Eq. (45). These are parameter-free because

and

cancel

in each.

Converting

1/14

into a statement about the physical speed of light requires the further

identication

sig

= c

. That identication is conditional, not established here. It rests

on the GW

light constraint of Eq. (26), which xes only the elastic

trace

speed, and the

no-go theorem of Section 23 shows that no stable elastic medium carries all polarizations

at a single speed. Until a sector is identied that does,

sig

is the lattice's own causal

bound, and 

0.07 c

 should be read as

0.07 v

sig

20 Exponential branching growth: an ination-shaped

geometric conversion

Sheets seed sheets.

A lift does more than advance the stacking front by one node. It

seeds an entire new plane, which then grows laterally at the fast in-plane speed

. Each

growing sheet is itself a substrate for further lifts, so sheets seed sheets, a self-exciting

(branching) process. Writing the population in moments (sheet number

, total radius

, total area

, with seeding rate

∝ σA

σ ∝ P

lift

), the chain closes as

...

A = 2πv

σ A =⇒ V (t) ∼ e

Γt

, Γ ∼



2πv



1/3

∝ P

1/3

lift

(46)

exponential

growth of the crystallized volume (Figure 8), conrmed numerically over

many orders of magnitude in

. The cluster aspect ratios of Table 2 of Ref. [22] test this

independently: they exceed the single-column prediction of Section 19 by a factor that

falls from

1.2

lift

rises from

0.01

0.85

, which is the signature of parallel sheet

nucleation. The rate inherits the lift-rate freedom (as a cube root, the seedingradius

area chain being three steps), so it is not a xed number; the number of

-folds is set by

the foam reservoir,

ln(V

foam

seed

)

, because new sheets nucleate only into the metastable

foam and growth halts exactly when the foam is consumed, a built-in graceful exit.

Why this is labeled speculative.

This exponential, self-terminating conversion of

metastable foam into crystal is a geometric analog of ination-with-reheating, accelerated

growth driven by a metastable phase, ending cleanly on reservoir depletion. It is an

analog

: it concerns the build-up of lattice volume, not the cosmological scale factor.

It is labeled speculative for two reasons. First, the spatial branching is scale-free. The

clustering spectrum follows a power law with

= 0.96

over the inertial range. That

is the prerequisite for an inationary spectrum, but not the spectrum itself. The raw

nucleation-clustering slope is steeply red (

n ≈ −3.3

), and we have not computed the

frozen, horizon-crossing curvature spectrum that would be compared to the observed

≈ 0.965

[19]. Second, identifying exponential growth of

lattice volume

with metric

ination requires the equation of physical space with the lattice, which remains unproven

(Section 26). The mechanism is genuine and ination-shaped; the claim that it

ination

is not established.

21 End-to-end coherence and robustness

One rule, end to end.

The preceding mechanisms were derived piecewise. We now test

whether they cohere under a

single

xed rule. We run one Metropolis-kinetic dynamics,

propose stitch and lift wherever geometrically available, accept a birth with Metropolis

probability

min(1, e

−β∆E

) = 1

for

∆E = −ε ∆n < 0

(

∆n

the bonds it forms; births are

0 50 100 150 200

time

/τ

crystallized volume

(

)

Branching growth: exponential, self-terminating

lift amp

= 0

0050

lift amp

= 0

0005

foam reservoir (exit)

Figure 8: The branching phase: crystallized volume grows exponentially (note the log-

arithmic axis) and saturates exactly at the foam reservoir, an ination-shaped, self-

terminating conversion. The rate scales as

1/3

lift

and so inherits the lift-rate freedom.

always accepted, the thermal factor entering through dissolution), and dissolve each node

with probability

∆t P

dissolve

(K)

from the derived stability, starting from a bare Bell pair,

with no per-stage switches. The same rule nucleates a triangle, grows sheets, lifts, and

forms a

K=12

bulk: under the bulk-interior diagnostic the interior reaches modal

K=12

early and holds it, with the bulk node count rising monotonically. The pieces cohere; they

do not ght at the seams.

Robustness across temperature.

Robustness is conrmed by sweeping the bath tem-

perature over a factor of eight (

β = k

T/ε

from

0.4

3.0

, i.e.

from

0.40

0.047

) at

four seeds each:

every

run reaches bulk modal

K=12

, with only one of four landing on a

grain-boundary

K=11

at the coldest point (Figure 9).

K=12

is the generic outcome of

the dynamics, not a ne-tuned one, and the derived operating point

q = 1/(1 +e)

(

β = 1

)

sits in the middle of the working range, not at its edge. The all-node mean coordination

plateaus near

(a nite, polycrystalline crystallite with modal

K=12

interior, match-

ing the bulk mean

11.40

of Ref. [22]); reaching single-crystal

⟨K⟩ → 12

is an annealing

question, not a missing parameter.

22 Derived scale relations

Several quantitative relations follow directly from the results above; we collect them here.

Each is labeled by what it rests on.

Node inertia bound (D).

The zero-point oor of Section 18,

√

κm L

/ℏ > 28.6

rearranges to a lower bound on the node inertia,

m > 28.6

ℏ

/(κL

)

. Using the rigidity

requirement

κL

≳ 36 ε

and

lat

= Lε/ℏ

gives

m ≳ 22.7 ε/v

lat

; with the light cone

c = 4

√

2 v

lat

m c

≳ 7.3 × 10

ε.

(47)

0.5 1.0 1.5 2.0 2.5

reduced bath temperature

T/ε

100

= 12

success (\%)

= 1

(1 +

)

operating point

= 12

is generic across temperature (single end-to-end rule)

bulk reaches

= 12

(\% of seeds)



Figure 9: Robustness of the assembly: under one end-to-end rule, the bulk interior reaches

K=12

for every seed across a factor-of-eight temperature range (green), with

⟨K⟩

(blue)

plateauing at the polycrystalline crystallite value. The derived point

q = 1/(1 + e)

lies

mid-range, not at a cli.

A stable close-packed node must carry an eective inertia of order seven hundred times

the binding scale, a derived massenergy hierarchy, conditional on the

c = 4

√

2 v

lat

light

cone and the pinned Lindemann convention of Section 18.

Critical rigidity temperature (D).

The melting condition

kT/(κL

) < 0.031

of Sec-

tion 18 sets a critical temperature

≈ 0.031 κL

. Imposing the derived stiness

κL

≳ 36 ε

≳ 0.031 × 36

≈

= T

(48)

The melting threshold sits right at the bath scale

= ε/k

that drives assembly. This

explains why the model requires the factor

∼ 36

: it makes the close-packed phase just

rigid enough to survive the same thermal scale at which it is built.

A geometric small parameter (T).

The Regge decit supplies a genuine small di-

mensionless number,

geom

≡

2π

2π − 5 arccos

2π

≈ 2.0 × 10

−2

(49)

geometric rather than tuned. It controls the residual anisotropy at

O(ε

geom

)

and the foam

instability strength at

O(ε

geom

)

Branching rate and exit time (S).

With the construction estimate

≈ 0.40 v

lat

of Section 19, the branching rate of Section 20,

Γ ∼ (2πv

σ)

1/3

with seeding density

σ = α

lift

/(L

)

, reduces to

Γτ

∼



2π(0.40)

lift



1/3

≈



lift



1/3

(50)

the

O(1)

prefactor collapsing to near unity, so the exponential rate is essentially the cube

root of the eective lift-seeding rate (and inherits its freedom). The exponential phase

then has a denite exit time,

exit

= Γ

−1

foam

seed

(51)

separating

how much

expansion (the reservoir ratio, the

-fold count) from

how long

lasts (the seeding rate). Growth halts when

crystal

= V

foam

and

Γ → 0

, a graceful exit

requiring no added decay mechanism.

A polycrystallinity order parameter (C).

The all-node mean coordination plateaus

near

⟨K⟩ ≈ 9

while the bulk interior is modal

K=12

, suggesting the order parameter

defect

∼ 1 −

⟨K⟩

≈ 0.25,

(52)

a measure of polycrystallinity (under-coordinated surface and grain-boundary nodes),

not a literal defective fraction. We note that this plateau is a grain-structure (nucleation)

eect: direct tests show post-growth relaxation does not drive

⟨K⟩

toward

, so reaching

single-crystal coordination is a nucleation/annealing question and not achieved by local

relaxation alone.

23 Further consequences. The acoustic spectrum and

the vacuum energy

Two quantities follow directly from the geometry and the GW-xed stiness

κ =

16 mε

/ℏ

of Section 12. Both sharpen, instead of soften, the constraints the model

must meet, and we state them as such.

The acoustic spectrum and its anisotropy (T).

The FCC phonon spectrum of

Section 18 (the dynamical matrix (40)) is a nearest-neighbor central-force model; in the

long-wavelength limit it is governed by the

rank-four

bond tensor [11],

αβ

(q) ≃

κL

µναβ

, T

µναβ

j=1

ˆn

= δ

µν

αβ

+δ

µα

νβ

+δ

µβ

να

−∆

µναβ

(53)

with

∆

µναβ

= 1

only when all four indices coincide. Unlike the rank-two

µν

= 4δ

µν

that

governs the scalar channel,

carries the cubic term

∆

, so the three acoustic branches are

isotropic only in their trace. Diagonalizing along the symmetry directions gives, in units

κ/m

longitudinal transverse transverse

[100] 1 1/

√

2 1/

√

[110]

√

5/2 1/

√

2 1/2

[111]

4/3 1/

√

3 1/

√

The longitudinal speed runs from

κ/m

along

[100]

4/3 L

κ/m

along

[111]

a parameter-free anisotropy of

4/3 − 1 ≈ 15.5%

set entirely by the FCC bond ge-

ometry. With

xed by GW

light (Eq. (26)) these become

∈ [0.71, 0.82] c

and

∈ [0.35, 0.50] c

. The three branch speeds obey the trace identity

+ v

= c

in every direction, which is the content of Eq. (26). The bare lattice therefore carries

sub-luminal, anisotropic

sound. The isotropic scalar speed

= L

2κ/m = c

of Sec-

tion 12 survives only in the trace channel. This is the concrete form of the residual cubic

anisotropy anticipated there. A nearest-neighbor central-force vacuum is thus Lorentz-

violating at the lattice scale, at the

4/3−1 ≈ 15.5%

level in its longitudinal branch and

the

√

2 − 1 ≈ 41%

level in its transverse branches (spread

[

√

] L

κ/m

across direc-

tions), the transverse sector being the one relevant to tensor (gravitational-wave) modes;

full continuum isotropy, with a single

shared by all polarizations, requires either angular

(bond-bending) couplings beyond nearest-neighbor central forces or a renormalization-

group ow to an isotropic xed point. We record the bare anisotropy as a sharp quanti-

tative target, not a solved problem.

Polycrystalline isotropization (D).

The

directional

anisotropy above (

15.5%

lon-

gitudinal,

41%

transverse) is a single-crystal property, and the bond tensor obeys the

Cauchy relation

= C

(Zener ratio

A = 2C

/(C

− C

) = 2

) characteristic of

a central-force lattice. The vacuum described here, however, is a polycrystal of ran-

domly oriented grains. Averaging the cubic tensor over orientations removes the direc-

tional dependence entirely. We use the standard VoigtReussHill construction, in which

Voigt and Reuss give the uniform-strain and uniform-stress bounds and Hill takes their

mean. The result is an isotropic eective medium with

≃ 1.08 L

κ/m ≃ 0.76 c

and

≃ 0.62 L

κ/m ≃ 0.44 c

at the Hill average.

Two consequences.

Above the grain size, the misoriented-crystallite scale that produces

the coordination scatter

≈ 0.99

reported in Ref. [22], the vacuum has no preferred di-

rection, so any directional gravitational-wave birefringence is conned to sub-grain scales,

a falsiable prediction. But isotropization does not merge the polarizations. The longi-

tudinal and transverse speeds remain split,

≃ 1.76

. The polycrystal thus removes

the

directional

Lorentz violation while leaving the

polarization

splitting, which a single

shared cone must still close. The next paragraph shows what that closing can and cannot

look like.

No-go theorem for an elastic universal cone (D).

The theorem has two halves.

First: nearest-neighbor pair forces cannot produce a stable isotropic elastic

medium.

Take the general axially symmetric nearest-neighbor force matrix

Φ = α ˆnˆn

β (⊮ −ˆnˆn

)

, with radial stiness

and tangential stiness

. A direct computation of the

FCC elastic constants gives

∝ α + β

∝ (α − 5β)/2

, and

∝ (α + 3β)/2

. The

isotropy condition is then

−C

−2C

∝ (β −α)/2

, which vanishes

only

β = α

. At

that point the force matrix degenerates to a scalar spring

Φ = α⊮

and all three acoustic

branches coincide. The constants stand in the ratio

: C

= 2 : −2 : 2

, so

λ = −µ

and the bulk modulus is

= λ +

µ = −

< 0

. The isotropic point is mechanically

unstable.

Second. No stable elastic medium carries all polarizations at one speed.

This

holds independently of any force model. Equality

= v

requires

λ + 2µ = µ

, hence

λ = −µ

, hence

< 0

for any nonzero shear rigidity.

The two conditions meet at the same forbidden point. Within nearest-neighbor pair forces,

the unique isotropic conguration

the unique equal-speed conguration, and stability

excludes it.

The consequences are constructive.

Restoring isotropy

with

stability requires three-

body angular terms,

bend

⟨ijk⟩

(ˆr

· ˆr

− cos θ

)

, or couplings beyond nearest

neighbors. Solving for the ratio

/κ

that cancels the cubic anisotropy is a dened target

calculation. Separately, a shared single speed across polarizations cannot be carried by the

displacement eld of

any

stable medium. The photon and gravitational sectors therefore

cannot both be elastic polarizations, and a distinct gauge-like sector is required. This

sharpens the open problem. The branch-level cone is closed to the elastic sector as a

matter of theorem, and open only to new sectors. If the framework contains a relativistic

tensor excitation, it must be a gauge-like, topological, or defect-based collective degree of

freedom of the bond network, not the displacement eld itself.

The vacuum energy density (D/S).

The FCC bond density is

√

2/L

(six bonds

per node, node density

√

2/L

), so the binding term contributes

bind

= −ε

√

≈ −8.49

(54)

negative, as a bound crystal requires. Phonon zero-point motion adds a positive

∼

ℏω

(

√

2/L

)

with

ℏω

∼ 4 ε

from the xed

, i.e.

∼ +O(1) ε/L

, of the same order.

The net density is therefore

|ρ

vac

| ∼ ε/L

. With the binding oor

ε ≳ 2.3 ℏ

/(mL

)

and a Planck-scale lattice (

L ∼ ℓ

) this is of order the Planck density, roughly

120

orders of magnitude above the observed cosmological constant: the model reproduces the

cosmological-constant problem [23] in its standard, unsolved form. The single structural

feature it oers is that

bind

< 0

and

> 0

are of the same order and opposite sign,

so a cancellation is dimensionally available, though nothing in the present construction

enforces it to the observed level.

24 The cascade as a sphere-packing variational ladder

The binding term, over pairs, is

−ε

times the contact number of a unit-sphere packing;

its minima, dimension by dimension, are the cascade's rungs (Table 6, Figure 10). The

K=6

sheet is the planar contact maximum; the

K=12

packing is the spatial contact

maximum, the kissing number [18] achieved everywhere only by the densest packing (Ke-

pler/Hales [3]); the

K=4

foam pays both a lower coordination and the Regge frustration,

sitting

1.37 ε

per node

above

the sheet, a barrier and, being rigid, a kinetic trap (the

glass). Registered lifts raise coordination monotonically

K : 6 → 9 → 12

; unregistered

buckling is trapped in the foam.

Table 6: Energy per node along the cascade, in units of the binding scale

rung role

binding total

E/N (ε)

Bell pair / triangle rigid seed (nucleus) 2

−1.00 −1.00

hexagonal sheet 2D contact maximum 6

−3.00 −3.00

tetrahedral foam 3D frustration barrier (glass) 4

−2.00 −1.63

close packing 3D contact maximum

code 12

−6.00 −7.00

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

assembly coordinate

energy per node (

)

triangle

= 2

hex sheet

= 6

(2D max)

foam

= 4

(barrier)

close pack

= 12

(3D min)

Energy ladder (units of the binding scale

)

registered path:

→

, downhill

unregistered: through foam barrier (glass)

Figure 10: The cascade as an energy ladder (

): the

K=6

sheet (2D maximum), the

K=4

foam (barrier

1.37ε

above, the glass), the

K=12

packing (

−7ε

). Registered lifts (solid)

descend monotonically; unregistered buckling (dashed) is trapped.

25 The CSS code on the close-packed lattice

The code term of Eq. (21) is the stabilizer Hamiltonian of the

[[192, 130, 3]]

CSS code [21]

introduced in Section 5: vertex-star

-checks and weight-12 octahedral

-checks that

commute over

GF(2)

, with weight-3 logical faces giving

d = 3

and rate

130/192 ≈ 0.677

What matters for the dynamics is that this term is nonzero only where octahedral voids

exist, on the cuboctahedral close packing, which is why it appears at the top rung of the

variational ladder (Section 24) and selects that registration (Section 9).

26 Scope and open problems

Derived here:

the emergent light cone

c = 4

√

2 v

lat

and, from the GW

light (trace-

channel) constraint, the xed stiness

κ = 16 mε

/ℏ

; the detailed-balance

q = 1/(1 + e)

and the dissolution ladder; the forced

K=12

terminus and

∗

2 −

√

2 L

; the lift rate's

freedom and the non-existence of a derivable value; the rigidity bounds Eqs. (42)(43);

the energy ladder; the code parameters; and the FCC/HCP selectors, including the iden-

tication of the single transported frontier bit that suces to exclude HCP (Section 16).

Established in Part I:

the reversibility and stationary measure of the assembly, its

concentration on maximal

construction

bonding under a stated entropy bound, the exact

nite-state enumeration, and the structure-tensor isotropy of a close packing.

Imported as established mathematics:

the kissing and Kepler theorems [18, 3] un-

derlying the contact maxima, and the Regge decit-angle construction [9] used for the

frustrated foam.

Open:

1. Constructing the embedded model

emb

of Section 3.9 explicitly: a rule identifying

geometrically congruent histories, contact updates after each move, reverse moves

that remove contacts consistently, and then irreducibility of the resulting transition

graph and the decit-entropy bound on it. Detailed balance transfers algebraically;

ergodicity and concentration do not, and the thermodynamic statement for the

physical energy rests on them.

2. The registration bias of the births is motivated by the code term lowering registered-

conguration energy, but it is not proven quantitatively from BornMarkov rates.

3. The rigidity factor

≳ 36

and the zero-point oor

28.6

follow from the central-force

harmonic spectrum. A fully anharmonic well would sharpen them.

4. Reaching single-crystal

⟨K⟩ → 12

, rather than the polycrystalline modal-

crys-

tallite, is an annealing and quench-rate question, not a missing parameter.

5. The branching phase (Section 20) is exponential and self-terminating, but its frozen

perturbation spectrum is uncomputed and its raw clustering is steeply red. Its

identication with ination is therefore unestablished.

6. The light cone is computed in the single-excitation sector, whose band is quadratic at

long wavelength, while the linear relativistic cone belongs to the harmonic (elastic)

sector. Unifying the two, and closing the branch-level single-cone gap of Section 23,

is open.

7. The construction is positional. It presupposes node positions and a metric exclu-

sion. A fully pre-positional dynamics, and with it any genuine claim about metric

expansion, remains the deepest open problem.

27 Conclusion

The Selection-Stitch vacuum has a single dynamical origin: a Lindblad master equation

whose Hamiltonian carries one energy scale and one length. Its coherent limit propa-

gates an emergent light cone of speed

lat

= Lε/ℏ

, reproducing the lattice isotropy, with

LiebRobinson front

c = 4

√

2 v

lat

; its dissipative limit assembles the structure, its rate

q = 1/(1 + e)

derived by detailed balance and its quantum-trajectory representation

reproducing the cascade.

What turns out not to be an input.

Three apparent inputs to that cascade turn

out not to be inputs at all. First,

K=12

is a forced terminus. The kissing bound forbids

a thirteenth neighbor for any exclusion

> R

∗

2 −

√

2 L

. Second, the lift rate

factorizes into a free geometric acceptance and a thermal

−3

xed by

= ε

, so no

single value is derivable. Third, the genuine parameter is a rigidity scale. The binding

curvature must exceed the thermal scale by a derived factor

≳ 36

, with a quantum oor on

the node inertia, so that the forced lattice resists its own zero-point and thermal motion.

A single-scale vacuum melts; a two-scale vacuum crystallizes.

Clock, ruler, and the assembled whole.

Time is the Schrödinger parameter and

length is

; their ratio sets the lattice signal scale, identied here with the emergent

light-cone velocity in the single-excitation sector. The journey from a single triangle

to the rigid

K=12

vacuum is one coupled dynamical framework, coherent, collective,

elastic, and dissipative sectors of a single open-system Hamiltonian, in real time and

space, with

K=12

forced, the lift rate free, and the rigidity scale derived. A single

end-to-end rule carries a bare Bell pair to a bulk

K=12

crystallite generically across

temperature, so the assembly is robust instead of ne-tuned. The lift-seeded branching

of sheets then makes the foam-to-crystal conversion exponential and self-terminating, an

ination-shaped geometric conversion.

The frontier.

The assembled lattice thus carries several of the features we associate

with physical space: spatial isotropy, a causal signal cone, and a transient homogeneous

curvature carried by the two-sheet foam. What it does not carry is the substrate itself.

The construction is positional throughout, presupposing node coordinates and a metric

exclusion, so the emergence of that positional substrate, from which a rst-principles

metric would follow, remains the frontier.

Data Availability Statement

The kinematic operators, the codimension suppression, and the compound-QEC argument

are specied in Sections 2.12.4; the Markov formulation, its acceptance rule, and the enu-

meration protocol are specied in Sections 33.6. The nite-state results are exhaustive

enumerations rather than samples, so they carry no random seed and any equivalent im-

plementation reproduces them exactly. Reference Python implementations are provided

as a verication aid.



finite_ssm_verification_v3.py

 enumerates the reachable stitchlift com-

plexes of Section 3.1, deduplicates states by colored incidence-graph isomorphism,

builds the full MetropolisHastings transition matrix, and veries detailed balance,

stationarity, and the decit-sector counts of Section 3.6:

https://github.com/rag

hu91302/ssmtheory/blob/main/finite_ssm_verification_v3.py



run_finite_verification_v3.py

 driver for the above, sweeping

βε

and the

forward lift proposal weight

to produce the concentration and

-independence

results of Section 3.6:

https://github.com/raghu91302/ssmtheory/blob/main/

run_finite_verification_v3.py



embedded_crossover_v1.py

 exhaustive enumeration of embedded congura-

tions for

n ≤ 12

, giving the construction-bond and contact-bond maxima and their

crossover in Section 3.7:

https://github.com/raghu91302/ssmtheory/blob/mai

n/embedded_crossover_v1.py



verify_ssm_integrated.py

 numerical verication of the analytic constants of

both parts. For Part I it checks the bond-set isotropy

µν

= 4δ

µν

, the vanishing rank-

three tensor, and the bond bookkeeping. For Part II it checks the LiebRobinson

front, the acoustic branch speeds, and the scalar-dispersion anisotropy. It also checks

the elastic constants and the isotropy no-go, the Lindemann constants, the disso-

lution probabilities, the FCC/HCP shell degeneracy, and the construction-velocity

anisotropy:

https://github.com/raghu91302/ssmtheory/blob/main/verify_s

sm_integrated.py

The history-resolved enumeration through

N = 8

runs in about ten seconds and the

embedded enumeration through

n = 11

in under two minutes on a current laptop; the

n = 12

level requires roughly twenty minutes and several gigabytes. Scripts for the growth

simulation of Ref. [22], which this paper cites but does not repeat, are listed in that paper's

data statement. The remaining results of Part II are analytic and are reproducible directly

from the equations as stated.

Declarations

Conict of interest:

The author declares that they have no conict of interest.

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