Emergent Face-Centered Cubic Vacuum from Discrete Entanglement Networks Forced K=12 Saturation and an Emergent Light Cone

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 macroscopic geometry follows if the vacuum is a discrete entangle-

ment network rather than a continuum, and show that a close-packed (

K=12

, face-

centered-cubic) structure, an emergent light cone, and approximate Lorentz invari-

ance arise from the network's own growth and dynamics instead of being imposed.

The treatment has two parts.

Part I (kinematic construction; consolidates and extends the growth simulation in-

troduced in Ref. [21]).

Local probabilistic assembly rules for discrete space tend to

jam into porous, geometrically frustrated graphs rather than saturated volumetric

bulks [1]. We show computationally that a two-operator growth lawa planar stitch

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

e

−3

(motivated kinematically by

the codimension dierence between the operators, and obtained in Part II from a

transition-state ansatza

3ε

exposed-bond barrieras the thermal factor of the

lift rate, with a free geometric prefactor)drives a discrete entanglement network

through the cascade

K=1 → 6 → 4 → 12

, past the frustrated

K=4

tetrahedral

foam (Regge decit

δ ≈ 7.36

◦

), and crystallizes a polycrystalline face-centered-cubic

(FCC) lattice at the Kepler kissing bound [3]. A strict bulk-interior diagnostic

(

≥ 42

neighbors within

2L

) gives modal coordination exactly

K=12

at every system

size from

N=250

updirect, non-extrapolated evidence of a close-packed, predom-

inantly twelve-coordinated bulk consistent with FCC crystalliteswith a leading-

order surface-to-volume trend

f ≈ 1−αN

−1/3

(eective amplitude

α = 6.8±0.6

over

the measured range

N=250



1000

) and a wide exclusion-radius plateau conrming

the operating values are not ne-tuned. An ideal (defect-free, periodic) bulk patch

of the emergent edge lattice carries an explicit

[[192, 130, 3]]

CSS code at

67.7%

rate [20].

Part II (dynamical foundations).

We develop the open-system dynamics beneath

the growth: a single bond Hamiltonian whose dissipative limit assembles the lat-

tice at the detailed-balance rate

q = 1/(1 + e)

at the operating point

k

B

T

0

= ε

,

and whose coherent limit propagates a ballistic signal cone with front speed

c =

4

√

2 v

lat

. We prove

K=12

is a

forced

terminus (largest cuboctahedral vacancy at

R

∗

=

p

2 −

√

2 L

), factorize the lift rate into a xed thermal factor

e

−3

and a free ge-

ometric amplitude, and derive a rigidity scale

κL

2

≳ 36 ε

that a single-scale vacuum

cannot meet. Requiring the light cone to coincide with the elastic trace speedthe

closest lattice statement of the observed equality of gravitational-wave and photon

1

speedsxes the stiness,

κ = 16 mε

2

/ℏ

2

, and with rigidity oors the binding scale

at

ε ≳ 2.3 ℏ

2

/(mL

2

)

. A conditional Regge-curvature estimate of the scalar spectral

tilt closes the paper.

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]. A persistent obstacle is geometric frustration.

When discrete nodes are assembled via purely randomized three-dimensional probabilis-

tic 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 coordination 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 structural saturation. In three dimensions, optimal spatial saturation is for-

mally 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 with-

out imposing a global background coordinate system remains a signicant computational

challenge. In this paper, we explore a resolution to this geometric bottleneck by altering

the dimensional probabilities of the generative kinematics. Motivated by holographic mod-

els of boundary-volume correspondence [4, 5], we hypothesize that saturated 3D volumes

can be deterministically generated if the underlying assembly process is overwhelmingly

two-dimensional. We introduce a discrete simulation based on the Selection-Stitch Model

(SSM), seeded by a Bell-pair triangle (three unit-distance bonds joining three nodes, the

minimal closed simplicial unit) and grown through two primary 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 as

P

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 4); Part II derives this value as the thermal factor of the lift

rate, the geometric prefactor remaining free (Section 16). 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 [7] and crystallizing into a polycrys-

talline FCC lattice that saturates the Kepler bound. We computationally verify that this

kinematic regime naturally drives the network through the cascade, bypassing geometric

frustration to yield a highly ordered, polycrystalline FCC geometry.

Relation to Ref. [21].

The growth simulation at the center of Part I was introduced in

Ref. [21], where it served as the supporting substrate for that paper's primary subject

the identication of matter with defects trapped in the close-packed vacuumand was

2

accordingly compressed. The simulation and its derivatives warrant a consolidated treat-

ment of their own, which this two-part paper provides. Part I therefore deliberately

reproduces, with expanded methodology, the kinematic denitions, the

e

−3

codimension

argument, the 30-seed dataset with its nite-size scaling, and the sensitivity sweeps of

Ref. [21], so that the dynamical derivations of Part II rest on a self-contained foundation

rather than a citation chain. All previously reported material is reused by the author with

attribution; Table 3 states exactly which columns are reproduced unchanged and which

are re-analyzed here (the revised bulk-interior columns supersede the published values).

The quantum error-correction reading of the lift amplitude (Sections 2.3 and 2.4) is new

to this paper, and Part II is entirely new.

Interactive 3D Visualization.

A 31-frame animated WebGL application

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

expansion to

e

−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



11

orange,

K <

10

red), matching the convention used in Figures 36. Controls: play/pause,

frame stepping, drag to orbit, scroll to zoom.

2 Methodology

The simulation algorithm is fully specied in this section; a reference implementation

is available at the URLs in the Data Availability statement. All simulations use

open

boundary conditions

: the lattice grows freely from the seed triangle without periodic

wrapping, xed walls, or any imposed simulation cell. The cluster terminates at a free

surface determined dynamically by the kinematics, and under-coordinated nodes therefore

reside at this free surfacea structural feature exploited in the surface-to-volume scaling

analysis of Section 3.1 and the bulk-interior diagnostic of Section 3.3. Statistical tables

(Tables 23) report means

±

standard deviations over 30 independent seeds. Illustrative

single-run gures use seed 42.

2.1 Generative operators and proximity bonding

The Bell-pair primitive and the triangular seed.

We treat the unit Bell-pair

bond(two nodes joined at distance

L

, the discrete structural analog of a maximally entan-

gled pair)as the primitive entanglement element of the network. The simulation begins

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

L

; 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: the simulation is a classical graph 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.

3

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 =

p

2/3 L

. This height is not a free

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

L

the only

distance at which all four inter-nodal separations equal

L

. 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

6(

in-plane

) + 3(

above

) + 3(

below

) = K = 12

cuboctahedral coordination of the FCC lattice (Figure 4).

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

is

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 [7]

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 [8, 9] 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 (Figure 4).

Tolerance window from the Regge decit.

The exclusion radius

R

ex

= 0.95 L

and

4

proximity bond radius

R

b

= 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

5%

is of

this natural size (slightly tighter than the full jitter), and the wide plateau of Section 3.8

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

K = 12

saturation

across the entire band

R

ex

∈ [0.58, 0.99] L

(Section 3.8, Figure 8) further conrms that

the specic value

R

ex

= 0.95 L

is not a tuned parameter.

2.2 Codimension suppression and the

e

−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

P

lift

/P

stitch

is exponentially suppressed in the codimension

dierence:

P

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 16).

What the lift rate controls is the growth

morphology

: smaller

P

lift

produces at, poorly

interlocked sheet structures, while larger

P

lift

produces foams with poor crystalline order

(Table 2). We treat equation (2) as a structural/kinematic motivation rather than a

derivation from a Euclidean tunneling action; the simulation's value of

p

lift

= 0.05

is

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

Section 3.1.

2.3 Quantum error-correction interpretation of

e

−3

The codimension-suppressed amplitude (2) receives a second, independent motivation

from the quantum error-correcting code that the emergent 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 [10, 20], with 192 phys-

ical qubits (edges), 130 logical qubits, encoding rate

k/n = 67.7%

, and code distance

d = 3

. The X-stabilizers act on octahedral voids (each touching the 12 edges of one oc-

tahedral cell) and the Z-stabilizers act on vertices (each touching the 12 incident edges);

both families have uniform weight 12. 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. A node participating in

t

triangular

stabilizer checks is protected by

t

independent error-detection circuits. Below the correc-

tion threshold, the error suppression scales exponentially with the decit. This motivates

the survival ansatz (a heuristic, not a consequence of code distance:

d = 3

guarantees

correction of arbitrary weight-one errors, and no decoding model, noise channel, syndrome

5

dynamics, or threshold calculation is claimed to produce Eq. (3)):

P (

survive

|t) =

(

1

if

t ≥ d + 1

e

−(d+1−t)

if

t < d + 1

(3)

The exponential form is qualitatively consistent with threshold behavior in topological

codes [11], 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 specic unit-coupling

exponential

e

−(d+1−t)

is the minimal one-parameter form that (i) saturates to unity at the

protection threshold

t = d+1

, (ii) decays monotonically with the decit

d+ 1−t

, and (iii)

recovers the textbook below-threshold scaling

P ∝ e

−∆

in the large-decit limit, while

placing the structural prediction

P

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 struc-

turally robust to the specic functional form, depending only on the monotonic 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

:

P

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 4). 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

t

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-

6

ary. A node

n

hops into the sheet interior accumulates compound detection paths from all

triangles within

n

-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 Summary of kinematic parameters

Every variable in the simulation is derived 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 is derived in

Part II as a thermal factor with a free geometric prefactor), and the proximity-bond

and exclusion radii together form the symmetric

±5%

Regge-decit tolerance window

derived in Section 2.1. Table 1 provides the complete kinematic parameter space. The

robustness of the operating values is conrmed by the exclusion-radius sensitivity analysis

(Section 3.8, Figure 8):

K = 12

is maintained across the entire band

R

ex

∈ [0.58, 0.99] L

,

far wider than any plausible tuning window, so the specic value

R

ex

= 0.95 L

is not a

ne-tuned parameter.

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

termined; the proximity bond and exclusion radii together form the symmetric

±5%

Regge-decit tolerance window of Section 2.1.

Parameter Value Derivation

Unitary Metric (

L

) 1.0 Invariant relational distance

Lateral Height

√

3

2

L ≈ 0.866 L

Equilateral triangle altitude

Lift Height

q

2

3

L ≈ 0.816 L

Regular tetrahedron altitude

Lift Probability

e

−3

≈ 4.98%

Codimension suppression /

QEC (Sections 2.22.3)

Proximity Bond (

R

b

)

1.05 L

Regge decit (

δ ≈ 7.36

◦

,

Eq. (1))

Hard Shell (

R

ex

)

0.95 L

Regge decit (Section 2.1);

plateau veried in Section 3.8

Observed kinetic cuto

1

√

3

L ≈ 0.577 L

Circumradius of unit trian-

gle; operator-kinetic wall (Sec-

tion 3.8)

3 Results: Kinematic Saturation and FCC Registry

3.1 Morphological dependence on lift probability

We conducted a parameter sweep across the probability of the 3D Lift operator. As shown

in Table 2 and Figure 1, all entries are means

±

standard deviations over 30 independent

7

random seeds.

Table 2: Lattice saturation, volumetric yield, and cluster shape as a function of lift prob-

ability (

N = 1000

, 30 seeds).

Φ = f

K=12

×(z

ext

/xy

ext

)

penalizes both poor crystallization

and failure to percolate into a 3D bulk.

Regime

K=12

(%) Layers Aspect

z/xy Φ

1% Lift

24.5 ± 5.8 13.4 ± 2.1 0.59 ± 0.12 0.145 ± 0.045

3% Lift

27.3 ± 6.1 15.4 ± 3.2 0.76 ± 0.16 0.207 ± 0.064

5% Lift (

e

−3

)

25.4 ± 5.4 19.2 ± 6.7 0.83 ± 0.14 0.211 ± 0.057

10% Lift

20.7 ± 6.2 25.2 ± 9.8 0.84 ± 0.10 0.174 ± 0.056

15% Lift

18.0 ± 4.5 33.3 ± 9.3 0.90 ± 0.11 0.162 ± 0.045

30% Lift

7.9 ± 3.1 48.3 ±5.7 0.92 ± 0.10 0.073 ± 0.030

50% Lift

1.5 ± 0.7 59.4 ±4.2 0.96 ± 0.09 0.014 ± 0.007

85% Lift

1.5 ± 0.4 64.8 ±4.6 0.99 ± 0.11 0.015 ± 0.004

Raw

K=12

percentage alone does not capture the quality of the 3D structure. At 1%

lift, the lattice achieves

24.5 ±5.8% K=12

(comparable to 5%) but only 13 layers with an

aspect ratio

z/xy = 0.59

: the structure is a at pancake that has failed to percolate into a

volumetric bulk. At 85% lift, the structure is nearly spherical (

z/xy = 0.99

) with 65 lay-

ers, but poorly crystallized (

1.5% K=12

). The volumetric yield

Φ = f

K=12

× (z

ext

/xy

ext

)

captures both requirements: high crystallization

and

3D volumetric extent. The prod-

uct (rather than sum) form is essential.

Φ

vanishes whenever

either

factor vanishes: a

perfectly crystallized but at sheet (high

f

K=12

, aspect

→ 0

, the low-lift regime) and a

spherical but uncrystallized foam (aspect

→ 1

, low

f

K=12

, the high-lift regime) are both

correctly identied as failures. A sum

f

K=12

+(z

ext

/xy

ext

)

would reward either achievement

independently and could rank a at sheet of perfect crystallinity above a moderately crys-

tallized 3D bulk, missing the physical point that both criteria are required simultaneously.

The multiplicative form is the simplest functional that vanishes whenever either require-

ment fails and grows monotonically when both improve, making it a natural composite

gure of merit for crystallized 3D bulk. The peak in

Φ

falls in the band

p ∈ [3%, 5%]

(with

Φ = 0.207 ± 0.064

at 3% and

0.211 ± 0.057

at 5%, statistically indistinguishable

within the 30-seed error bars), and the codimension-suppressed value

e

−3

≈ 4.98%

sits

squarely inside this optimal band.

Φ

degrades clearly below this band (

0.145 ± 0.045

at

1%, dominated by at-pancake structure) and above it (

0.073 ±0.030

at 30%, dominated

by poor crystallization), establishing

e

−3

as a value consistent with the structural opti-

mum: the lift rate at which the system simultaneously achieves crystalline order and 3D

bulk percolation. With only a coarse grid of tested lift rates, the data locate the optimum

only within a broad band of roughly

2



7%

; a denser sweep with condence intervals on the

optimum's position, alternative objective functions, and larger

N

is left to future work.

3.2 Finite-size scaling and the thermodynamic limit

The simulation was executed at four system sizes (

N = 250



1000

) with identical parame-

ters, each repeated over 30 independent random seeds. All results in Table 3 report means

±

standard deviations.

8

0 10 20 30 40 50 60 70 80 90

Lift Probability (%)

0

10

20

30

40

50

Fraction of Nodes (%)

(a) Lattice saturation (N=1000, 30 seeds)

e

3

5.0%

K = 12 (%)

K 10 (%)

0 10 20 30 40 50 60 70 80 90

Lift Probability (%)

0.00

0.05

0.10

0.15

0.20

0.25

Volumetric Yield

(b) Volumetric yield peaks at

e

3

e

3

(Vol. Yield)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Aspect ratio

z

/

xy

Aspect

z

/

xy

Figure 1: (a)

K=12

saturation vs lift probability (30 seeds,

1σ

error bars). Raw

K=12%

is similar for

p ≤ 5%

. (b) Volumetric yield

Φ

(green diamonds) reaches its optimal band

at

p ∈ [3%, 5%]

encompassing

e

−3

; aspect ratio

z/xy

(purple triangles) reveals that low-

lift structures are at pancakes.

At leading order the data follow the surface-to-volume form [21]

f(K=12) = 1 −

α

N

1/3

, α = 6.8 ± 0.6

(5)

as shown in Figure 2. This functional form has a direct geometric interpretation: under-

coordinated nodes (

K < 12

) reside exclusively at the cluster boundary, whose node count

scales as

N

2/3

. The bulk interior, scaling as

N

, achieves full

K = 12

saturation. The

under-coordinated fraction therefore scales as

N

2/3

/N = N

−1/3

.

Equation (5) is a leading-order description over a factor of four in

N

rather than a con-

verged asymptotic law: the eective amplitude

α

eff

(N) = (1−f) N

1/3

rises monotonically

from

5.8

at

N=250

to

7.5

at

N=1000

, and the one-term form would give

f < 0

below

N ≈ α

3

≈ 310

; a subleading

N

−2/3

(edge) correction brings the t within the quoted

errors. We therefore quote

α = 6.8 ± 0.6

as the mean eective amplitude across the

measured range; larger systems (

N

up to

∼ 1.6 × 10

4

) would materially strengthen the

extrapolation and are the natural next step. Extrapolating the leading-order form gives

f → 1

in the thermodynamic limit (

N → ∞

), consistent with complete FCC saturation

as the asymptotic ground state of the kinematics. Power-law ts over so narrow a range

are more easily believed when the functional form is geometrically motivated, as it is

here, but the central evidence that the bulk is close packed at modal coordination

K=12

comes not from extrapolation but from the bulk-interior diagnostic introduced next. The

last column of the all-node summary in Table 3 shows the macroscopic shape isotropy

λ

min

/λ

max

of the spatial covariance matrix, rising steadily toward 1.0 with increasing sys-

tem sizeconsistent with the expectation that larger polycrystalline clusters average over

more grain orientations.

9

Table 3: Coordination statistics across system sizes (

p

lift

= 0.05

,

R

ex

= 0.95 L

, 30 seeds

each). All columns are computed from the same simulation runs: the bulk-interior

columns are obtained by re-analyzing the lattice output under the criterion of Sec-

tion 3.3a node is classied as bulk-interior i at least 42 other nodes lie within radius

2L

(the count of an ideal FCC interior node's neighbors in its rst three coordination

shells,

12 + 6 + 24

, at distances

L

,

√

2 L

, and

√

3 L

). The threshold 42 is geometrically

derived from FCC crystallography, not tuned. The bulk-restricted columns supersede the

corresponding columns of Table 2 in Ref. [21]: the all-node columns are identical (com-

puted from the same 30-seed simulation output), while the bulk-interior analysis script

was revised after that publication.

N

¯

K K=12

(all, %)

λ

min

/λ

max

K=12

(bulk, %) Bulk modal

K

250

7.41 ± 0.33 8.2 ± 3.7 0.39 ± 0.17 49.3 ± 13.0

12

500

8.20 ± 0.28 15.9 ± 4.7 0.46 ± 0.17 56.0 ± 11.1

12

750

8.64 ± 0.26 21.8 ± 5.1 0.51 ± 0.16 61.7 ± 8.9

12

1,000

8.90 ± 0.26 25.4 ± 5.4 0.54 ± 0.15 64.2 ± 8.8

12

3.3 Bulk-interior diagnostic: direct evidence of close-packed bulk

The all-node

K = 12

percentages in Table 3 are mixed measurements: they include free-

surface nodes that are intrinsically under-coordinated and that have no physical analog

in the cosmological setting, where the observable universe sits in a bulk regime far from

any boundary. The physically relevant question is not what fraction of

all

nodes have

reached

K = 12

, but whether the

bulk interior

has done so.

We extract this directly via a strict geometric criterion: a node is bulk-interior i at least

42 other nodes lie within radius

2L

of it, the population of an ideal FCC interior node's rst

three coordination shells. This is a population criterion, not a crystallographic classier:

it matches the shell count of ideal FCC but does not by itself verify shell radii, angular

registry, or ABC (rather than ABAB) stacking, and coordination

K=12

is shared by FCC

and HCP. The results below should accordingly be read as establishing a close-packed,

predominantly twelve-coordinated bulk; the crystallographic FCC assignment rests on the

interlocking construction and awaits conrmation by a standard classier. The threshold

42 is not tuned: it is exactly

12+6+24

, the count of an ideal FCC interior node's neighbors

in its rst three coordination shells (at distances

L

,

√

2 L

, and

√

3 L

, respectively). This

is a strict crystallographic criterion derived from FCC geometry; it excludes any node

whose surroundings are not yet a complete bulk environment.

Re-analyzing the same 30-seed simulation output under this criterion gives the rightmost

two columns of Table 3. Three observations:

1. The bulk modal

K

is exactly 12 at every system size from

N = 250

upward: the

close-packed character of the bulk is a direct measurement, not an extrapolation.

The crystallographic FCC label awaits the classier follow-up (Section 3.3).

2. The bulk-restricted

K = 12

fraction (e.g.,

64.2±8.8%

at

N = 1000

) is roughly

2.5×

the all-node value (e.g.,

25.4 ± 5.4%

), consistent with the surface-volume interpre-

tation of the scaling law: the under-coordinated population that drags the all-node

mean down is concentrated at the boundary.

10

200 400 600 800 1000 1200 1400

N

20

10

0

10

20

30

40

50

K=12 (%)

(a) Scaling: = 6.8 ± 0.6

f

= 1 6.8/

N

1/3

K=12 (%)

300 400 500 600 700 800 900 1000

N

5

6

7

8

9

10

11

12

13

K

(b)

K

Kepler bound

K=12

K=6

K

Figure 2: Finite-size scaling (30 seeds,

1σ

error bars, shaded band). (a)

K=12

fraction vs

system size with the leading-order form

f = 1 −6.8/N

1/3

. (b)

¯

K

approaching the Kepler

bound.

3. The bulk standard deviation is

σ

K

≈ 0.99

at

N = 1000

with bulk mean

¯

K

bulk

≈

11.40

. The bulk-interior nodes that fall short of

K = 12

are concentrated at

K =

10

,

11

rather than spread broadlya ngerprint of grain-boundary nodes between

FCC crystallites of dierent orientations, not of a coordination distribution centered

below 12. This is consistent with the polycrystalline character of the emergent

lattice.

The all-node nite-size scaling and the bulk-interior modal-

K

measurement therefore

provide independent and complementary evidence: the rst establishes that under-

coordinated nodes are conned to the surface (where they decay as

N

−1/3

), the second

establishes that the bulk itself is close packed with modal coordination exactly

K=12

at

every system size measured (the crystallographic FCC label awaiting the classier follow-

up above). Neither relies on extrapolation to

N → ∞

.

3.4 Three-dimensional lattice structure

Figure 3 shows the emergent

N = 3000

cluster. The

K = 12

saturated core (green) is

surrounded by a thin under-coordinated shell (red), consistent with the surface-to-volume

scaling law. The central hexagonal layer (

z ≈ 0

) displays the emergent hex bond topology

with perfect structural planarity.

Figure 4 illustrates the cuboctahedral coordination shell: panel (a) shows the ABC stack-

ing of three hexagonal sheets at

z = 0

,

z = +h

,

z = −h

with the focal node and its

6 + 3 + 3 = 12

neighbors, and panel (b) shows the same neighbor set as the cuboctahe-

dron polyhedron with its 8 triangular and 6 square faces.

11

10.0

7.5

5.0

2.5

0.0

2.5

5.0

7.5

10.0

x

10.0

7.5

5.0

2.5

0.0

2.5

5.0

7.5

10.0

y

10.0

7.5

5.0

2.5

0.0

2.5

5.0

7.5

10.0

z

(a) N=3000 cluster, color by K

7.5 5.0 2.5 0.0 2.5 5.0 7.5 10.0

x

7.5

5.0

2.5

0.0

2.5

5.0

7.5

y

(b) Central layer z 0 (257 nodes)

Figure 3: (a) Full

N = 3000

cluster colored by coordination: green (

K = 12

), orange

(

K = 10

11), red (

K < 10

). (b) Central hexagonal layer at

z ≈ 0

.

3.5 Layer planarity and FCC registry

A

z

-clustering analysis of the

N = 3000

lattice identies 28 well-dened planar layers

(Figure 5).

Exact structural atness.

Of the 23 layers containing

≥ 10

nodes, 22

exhibit

σ

z

< 10

−10

L

(Figure 5c). The stitch operator places each node at the exact

equilateral apex, yielding perfect planarity by construction.

Ideal FCC layer spacing.

The measured inter-layer spacing is

0.8165 ± 0.0011 L

, matching the ideal FCC value

p

2/3 L = 0.8165 L

to 0.01% (Figure 5b).

Surface shell thickness.

All

K < 12

nodes

are conned to a boundary shell of constant thickness

t

shell

≈ 1.6 L

, corresponding to

approximately 2 FCC layer spacings.

3.6 Coordination distribution

Figure 6 shows the coordination distribution at

N = 3000

. The peak at

K = 12

contains

1,141 nodes (38.0%). Under-coordinated nodes reside exclusively at the cluster surface.

3.7 Inter-layer bonding: welds vs. rivets

The 5% lift events act as topological rivets, but proximity bonding ensures rapid percola-

tion. At

N = 3000

, 99.0% of layer nodes possess inter-layer bonds, averaging 4.9 per node

(Figure 7). This far exceeds the

∼ 20%

threshold for rigidity percolation. The layers are

structurally welded, approaching the theoretical maximum of 6 inter-layer bonds for an

ideal

K = 12

site.

12

1.5

1.0

0.5

0.0

0.5

1.0

1.5

x (L)

1.5

1.0

0.5

0.0

0.5

1.0

1.5

y (L)

0.8

0.6

0.4

0.2

0.0

0.2

0.4

0.6

0.8

z (L)

Layer A (z = h)

Layer B (z = 0)

Layer C (z = +h)

(a) ABC stacking of 3 hexagonal sheets:

6 in-plane (Layer B) + 3 above (Layer C) + 3 below (Layer A) = 12 NN

Focal node (Layer B)

6 NN in plane (Layer B)

3 NN above (Layer C)

3 NN below (Layer A)

1.00

0.75

0.50

0.25

0.00

0.25

0.50

0.75

1.00

x (L)

0.75

0.50

0.25

0.00

0.25

0.50

0.75

y (L)

0.8

0.6

0.4

0.2

0.0

0.2

0.4

0.6

0.8

z (L)

(b) Cuboctahedral coordination shell:

8 triangular + 6 square faces

Triangular faces (8)

Square faces (6)

Focal node (interior)

Figure 4: The

K = 12

cuboctahedral coordination shell of an interior FCC bulk node.

(a) ABC stacking: the focal node sits in the central hexagonal sheet (Layer B,

z = 0

),

with 6 in-plane neighbors (green) plus 3 in the upper sheet (Layer C, blue,

z = +h

with

h =

p

2/3 L

) and 3 in the lower sheet (Layer A, orange,

z = −h

), reached by the rare

e

−3

Lift events. The

6+3+3 = 12

decomposition saturates the kissing-number bound. (b) The

same 12 vertices viewed as the cuboctahedron, with its 8 triangular faces (green) and 6

square faces (blue). The triangular faces are the closed gauge-invariant loops underlying

the compound-QEC argument of Section 2.4; the square faces, representing the absence

of a diagonal bond, are topologically distinct.

3.8 Sensitivity sweep and the

R

ex

= 1/

√

3

geometric cuto

To ensure

K

max

= 12

is not an artifact of a tuned exclusion radius, a ne-grain sweep

was conducted across

R

ex

(reference implementation:

plot_rex_sweep.py

in the Data

Availability statement). As shown in Figure 8, the maximum coordination of 12 is main-

tained across the entire band

R

ex

∈ [0.58, 0.99]

. The observed breakdown at

R

ex

≈ 0.58

corresponds to the circumradius of the equilateral triangle (

1/

√

3 ≈ 0.577 L

); below it the

exclusion is too weak and unit-radius packing violations appear (

K > 12

). This observed

wall is an

operator-kinetic

threshold, not the absolute geometric one: Part II proves that

a thirteenth-neighbor site exists geometrically for all

R

ex

< R

∗

=

p

2 −

√

2 L ≈ 0.765 L

(Section 15). In the intermediate band

R

ex

∈ (0.577, 0.765) L

the persistence of

K

max

= 12

is therefore kineticthe stitch and lift operators never propose the thirteenth-neighbor

site, which sits at unit distance from only one existing node. At 1.00 and above, strict

rigidity causes lattice freezing. The wide stability plateau conrms that the specic value

R

ex

= 0.95

is not ne-tuned.

13

0 5 10 15 20 25

Layer index

0

50

100

150

200

250

Nodes

(a) 28 layers

0.813 0.814 0.815 0.816 0.817 0.818 0.819 0.820

Spacing (L)

0.0

2.5

5.0

7.5

10.0

12.5

15.0

17.5

20.0

Count

(b) 0.8165 ± 0.0011 L

FCC = 0.8165

50 100 150 200 250

Layer pop.

10

14

10

12

10

10

10

8

10

6

10

4

10

2

z

(L)

(c) 22/23 exactly flat

10

10

Figure 5: Layer structure (

N = 3000

). (a) Nodes per layer. (b) Inter-layer spacings match

the FCC ideal to 0.01%. (c) 22 of 23 substantial layers have

σ

z

< 10

−10

L

: exact atness.

2 4 6 8 10 12

Coordination K

0

5

10

15

20

25

30

35

Fraction (%)

K=12: 1140

(38.0%)

Coordination distribution (N = 3000)

Figure 6: Coordination distribution (

N = 3000

). Green:

K = 12

(38.0%). Orange:

K = 10

11. Red:

K < 10

.

4 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 [10], 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 [20], are:

14

0 2 4 6 8

Inter-layer bonds per node

0

200

400

600

800

1000

1200

1400

Count

(a) Mean=4.9, 99% bonded

0 1 2 3 4 5 6 7

Inter-layer bonds

2

4

6

8

10

12

K

(b) K=12 requires ~6 IL bonds

Figure 7: Inter-layer bonding (

N = 3000

). (a) Distribution of inter-layer bonds per node;

modal value = 6 (FCC maximum). (b)

K = 12

requires

∼ 6

inter-layer bonds.

Physical qubits (

n

) 192 (edges of the FCC lattice)

Logical qubits (

k

) 130

Code distance (

d

) 3

Encoding rate (

k/n

) 67.7%

Vertices (

V

) 13 per coordination cluster

Edges (

E

) 36 per coordination cluster

Faces (

F

) 38 (

= 32

triangles

+6

squares)

The 13-node coordination cluster (the central node together with its 12 cuboctahedrally

arranged neighbors of Figure 4) 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 (visible

in Figure 4b), 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. 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

as

n → ∞

. 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 tetrahedral 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

d

.

15

0.5 0.6 0.7 0.8 0.9 1.0

Exclusion radius

R

ex

(units of

L

)

0

4

6

8

12

16

20

24

Maximum coordination

K

max

Unphysical

overlap

(

K >

12

)

Lattice

freezing

FCC saturation

(Kepler bound)

Geometric phase transition at the metric wall

R

ex

=

L/

p

3

K = 12 plateau

R

ex

=

L/

p

3

≈

0

.

577

L

K

max

(simulation)

Figure 8: Observed (operator-kinetic) breakdown at

R

ex

= L/

√

3

(

N = 500

, 30 seeds,

p

lift

= 0.05

). The maximum coordination

K

max

= 12

is maintained across the entire stabil-

ity plateau

R

ex

∈ [0.58, 0.99] L

. Below

L/

√

3 ≈ 0.577 L

(the equilateral-triangle circum-

radius), the exclusion is too weak and unit-radius-packing violations appear (

K

max

> 12

);

above

1.0 L

, strict rigidity causes lattice freezing. The absolute geometric threshold for

a thirteenth neighbor is

R

∗

≈ 0.765 L

(Part II); the plateau segment between

0.577 L

and

0.765 L

reects kinetic protection by the operator basis (see text). The wide plateau

conrms that the operating value

R

ex

= 0.95 L

is not ne-tuned.

16

5 Discussion

5.1 Exact rank-two spatial isotropy and conditions for an emer-

gent Lorentz limit

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 [12],

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

n

j

∈



(±1, ±1, 0), (±1, 0, ±1), (0, ±1, ±1)



/

√

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

(6)

Dene the rank-2 structure tensor

S

µν

≡

P

12

j=1

n

µ

j

n

ν

j

. By direct enumeration:

S

xx

= 4 ×

1

2

|{z}

(±1,±1,0)

+ 4 ×

1

2

|{z}

(±1,0,±1)

+ 0

|{z}

(0,±1,±1)

= 4,

(7)

S

xy

=

1

2



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



+ 0 + 0 = 0,

(8)

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

S

yy

= S

zz

= 4

and

S

xz

= S

yz

= 0

. Therefore

S

µν

= 4 δ

µν

(exact, by enumeration)

.

(9)

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

the scalar sector at leading order in

k

. The odd-rank tensor

T

µνλ

≡

P

j

n

µ

j

n

ν

j

n

λ

j

vanishes

exactly because the FCC bond set is centrosymmetric: every

n

j

has a partner

−n

j

, so all

odd-power sums cancel,

T

µνλ

= 0

(exact, by inversion symmetry)

.

(10)

Equation (10) forbids any preferred direction and any linear-in-

k

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)

2

= κ

P

12

j=1



1 − cos(k · n

j

a)



. At long wavelengths (

|k|a ≪ 1

),

expanding the cosine,

ω(k)

2

≈

κa

2

2

12

X

j=1

(k · n

j

)

2

=

κa

2

2

k

µ

k

ν

S

µν

= 2κa

2

|k|

2

,

(11)

so

ω = c

s

|k|

with

c

s

= a

p

2κ/m

, where

κ

is the bond stiness that xes rigidity in

Part II and

m

the node inertia. This scalar speed is the

trace

channel of the displacement

17

dynamics: the three acoustic branches of the full vector (phonon) problem satisfy

v

2

L

+

v

2

T

1

+ v

2

T

2

= c

2

s

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

c

s

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 (9),

not an approximation. Anisotropic corrections to the scalar dispersion enter at relative

order

(|k|a)

2

the

[111]



[100]

splitting of

ω

2

is

−(|k|a)

2

/72

at leading ordersuppressed

by

(E/M

P

)

2

for Planck-scale lattice spacing

a ∼ ℓ

P

.

Step 3: Emergence of Lorentz boosts.

The isotropy (9) guarantees equal propaga-

tion speed in every direction for any

scalar

excitation on the bond setthe geometric

prerequisite for emergent Lorentz invariance; vector (displacement) excitations are gov-

erned 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

c

is derived in Part II; the elastic speed

c

s

above is a

distinct, metric-perturbation mode, and a single light cone requires

c

s

= c

. That equal-

ity is not a tuning but an observed factgravitational-wave and photon speeds agree

to one part in

10

15

and Part II shows that imposing it at the level of the elastic trace

channel xes the stiness

κ

; the sharper branch-level statement remains an open target

(Part II, Section 23). With one cone, an isotropic linear dispersion gives rise to a rela-

tivistic eective eld theory in the standard continuum limit [12]. Lattice corrections at

the cuto scale

1/a

are suppressed by

(E/M

P

)

2

and lie far below all current experimental

bounds on Lorentz violation [13, 14], which constrain departures at the level of

10

−20



10

−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) re-

main Lorentz-violating until the branch-level cone closes, and protection of the invariance

against radiatively generated lower-dimension Lorentz-violating operators [12] is an open

requirement rather than a result obtained here. Residual

O

h

anisotropy is absent from the

scalar

sector at leading order by Eq. (9); 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 (9)

(11) 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 3 reports the macroscopic shape isotropy

λ

min

/λ

max

of the spatial co-

variance matrix

C

ij

= ⟨(x

i

−¯x

i

)(x

j

−¯x

j

)⟩

. 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 (9).

5.2 Informational driver of dimensional projection

The simulation is consistent with an information-theoretic interpretation of dimensional

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

K = 6

, yielding

3N

total bonds. A 3D FCC lattice reaches

K = 12

, yielding

6N

bonds.

18

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

room. By projecting into a third dimension via the

e

−3

codimension-suppression limit, the

system doubles its entanglement capacity. The causal chain is: minimal gauge-invariant

loops (triangles) [6] assemble into

K = 6

sheets

→

the 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 [7] drives the cascade upward

→

stacking

to

K = 12

saturates the Kepler bound [3]

→

the

[[192, 130, 3]]

CSS code provides the

error-correction framework whose distance

d = 3

is consistent with the same suppression

amplitude under the survival ansatz of Section 2.3.

6 Conclusion

We have veried computationally that the FCC vacuum emerges naturally from the

Selection-Stitch Model through the kinematic cascade

K = 1 → K = 6 → K = 4 →

K = 12

, with the principal phase transition

K = 4 → K = 12

resolving the geometric

frustration of the tetrahedral-foam intermediate. By enforcing a 2D-dominant growth

phasedriven by the

e

−3

-suppressed lift probability, motivated by codimension counting

and by QEC survival at code distance

d = 3

, and derived in Part II as a thermal factor

combined with proximity bonding, the discrete vacuum circumvents geometric frustration.

The principal results are: (1) a bulk-interior diagnostic (Section 3.3) showing that the bulk

modal coordination is exactly

K = 12

at every system size from

N = 250

upward, provid-

ing direct (non-extrapolated) evidence of close-packed, predominantly twelve-coordinated

bulk crystallization consistent with FCC crystallites (crystallographic conrmation via a

standard classier is a stated follow-up, Section 3.3), complemented by an all-node nite-

size scaling trend

f ≈ 1 −(6.8 ±0.6) N

−1/3

that connes under-coordinated nodes to the

cluster surface; (2) a volumetric yield analysis showing that the combined crystallization-

plus-bulk metric

Φ

is consistent with a maximum near

e

−3

(statistically indistinguish-

able from

p = 3%

given the 30-seed error bars, but clearly degraded at

p ≤ 1%

due

to at-pancake structures with aspect ratio

z/xy = 0.59

, and at

p ≥ 10%

due to poor

crystallization); (3) exact structural atness (

σ

z

< 10

−10

L

) of internal layers with spac-

ings matching FCC to 0.01%; (4) a sharp operator-kinetic coordination breakdown at

R

ex

= 1/

√

3 L

(the absolute geometric threshold lying at

R

∗

≈ 0.765 L

, Part II), with a

wide stability plateau across

[0.58, 0.99] L

conrming that operating values are not ne-

tuned; (5) a macroscopic shape isotropy

λ

min

/λ

max

rising from 0.39 at

N = 250

to 0.54

at

N = 1000

(30 seeds), conrming polycrystalline averaging that complements the ex-

act algebraic spatial isotropy of the FCC bond set (Section 5.1). The emergent lattice

supports a

[[192, 130, 3]]

CSS code with 67.7% encoding rate, whose code distance

d = 3

matches the exponent of the lift suppression under the survival ansatz of Section 2.3: the

diculty of creating 3D volume mirrors the strength of the code's error protection.

19

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

). A stitchlift growth simulation of Part I reproduces this

K=4 → K=12

transition, exhibiting

K=12

saturation at the Kepler bound, a sharp

operator-kinetic breakdown at the

L/

√

3

wall, nite-size scaling toward full saturation,

and an isotropic linear dispersion. The structural lattice carries an explicit quantum

error-correcting code, the

[[192, 130, 3]]

CSS code on the FCC edges. The contribution

of Part II is the dynamical law beneath that growth and the assembly it produces; the

cosmological material in Sections 19 and 20 is a consistency 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, but not the

dynamical

law

: 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 and assembles every element in one place. The organizing

result is that one open-system equation (Figure 9) 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. Along the

way three implicit features of the simulation are settled: 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

e

−3

xed by

T

0

= ε

(Section 16), and the parameter

the model requires is a rigidity scale, derived self-consistently from the phonon spectrum

(Section 17). Sections 2426 give the energy ladder, the code, and the FCC-over-HCP

selection. The stitchlift growth picture and the

[[192, 130, 3]]

code are developed in Part I;

the geometric maxima use the kissing-number [15] 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 (Sec-

tion 12), a constant Regge curvature that drives expansion, and a conditional Regge-limit

estimate for the primordial tilt that is numerically close to the observed value (Section 20).

In this sense Part II is a program for

emergent geometry

: the lattice does not sit in space,

it

is

the space, and its geometric, causal, and cosmological properties are consequences of

the assembly dynamics rather than inputs. The boundary of the program is that the dy-

namics 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.

20

Lindblad master equation

˙

ρ

=

−

i

[

H, ρ

] +

X

b

(

γ

↑

D

[

σ

+

b

] +

γ

↓

D

[

σ

−

b

])

ρ

coherent limit

bath off:

˙

ρ

=

−

i

[

H, ρ

]

unitary evolution

⇒

light cone,

v

= 2

`

lat

ε/

dissipative limit

thermal jumps at

T

0

=

ε/k

B

quantum-trajectory assembly

⇒

growth,

q

=

1

1 +

e

one dynamics, two limits

Figure 9: One open-system Hamiltonian, several sectors: Eq. (13) coupled to a thermal

bath at

T

0

= ε/k

B

. Coherent limit

→

ballistic signal front and the collective sectors of

Table 6; dissipative limit

→

constructive assembly with detailed-balance rate

q = 1/(1+e)

.

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

R

∗

=

p

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).

(C)

Consistent but separately grounded

: the light cone, the relation

c = 4

√

2 v

lat

, and the

rigidity bounds, which follow from the posited Hamiltonian and are mutually consistent

but are not independently tested here.

(C/S) Observational contact, conditional on stated

assumptions

: the exponential branching-growth phase (Section 19) and the spectral index

n

s

= 0.9646

(Section 20); the tilt's magnitude is parameter-free (the Regge decit) and its

coecient

√

3

is selected by the Regge area-weighting under a stated

O(1)

normalization

convention, numerically consistent with Planck, but this is an estimate within Regge

calculus on a presupposed metric, and identication with cosmic ination awaits a pre-

positional theory.

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 =

4

√

2 v

lat

; the detailed-balance derivation of

q = 1/(1 + e)

; the forced

K=12

terminus and

the exact threshold

R

∗

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

and a xed thermal factor

e

−3

; the rigidity scale

κ ≳ 36 ε/L

2

; the end-to-end single-

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

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

the Regge-limit estimate of the spectral tilt

n

s

= 0.9646

(Section 20). 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 [15, 3].

21

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

L

(Section 14, Eq. (22))and a fully pre-positional formulation, in which

L

itself emerges,

is left as the frontier (Section 27).

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

The ele-

mentary 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

L

that sets the unit of length. All rates and lengths in Part II are ex-

pressed 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

ε

, the

single energy parameter of the Hamiltonian (Section 10). 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.

The kinematics on which these premises actthe stitch and lift operators, the cascade

K=1 → 6 → 4 → 12

with its frustrated

K=4

foam and Regge decit

δ = 2π−5 arccos

1

3

≈

7.36

◦

(Eq. (1)), and the

K=12

saturation at the Kepler boundare established in Part I

and are treated here as the classical limit of the dynamics, not repeated.

9 Code-selected registration and the two-sheet foam

The dynamics below need two features of the close packing that the growth model of

Part I does not address: how the layer-by-layer arrival of sheets produces a transient,

spatially homogeneous curvature, and why the code selects the regular FCC registration

over its competitors. Both are developed here.

The two-sheet tetrahedral foam.

A point that matters for the curvature content

of Section 20 concerns

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. We verify this directly: stacking two at

K=6

sheets

at the close-packed oset and spacing

h =

p

2/3 L

, every interior node of the lower

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

coordination

K=9

, and the node together with the triangle above it forms a

regular

22

tetrahedron

(all six edges equal to

L

;

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 in addition to

the four-coordinated tetrahedral motif, so the two-sheet contact realizes the tetrahedral

foam as an extended, registered layer rather than a disordered bulk. The value of the

decit is xed, not incidental. A new sheet can be seeded only where the lifted apexes

project to

equidistant

points, since those apexes must themselves form the next

K=6

sheet; we verify that the apexes of the lifted up-triangles do form a regular triangular

lattice (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 relaxing to an irregular packing: the frustration is paid as curvature, not

absorbed by distorting the bond angles. That the code prefers this registration is not

assumed but computed: Table 4 compares three candidate stackings of the second sheet

the regular close packing, the eclipsed (AA) stacking, and a relaxed irregular packing

by the fraction of triangular plaquettes that are genuine cell faces, i.e. the fraction of

Z

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

p

2/3 L

an eclipsed second sheet is excluded outright by the hard core, aligned nodes

sitting at

0.8165 L < R

ex

; the eclipsed row is therefore meaningful only at unit vertical

spacing.)

FCC versus HCP: what selects the stacking.

Local

K=12

packing admits two

registrationscuboctahedral (FCC, ABC) and anticuboctahedral (HCP, ABAB)and

at nearest-neighbor level they are

exactly

degenerate: both coordination shells contain 24

unit-distance pairs and 8 unit triangles (direct enumeration), so the pair energy

and

any

three-body triangle term

H

3

= −J

3

P

△

n

ij

n

jk

n

ki

are blind to the dierence. Selection

must come from structure the shells do not share: (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

c

axis while FCC's share only edges, so

the octahedral

X

-stabilizers of the emergent code see inequivalent environments, making

the code term itself a candidate selection energy; or (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; pending it and the classier follow-up of Part I,

the honest statement is FCC-or-HCP close packing with a candidate selection mechanism.

registration forms tetrahedral cells at all (

32

versus none) and closes its plaquettes (

73%

faced versus

0%

); the alternatives are stabilizer-frustrated. Among the candidates com-

pared, 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 4 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 compar-

ison across the physical competitors, a structural proxy for code distance rather than a

full distance calculation; the margin is wide enough that the conclusion is insensitive to

the proxy.)

Five regular tetrahedra about a hinge subtend

5 arccos

1

3

= 352.64

◦

, leaving exactly the

Regge decit

δ = 2π − 5 arccos

1

3

= 7.36

◦

(Eq. (1)); a relaxed, irregular foam would

instead distribute a much larger and variable decit. Because the seeding-plus-code con-

straint enforces the

same

regular registration everywhere along the contact, the decit

23

Table 4: Comparative code metrics for three registrations of the second sheet onto the

rst (a

50

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

cellthe fraction of plaquette (

Z

) 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

Table 5: Claim ledger. The forced terminus, the dissolution ratio, and the robust assembly

are the secure pillars; the spectral tilt is a conditional estimate.

Claim Status Basis

K=12

forced for

R

∗

<

R

ex

< L

T cuboctahedral square-face vacancy

+ kissing bound

q = 1/(1 + e)

D detailed balance at

T

0

= ε/k

B

robust modal

K=12

as-

sembly

D end-to-end stochastic rule,

temperature-generic

e

−3

lift suppression C thermal factor at the operating

point

βε = 1

; geometric prefactor

free; QEC reading heuristic

κL

2

≳ 36 ε

rigidity C/D FCC harmonic phonon estimate

n

s

= 0.9646

C/S Regge dual-area transfer assump-

tion

is spatially homogeneous and takes this single xed value (Figure 10). The third sheet

converts

K=9 → 12

and the decit vanishes, the cuboctahedral shell tiling space without

frustration. 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).

(12)

This is the dynamical realization of the homogeneous decit that the curvature estimate

of Section 20 requires: the foam is not assumed, it is the two-sheet stage that the layer-

by-layer arrival necessarily produces.

Claim ledger.

Table 5 summarizes the principal claims and their status, so that the

hard geometric results can be separated at a glance from the conditional cosmological

estimate. Tiers:

(T)

theorem-level;

(D)

derived and numerically tested;

(C)

consistent

within the posited dynamics;

(S)

speculative.

24

δ

= 7

.

36

◦

Five regular tetrahedra about a hinge:

frame held rigid, frustration

=

δ

= 7

.

36

◦

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 10: Left: the xed value of the decit. Five

regular

tetrahedra about a shared

hinge subtend

5 arccos

1

3

= 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

δ

, and vanishes when the third sheet closes

the cuboctahedral shell (

K=9 → 12

).

10 The dynamical Hamiltonian and its scales

Let each potential bond

b

be a qubit,

|1⟩

b

entangled (energy

−ε

) or

|0⟩

b

failed (

0

),

n

b

=

|1⟩⟨1|

b

:

H = −ε

X

b

n

b

− Γ

X

⟨b,b

′

⟩

1

2

(X

b

X

b

′

+ Y

b

Y

b

′

) − ε

h

X

v

A

v

+

X

o

B

o

i

, Γ = ε.

(13)

The

binding

term rewards each entangled bond; the

kinetic

term lets entanglement hop

between bonds sharing a node and makes

H

generate motion; the

code

term is the CSS

stabilizer Hamiltonian (Section 25),

A

v

=

Q

b∋v

Z

b

,

B

o

=

Q

b∈o

X

b

. The bond-state Hamil-

tonian has a single binding depth

ε

(and we set the kinetic coupling

Γ = ε

); as Section 17

shows, the

positional

realization of the bond requires in addition a curvature scale

κ

,

distinct from the depth. The constants

ε

and

ℓ

lat

yield

τ

0

=

ℏ

ε

, v

lat

=

ℓ

lat

ε

ℏ

, T

0

=

ε

k

B

.

(14)

25

11 The open-system dynamics

An entangled bond decoheres against its environment, and that decoherence

is

bond

failure; the dynamics is Eq. (13) coupled to a thermal bath at

T

0

= ε/k

B

:

˙ρ = −

i

ℏ

[H, ρ] +

X

b



γ

↑

D[σ

+

b

] + γ

↓

D[σ

−

b

]



ρ, D[L]ρ = LρL

†

−

1

2

{L

†

L, ρ}.

(15)

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

T

0

= ε/k

B

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

ln 2

). This is 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 27).

12 Coherent limit: a single-excitation signal cone and

lattice isotropy

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

(line) graph of the lattice; a localized single excitation spreads ballistically (Figure 11). On

a one-dimensional bond chain the front velocity is

2ℓ

lat

ε/ℏ = 2v

lat

(veried as

2.01 ℓ

lat

ε/ℏ

on

401

bonds). On the FCC bond network itself the front is faster: for a

K

-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,

v

front

= 4

√

2 v

lat

≈

5.66 v

lat

, attained for

k

along

[100]

(the eight bonds with a

[100]

component contribute

|∇E|

max

= 4

√

2 Γℓ

lat

). Being a band-structure extremum, this front speed is direction-

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

wavelength scalar sector is isotropic (Section 5.1). We therefore give the bound its precise

name:

v

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

0

+ A|k|

2

+ O(k

4

)

, so the group velocity of long-wavelength

bond excitations vanishes as

k → 0

. The genuinely

linear

low-energy cone belongs to the

scalar collective sector: a lattice eld with canonical pair

(ϕ

i

, Π

i

)

and Hamiltonian

H

ϕ

=

1

2

X

i

Π

2

i

+

K

ϕ

2

X

⟨ij⟩

(ϕ

i

− ϕ

j

)

2

, ω

2

(k) =

K

ϕ

m

ϕ

X

j



1 − cos(k ·a

j

)



,

(16)

26

Table 6: The three propagation sectors of the lattice. One symbol

c

hides three inequiv-

alent speeds; this table keeps them separate.

Sector Excitation Dispersion Status

Bond hopping

H

hop

localized bond

excitation

quadratic at low

k

; bal-

listic front bounded by

v

sig

= 4

√

2 v

lat

, direction-

dependent

derived

Scalar collective

H

ϕ

emergent scalar

eld

linear and isotropic as

k → 0

, speed

c

ϕ

= c

s

identied here

(trace channel)

Elastic

H

u

displacement

modes

linear but polarization-

split,

v

L

= v

T

1

, v

T

2

derived; single cone

closed by no-go

(Section 23)

whose small-

k

dispersion is

ω

2

= c

2

ϕ

k

2

+ O(k

4

)

with

c

ϕ

= L

p

2K

ϕ

/m

ϕ

exactly the trace-

channel speed

c

s

of the displacement dynamics when

K

ϕ

= κ

,

m

ϕ

= m

. The honest

status of this sector: the close-packed geometry

supports

an additional scalar collective

eld with an isotropic linear continuum limit; deriving

ϕ

i

as a collective coordinate of

the bond qubits themselvesschematically

ϕ

i

∼

P

b∋i

f(n

b

)

, with

H

ϕ

obtained by coarse-

graining Eq. (13)is the dened target that would make the sector emergent rather than

adjoined. The three sectors and their status are collected in Table 6; the single-cone

problem is then stated exactly as the question of whether

c

ϕ

= v

T

= v

L

can be arranged,

answered in Section 23. 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 27). This is the dynamical counterpart of

the structure-tensor isotropy: the twelve bond vectors give

S

µν

=

P

j

n

µ

j

n

ν

j

= 4δ

µν

and

T

µνλ

= 0

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

anisotropy entering only at relative order

O((|k|ℓ

lat

)

2

)

(coecient

1/72

; Section 5.1). The

causal cone is the signal front itself:

c ≡ v

front

= 4

√

2 v

lat

= 4

√

2 ℓ

lat

ε/ℏ

, the fastest speed

at which any inuence propagates on the lattice; the clock is

τ

0

= ℏ/ε

and the ruler

ℓ

lat

.

This bond-excitation cone is the emergent light conedistinct from the elastic phonon

speed

c

s

= ℓ

lat

p

2κ/m

, which propagates metric perturbations rather than signals. A

single, shared cone is not optional: the measured equality of the gravitational-wave and

photon speeds to one part in

10

15

[19] requires the elastic and signal cones to coincide;

imposing this on the elastic trace channel,

c

s

= c

(see the mode-speeds paragraph below),

xes the otherwise-free stiness,

κ =

mc

2

2L

2

=

16 mε

2

ℏ

2

(

GW

=

light, trace channel

: c

s

= c).

(17)

This is sharpened by the rigidity requirement

κL

2

≳ 36 ε

of Section 17: the two together

impose a oor on the binding scale,

ε ≳ 2.3 ℏ

2

/(mL

2

)

the bond energy cannot lie below

the lattice kinetic scale

ℏ

2

/(mL

2

)

. 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.

27

200 150 100 50 0 50 100 150 200

position

x / `

lat

0

10

20

30

40

50

60

70

80

time

t /

(

/ε

)

v

= 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

10

|

ψ

|

2

Figure 11: Coherent limit: a bond excitation on a one-dimensional bond chain spreads

with front velocity

v = 2ℓ

lat

ε/ℏ

; on the full FCC bond network the corresponding Lieb

Robinson front is

4

√

2 ℓ

lat

ε/ℏ

(see text).

Mode speeds and the single cone.

The cone

c = 4

√

2 v

lat

used here is the max-

imum group velocity (LiebRobinson front) of the scalar bond excitation on the FCC

bond network. It is a distinct object from the elastic (metric-perturbation) branch

c

s

= L

p

2κ/m

: the former is a band-structure maximum of the signal sector, the lat-

ter the long-wavelength sound speed of the displacement eld. One necessary geometric

ingredient of emergent

SO(3, 1)

invariancethe exact bond-set isotropy

S

µν

= 4δ

µν

and

centrosymmetry

T

µνλ

= 0

, which force a rotationally isotropic scalar kinetic term at lead-

ing orderwas established in Section 5.1, conditional on a common propagation speed;

it is an ingredient, not the invariance itself, and the role of Part II is to examine whether

that common speed can be supplied dynamically. The GW

=

light identication (Eq. (17))

forces the elastic

trace

speed to coincide with the signal front,

c

s

= c

. This is weaker than

the branch-level statement the observation constrains: the three elastic polarizations sat-

isfy

v

2

L

+ v

2

T

1

+ v

2

T

2

= c

2

s

in every direction (Section 23), so xing

c

s

= c

places the branches

at

c

only in quadrature mean, the individual polarizations remaining sub-luminal and split

(

v

L

∈ [0.71, 0.82] c

,

v

T

∈ [0.35, 0.50] c

). Closing the gap to a genuine single cone cannot be

done within the elastic sector at all (the no-go results of Section 23): three-body angular

terms can restore isotropy but not speed equality, which requires a distinct excitation sec-

tor. What is not derived is the stronger, fully pre-positional statement that every sector

and polarization shares this cone without the GW

=

light input; we take

c ≡ 4

√

2 v

lat

as

the causal cone of the signal sector and note 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

ε

below the failed state, so detailed balance requires

γ

↑

/γ

↓

=

e

ε/k

B

T

0

; in general

q(β) = 1/(1 + e

βε

)

or

q = [1 + (g

1

/g

0

) e

βε

]

−1

with bound/unbound

degeneracies

g

1

, g

0

, taken equal here; a self-consistent derivation of the bath temperature

from reservoir thermodynamics,

T

−1

= ∂S

foam

/∂E

foam

, is the open route to replacing

the operating point with an equilibrium conditionand at the operating point

βε = 1

a

stated identication of the bath temperature with the bond scale, not a derived equality

28

the single-bond rate equation has stationary failure probability

q =

γ

↓

γ

↑

+ γ

↓

=

1

1 + e

≈ 0.2689,

(18)

with no free parameter beyond the chosen operating point

βε = 1

: the detailed-balance

ratio at

T

0

= ε/k

B

, where the only scale is the bond entropy

S = ln 2

. A node loses

rigidity when

K − 2

of its

K

bonds fail, so its dissolution probability is the binomial tail

P

dissolve

(K) =

K

X

j=K−2



K

j



q

j

(1−q)

K−j

= {0.61, 0.29, 0.048, 7.5×10

−5

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

(19)

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. (15),

not an input.

Geometry-resolved cross-check.

The rule behind Eq. (19)a node fails on losing

K − 2

of its

K

bondscan be tested against a criterion built from the actual bond

geometry: form the local stiness matrix

M

αβ

= κ

P

a∈B

ˆn

a,α

ˆn

a,β

over the surviving bond

set

B

and declare the node stable only if

M

has full rank (

λ

min

(M) > 0

: the survivors

span three dimensions). Exhaustive enumeration over survival patterns at

q = 1/(1 + e)

gives: for the tetrahedral

K=4

node the two criteria coincide

exactly

(

P = 0.294

; any

three of the four tetrahedral directions span

R

3

, so rank failure is precisely the loss of

K − 2

bonds); for the

K=12

cuboctahedral node the rank criterion gives

1.6 × 10

−3

against the binomial

7.5 × 10

−5

the dierence being the 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 negligible against the foam's

0.29

; and

the planar

K=6

sheet node is rank-decient out of plane for

every

survival pattern, which

quanties why sheets are metastable waypoints of the cascade rather than termini: two-

dimensionally rigid (in-plane rank-2 failure at

1.5 ×10

−2

), three-dimensionally soft. The

stability hierarchy driving the cascade is robust to the choice of criterion, and the simple

K − 2

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

test (

λ

min

> 0

) classies arbitrarily oppy congurations as stable; a physical threshold

λ

min

> λ

c

tied to the Lindemann windowsoft-mode uctuations

kT/λ

c

< (L−R

∗

)

2

give

λ

c

/κ = 1/(0.0551 × 36) ≈ 0.50

at

T

0



widens

the hierarchy rather than eroding it: at

λ

c

/κ = 0.50

the tetrahedral node dissolves at

P = 0.714

(it now requires all four bonds,

any three giving

λ

min

/κ =

1

3

< 0.50

) while the cuboctahedral node stays at

8.5×10

−3

, the

full-shell stinesses being

λ

min

/κ = 4

(

K=12

) against

4

3

(

K=4

). This joins the dissolution

rule and the rigidity bound of Section 17 into a single mechanical stability criterion.

14 The assembly as a quantum-trajectory representa-

tion

A quantum-trajectory unraveling of Eq. (15) replaces the density matrix by stochastic

paths with bond-creation (

σ

+

) and bond-destruction (

σ

−

) jumps; the Part I stitchlift

growth is one such path (Algorithm 1). Births are

σ

+

jumps placing nodes at unit

29

distance, accepted under exclusion; deaths are

σ

−

jumps dissolving a node with probability

∆t P

dissolve

(K)

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

τ

0

;

the core coordination then climbs and

holds

(Figure 12), 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. (15) 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

do

reproduce the rule can be written explicitly:

˙ρ = −

i

ℏ

[H, ρ] +

X

x∈S(C)

γ

s

(x) D[J

s

(x)]ρ +

X

x∈L(C)

γ

ℓ

(x) D[J

ℓ

(x)]ρ +

X

b

γ

−

D[σ

−

b

]ρ,

(20)

with composite birth jumps

J

s

(x) = P

adm

(x) a

†

x

σ

+

xi

σ

+

xj

, J

ℓ

(x) = P

adm

(x) a

†

x

σ

+

xi

σ

+

xj

σ

+

xk

,

(21)

where

a

†

x

creates the node degree of freedom at

x

,

P

adm

(x)

projects onto congurations

satisfying the hard-core and proximity constraints, and

S(C)

,

L(C)

are the conguration-

dependent sets of admissible stitch and lift sites of the current conguration

C

. Because

a

†

x

changes the particle number and the admissible jump set depends on

C

, Eq. (20) is

not an ordinary xed-dimensional Lindblad equation but a

hybrid Fock-space quantum

Markov process

: the state space is

H =

L

∞

N=0

H

N

over

N

-node embedded congurations,

a

†

x

: H

N

→ H

N+1

, and

P

adm

encodes classical geometric conditions on the embedding.

Equation (20) 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

s

, γ

ℓ

from a microscopic systembath cou-

pling (below they are xed by detailed balance on the bonds each jump creates). The map-

ping deserves a precise statement, because the Lindblad jumps act on

bond

states while

the moves are

node

placements. A

σ

+

b

jump sets a bond

b

from

|0⟩

to

|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

x

is the joint jump

Q

i∈A(x)

σ

+

(x,i)

over the anchor set

A(x)

(

|A| = 2

stitch,

3

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 re-

verse, dissolution channel through detailed balance); a

σ

−

b

jump lowers a bond, and a node

dissolves when its surviving bonds fall below rigidity, reproducing

P

dissolve

(K)

of Eq. (19).

The continuous bond-ip rates

γ

↑

, γ

↓

thus generate the discrete moves through the

ad-

missible anchor sets

: the geometry enters by restricting which coincident

σ

+

patterns

correspond to a unit-distance registered site.

30

Algorithm 1

Quantum-trajectory assembly (a trajectory representation of Eq. (15); see

text for its status)

1:

seed a Bell pair / triangle nucleus (edge

ℓ

lat

)

2:

for

each step

∆t

(units of

τ

0

)

do

3:

births (

σ

+

):

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

p

2/3 ℓ

lat

); reject on exclusion

R

ex

∈ (R

∗

, ℓ

lat

)

; bond within

R

b

4:

deaths (

σ

−

):

dissolve each node with prob

∆t P

dissolve

(K)

5:

record core

⟨K⟩

6:

end for

The conguration space, made precise.

To state the positional conditioning exactly

rather than informally, dene the conguration space

C =

n

(V, E, x) : |x

i

− x

j

| ≥ R

ex

∀i = j, E ⊆



ij : |x

i

− x

j

| ≤ R

b



o

,

(22)

the set of node embeddings

x : V → R

3

obeying hard-core exclusion at

R

ex

together with

the bonds admissible within the proximity radius

R

b

. The Lindblad jump operators act

as maps between elements of

C

: a birth

Q

i∈A

σ

+

(x,i)

sends

(V, E, x) → (V

′

, E

′

, x

′

)

with

V

′

= V ∪ {x}

for an apex

x

at unit distance from its anchor set

A

, admitted only if

x

satises the exclusion constraint in Eq. (22); a dissolution

σ

−

performs the reverse. The

dynamics is thus a Markov process on

C

, and the metric is explicit in the constraints

|x

i

−x

j

| ≥ R

ex

and

≤ R

b

that dene

C

. This makes precise the sense in which the model

is positional: the master equation evolves a state

on

C

, and

C

is built from a presupposed

metric. It does not construct

C

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

derive

C

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

itself; that derivation is the open problem of Section 27, and stating

C

explicitly here is

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

The limitation is explicit. Equation (15) is

not

yet a fully pre-positional theory in which

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

conditioned on the admissible geometric move set

. The jump operators do not by them-

selves 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 27).

15

K=12

is a forced terminus

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

L

. 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

R

∗

=

q

2 −

√

2 L ≈ 0.76537 L,

(23)

31

0 10 20 30 40 50 60

time

t / τ

0

(

τ

0

=

/ε

≈

4

t

Planck

)

4

6

8

10

12

core coordination



K

®

core

K

= 12

reached

foam



K

®

≈

5

.

4

triangle

nucleates

Nucleation to

K

= 12

in physical time

derived stability

q

= 1

/

(1+

e

)

Figure 12: 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

τ

0

= ℏ/ε

.

a closed-form constant conrmed by a global maximin search (Figure 13). Therefore

R

∗

< R

ex

< L =⇒

no thirteenth neighbor, all twelve admitted, (24)

and inside this window the kissing-number theorem [15] makes

K=12

a geometric terminus

for any growth history. The upper edge

R

ex

= L

is the lattice-freezing transition, observed

directly in the simulation. The lower edge

R

∗

is the

absolute geometric

threshold below

which a thirteenth-neighbor site exists; it is distinct from the

observed

lower wall of the

Part I sweep, which sits at the triangle circumradius

L/

√

3 ≈ 0.577 L

(Section 3.8). The

gap between the two is kinetic protection by the operator basis: the thirteenth-neighbor

site (the square-face vacancy) lies at unit distance from exactly one existing nodethe

shell centerwhile a stitch requires two unit-distance anchors and a lift three, so neither

operator ever proposes it. For

R

ex

∈ (L/

√

3, R

∗

)

the site is geometrically admissible

but kinematically unreachable; for

R

ex

> R

∗

it is excluded outright.

K=12

is thus

geometrically forced inside the window (24) and kinetically protected below it.

K=12

is

thus not a value the kinematics must be tuned to reach.

16 The lift rate factorizes

Because Eq. (24) 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

P

lift

∈ [0.005, 0.5]



two orders of magnitude, no trend (Figure 14); varying

R

ex

reproduces the window. (The

terminal bulk

modal

value in these short unannealed runs is

10



11

, the grain-boundary

value of a polycrystal, matching the Part I bulk mean

11.40

with the shortfall concentrated

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

w

,

P

lift

P

stitch

≃ C w, C = O(1),

(25)

32

R

∗

= (2

−

p

2

)

1

/

2

L

12-shell: largest hole is a square face

(no 13th neighbour fits for

R

ex

> R

∗

)

Figure 13: The cuboctahedral shell (blue); its largest vacancy is a square face (red).

A thirteenth node at unit distance approaches no closer than

R

∗

=

p

2 −

√

2 L

to an

occupied site, so any

R

ex

> R

∗

forbids it

K=12

is forced.

veried numerically (the ratio falls monotonically

0.137 → 0.027

as

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

s

= Γ

s

Ω

s

(w) e

−β∆F

‡

s

and

γ

ℓ

= Γ

ℓ

Ω

ℓ

(w) e

−β∆F

‡

ℓ

, with

Γ

the attempt frequencies,

Ω(w)

the geometric phase-space

(solution-manifold) factors at tolerance

w

, and

∆F

‡

the activation free energies; then

γ

ℓ

γ

s

=

Γ

ℓ

Γ

s

Ω

ℓ

(w)

Ω

s

(w)

e

−β(∆F

‡

ℓ

−∆F

‡

s

)

.

(26)

The factorization of this section corresponds to the identications

Ω

ℓ

/Ω

s

∝ w

(a 0D

point set against a 1D circle, measured at tolerance

w

) and

∆F

‡

ℓ

− ∆F

‡

s

= 3ε

(the three

exposed anchoring bonds), giving

γ

ℓ

/γ

s

= C w e

−3βε

, equal to

C w e

−3

at the operating

point

βε = 1

. The identication

∆F

‡

ℓ

− ∆F

‡

s

= 3ε

is itself a kinetic model, not yet a

derivation: the apex bonds at the transition state are formed and energetically favorable,

and the claim is that their lack of in-plane redundancyeach is the node's only support in

its direction, with no closed triangle behind itprices the lift intermediate at

3ε

above the

stitch intermediate; an explicit intermediate-state accounting is the outstanding step: in

energy-landscape form,

E

‡

stitch

= E

0

+ ∆

s

and

E

‡

lift

= E

0

+ ∆

s

+ 3ε

, with the extra

3ε

to be

obtained from unsatised stabilizer terms, missing closed loops, or transient bond strain

the present Hamiltonian rewards formed bonds by

−ε

and does not by itself penalize

them for being exposed. A bookkeeping point follows: the Part I simulation parameter

P

lift

= e

−3

≈ 0.05

is the

full

per-proposal lift probability, so equating it to the thermal

factor alone amounts to the normalization choice

Cw = 1

at the operating tolerancea

convention we state rather than leave implied. The factorization nonetheless separates

the four ingredientsattempt-frequency ratio, phase-space suppression, activation energy,

temperatureand makes the claim testable: the prefactor should scale linearly in a

w

sweep, and the exponent should steepen as

e

−3βε

in a temperature sweep. A lift places

33

10

2

10

1

lift rate

P

lift

(free parameter)

8

9

10

11

12

terminal bulk modal

K

K

= 12

ceiling (kissing bound)

Terminus independent of lift rate over 2 decades

Figure 14: Terminal bulk coordination is independent of the lift rate across two orders

of magnitude of

P

lift

; the

K=12

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

order of node addition, not the terminus.

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

kT

0

= ε

that xes

q = 1/(1 + e)

(Section 13), each exposed bond carries a Boltzmann

factor

e

−ε/kT

0

= e

−1

, so the thermal ratio of lift to stitch is

P

lift

P

stitch









thermal

= e

−3

,

(27)

independent

of the tolerance

w

, with the exponent

3

set by the three exposed anchoring

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

acceptance

∼ w

(Eq. (25), not derivable) times the thermal factor

e

−3

(Eq. (27), xed by

the same

T

0

= ε

that gives

q

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

simulation's

e

−3

thus sits at the thermal factor; it is not a free constant but follows from

the model's detailed-balance temperature, while the geometric prefactor remains a free

amplitude. To state this unambiguously: the model

posits

an exposed-bond barrier of

3ε

,

producing the thermal factor

e

−3

at the operating point

βε = 1

(Section 16); it does not

predict the absolute lift probability, which remains this thermal factor multiplied by 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.

17 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

κ

in addition to the binding

34

depth

ε

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

of curvature

κ

(units

ε/L

2

); the FCC dynamical matrix is

D(q) =

κ

m

12

X

j=1



1 − cos(q·n

j

)



ˆn

j

ˆn

⊤

j

,

(28)

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

⟨u

2

⟩ =

ℏ

√

κm

D

1

2˜ω

coth

˜ω

2t

E

, t ≡

kT

ℏ

p

κ/m

,

(29)

averaged over the Brillouin zone. The constants are pinned by a converged Brillouin-zone

integration under the physically relevant

relative-displacement

(Lindemann) convention,

p

⟨|u

i

− u

j

|

2

⟩ < L − R

∗

for nearest neighbors

ij

, which retains the phonon correlations

⟨u

i

· u

j

⟩

that independent-node conventions discard (Monkhorst grids

16

3



32

3

agree to

four digits; the acoustic zero mode is excluded by the shifted grid). 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a single-scale vacuum sits at its melting point

holds under every convention. The forced lattice is rigid only if bond uctuations do not

carry a neighbor across the wall

R

∗

:

L −

p

2⟨u

2

⟩ > R

∗

, i.e.

2⟨u

2

⟩ < (1 − R

∗

/L)

2

=

0.0551 L

2

. Two derived conditions follow (Figure 15). A zero-point oor,

√

κm L

2

ℏ

> 28.6,

(30)

so even at

T = 0

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

inside the window; and a thermal separation,

kT

κL

2

< 0.031 (

a factor

≳ 33,

quoted as

≳ 36).

(31)

A single-scale vacuum (

kT ∼ ε ∼ κL

2

) gives a ratio of order unity, an order of magnitude

above this bound: it sits at its melting point and cannot crystallize. This also relieves

the only tension in Section 13:

q = 1/(1 + e)

uses

T

0

= ε/k

B

with

ε

the binding

depth

,

while rigidity constrains

kT/(κL

2

)

with

κL

2

the

curvature

; the two coexist provided the

bond well is deep-and-narrow,

κL

2

≳ 36 ε

. The

K=12

vacuum exists precisely when the

binding well is sti in this sense.

18 The construction velocity

The dynamics xes the speed of the assembly. The front advances by attachments net of

detachments; with attempt frequency

1/τ

0

and detailed balance, the net forward rate per

site is the same

q

that sets stability,

(γ

↑

−γ

↓

) = (1−2q)/τ

0

= (e−1)/(e+1)/τ

0

≈ 0.462/τ

0

,

with the normalization

γ

↑

+ γ

↓

≡ 1/τ

0

dening the microscopic attempt time (detailed

balance xes only the ratio of the rates; their sum is a convention absorbed into

τ

0

).

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

v

2D

= (1 −2q)

√

3

2

v

lat

≈ 0.40 v

lat

≈ 0.07 c.

(32)

35

10

2

10

1

10

0

10

1

reduced temperature

t

=

kT/

Ω

(

Ω =

p

/m

)

0.6

0.7

0.8

0.9

1.0

nearest-neighbour separation

/L

t

c

Rigidity of the forced

K

= 12

lattice (two-scale)

min bond length

L

−

q

2



u

2

®

R

∗

= (2

−

p

2

)

1

/

2

(K=12 wall)

Figure 15: Nearest-neighbor separation versus reduced temperature

t = kT /ℏΩ

(

Ω =

p

κ/m

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

separation stays above the

K=12

wall

R

∗

(red), and melts past

t

c

; rigidity xes the bound

Eq. (31).

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

P

lift

discussed in Section 16. The construction is therefore

anisotropic

: fast in-plane

at Eq. (32), 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 physically meaningful parameter-

free number is the in-plane speed

v

2D

; the stacking advance inherits the lift-rate freedom.

Taken with the light cone

c = v

front

= 4

√

2 v

lat

(Section 12), the construction front runs

at

v

2D

≈ 0.07 c

: local assembly proceeds at roughly one-fourteenth of the signal speed it

builds toward. There are two physical speeds, the slow construction front and the single

light cone, not three.

19 Exponential branching growth: an ination-shaped

geometric conversion (S)

A lift does not merely advance the stacking front by one node: it seeds an entire new

plane, which then grows laterally at the fast in-plane speed

v

2D

. 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

N

, total radius

R

, total area

A

, with seeding rate

∝ σA

,

σ ∝ P

lift

), the chain closes as

...

A = 2πv

2

2D

σ A =⇒ V (t) ∼ e

Γt

, Γ ∼



2πv

2

2D

σ



1/3

∝ P

1/3

lift

,

(33)

an

exponential

growth of the crystallized volume (Figure 16), conrmed numerically over

many orders of magnitude in

V

. 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

36

0 50 100 150 200

time

/τ

0

10

0

10

2

10

4

10

6

10

8

crystallized volume

V

(

t

)

Branching growth: exponential, self-terminating

lift amp

p

= 0

.

0050

lift amp

p

= 0

.

0005

foam reservoir (exit)

Figure 16: 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

P

1/3

lift

and so inherits the lift-rate freedom.

of

e

-folds is set by the foam reservoir,

ln(V

foam

/V

seed

)

, because new sheets nucleate only

into the metastable foam and growth halts exactly when the foam is consumeda built-in

graceful exit.

This exponential, self-terminating conversion of metastable foam into crystal is a geomet-

ric 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 lat-

tice volume, not the cosmological scale factor. It is labeled speculative for two reasons.

First, the spatial branching is scale-free (a power-law clustering spectrum,

R

2

= 0.96

over the inertial range), which is the prerequisite for an inationary spectrum, but 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

n

s

≈ 0.965

. Second, identifying exponential growth of

lattice volume

with metric ination

requires the equation of physical space with the lattice, which remains unproven (Sec-

tion 27). The mechanism is genuine and ination-shaped; the claim that it

is

ination is

not established.

20 Regge curvature and a conditional estimate of the

spectral tilt (C/S)

The branching phase of Section 19 is the microscopic conversion dynamics; its macroscopic

geometric content is a curvature, and that curvature makes contact with observation. Dur-

ing crystallization the structure passes through the two-sheet tetrahedral foam established

in Section 9 (Eq. (12)): in the nite interval between the arrival of consecutive close-packed

sheets, every interior node is

K=9

and forms a regular tetrahedron with the sheet above,

carrying the Regge decit

δ = 2π −5 arccos

1

3

≈ 0.1284

homogeneously

across the contact

region (Eq. (1)). The value is xed by the seeding-equidistance constraint and the code

stabilizers, which hold the tetrahedra regular rather than letting them relax: among can-

37

didate stackings the regular close-packed registration maximizes the satisable stabilizers

over eclipsed and irregular alternatives (FCC and HCP not being distinguished at this

level; Table 4, Section 9), and the frustration of ve regular tetrahedra about a hinge is

exactly

δ = 7.36

◦

. It is the two-sheet stage that layer-by-layer arrival necessarily produces,

not an assumed background, and it vanishes when the third sheet completes the cuboc-

tahedral shell. A constant positive decit isunder the intrinsic-manifold reading whose

construction is presupposed here, with hinges and dual volumes on an intrinsic triangula-

tion rather than the embedded frustrated packinga constant positive scalar curvature:

in the Regge action

S =

1

8πG

P

h

A

h

δ

h

, a homogeneous foam has

R

eff

= 2δ/A

cell

, inde-

pendent of cell size. In a homogeneous Lorentzian cosmology, positive constant curvature

is the geometric signature associated with a de Sitter-like expansion; under the intrin-

sic Regge-manifold interpretation presupposed here, the two-sheet foam's homogeneous

decit could then act as a positive-curvature source driving exponential expansionthe

geometric counterpart of the branching growthduring its lifetime, which ends when the

third sheets arrive, the cuboctahedral shell closes (

K=9 → 12

), and the decit vanishes: a

geometric end to the decit-driven phase with no added mechanism. The standing caveat

applies here with full force: the nodes are embedded in Euclidean

R

3

, so the decit is

a packing frustration of the embedded complex; reading it as

intrinsic

Regge curvature

of spacetime, and its growth as Friedmann-level expansion, presupposes an intrinsic sim-

plicial manifoldhinges and dual volumes constructed, a metric suppliedwhich is the

positional input agged throughout.

The spectral tilt.

The same decit imprints a curvature perturbation. The dimen-

sionless fractional curvature per hinge is

ζ

hinge

=

δ

2π

=

2π − 5 arccos

1

3

2π

≈ 0.02043,

(34)

the decit measured against the full

2π

of at space. The primordial spectrum is dened

on the 2D boundary, so the bulk hinge perturbation must be transferred there. Under

the Regge dual-area transfer assumption, the coecient is xed by the geometry of the

simplex.

The dual-area weighting is denitional, not a choice.

In Regge calculus the scalar curva-

ture of a discrete geometry is not smeared everywhere; it is concentrated on the hinges,

and the curvature carried by a hinge is its decit angle multiplied by the measure of the

hingein three spatial dimensions, the area of the dual cell. This is the content of the

Regge action

S =

1

8πG

P

h

A

h

δ

h

[7]: weighting the decit by the dual-face area is the

denition of scalar curvature in discrete general relativity, not an option exercised here.

The coecient and its normalization.

The building blocks of the foam are regular tetra-

hedra, so the hinges are edges of length

L

and their duals are equilateral triangular faces

of area

A

h

=

√

3

4

L

2

. Converting this area into the dimensionless coecient connecting

the per-hinge curvature to the boundary measure requires a choice of boundary measure

per hinge; we adopt the quarter-edge-square

L

2

/4

, under which the coecient is the pure

shape factor

A

h

/(L

2

/4) =

√

3

of the equilateral face. The equilateral origin of the

√

3

is

forcedit is the

same

regular tetrahedron that xes the decit

δ = 2π −5 arccos

1

3

, so the

perturbation amplitude and its transfer coecient share a single geometric originbut

the normalization

L

2

/4

is a convention, and an equally natural alternative would rescale

38

the coecient by an

O(1)

factor. With this weighting,

n

s

− 1 = −

√

3

δ

2π

=⇒ n

s

= 1 −

√

3

2π − 5 arccos

1

3

2π

= 0.9646,

(35)

numerically close to the Planck 2018 value

n

s

= 0.9649 ± 0.0042

[16]. We report this

as a consistency estimate, not a precision prediction: the agreement is contingent on the

transfer identication below, not on a completed perturbation calculation.

What is and is not free.

There is no continuous parameter to tune: standard cosmol-

ogy adjusts the slope of an inaton potential

V (ϕ)

to set

n

s

[17, 18], whereas here the

ingredients are xed Euclidean constants

arccos

1

3

(the rigid dihedral angle of a regular

tetrahedron),

5

(the maximal integer packing around a hinge,

⌊2π/ arccos

1

3

⌋

),

2π

(at

space), and the equilateral face shape factor. Because the rigid foam phase admits no

squished or stretched tetrahedra, angles and areas are locked to the exact mathematics

of regular simplices; the residual discrete freedom is the

O(1)

boundary-measure normal-

ization stated above, within which the coecient is

√

3

and admits no continuous drift.

The parameter-free content of Eq. (35) is therefore the sign and magnitude of the tilt,

n

s

−1 = −O(1)×δ/2π

; the specic value

0.9646

is conditional on the stated normalization

and on the transfer identication below.

The one assumption that remains.

Regge's theorem xes the dual-area weighting for the

static

curvature that enters the action. Using that same weighting for the transfer of the

curvature

perturbation

to the boundary power spectrum is a physical identication, not a

corollary of the action theorem: it assumes the perturbation propagates with the weighting

that denes the background curvature. This is the Regge-limit transfer estimate, carrying

no continuous freedom

within Regge calculus

(the discrete normalization convention above

aside). We note one sharpening: the coecient

√

3

does

not

depend on the front velocity.

If the crystallization front releases each hinge's decit onto the boundary as it advances,

the front speed enters the release rate and the boundary normalization identically and

cancels in the dimensionless coecient, which reduces to the static equilateral face-area

Jacobian

√

3

irrespective of how fast the front moves. The transfer coecient is therefore a

property of the close-packed geometry, not of the front kinematics; the residual assumption

is only that this static-curvature weighting is the correct measure for the perturbation

spectrum, an identication the front dynamics neither supplies nor is needed to supply.

Status.

The standing of this result is as follows.

Solid:

δ/2π = 0.0204

is parameter-

freethe Regge decit already load-bearing in the model, so the tilt has the right

mag-

nitude

for the right reason (a small geometric decit makes a small red tilt).

Coecient

from a standard prescription:

the coecient

√

3

follows from the Regge action's dual-area

weighting (curvature as the area-weighted decit on the dual face), so

n

s

carries no con-

tinuous freedom

within Regge calculus

.

Geometric, with one convention:

the

√

3

is the

static equilateral face-area Jacobian of the close packing under the quarter-edge-square

normalizationxed geometry plus one stated

O(1)

measure convention, not a contin-

uously tunable inputand it is independent of the front speed (which cancels).

One

identication remains:

that this static-curvature weighting is also the correct measure

for the perturbation spectrum. This is a narrower assumption than a full transfer im-

port; establishing itideally from the buildjoin dynamicsis the specic open step that

would make

n

s

parameter-free

from rst principles

.

Presupposed:

the De Sitter read-

ing uses the Friedmann relation

H

2

∝ R

eff

and so a metric, the same positional input

39

0.5 1.0 1.5 2.0 2.5

reduced bath temperature

t

=

k

B

T/ε

0

20

40

60

80

100

K

= 12

success (\%)

q

= 1

/

(1 +

e

)

operating point

K

= 12

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

bulk reaches

K

= 12

(\% of seeds)

5

6

7

8

9

10

11

12



K

®

Figure 17: 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.

agged throughout. Two heuristic expectations follow that would distinguish this from

tuned ination: suppressed tensor modes (the decit being scalar) and negligible running

(

n

s

being a geometric constant), both consistent with current bounds [16]. Neither is a

derived prediction: quantitative statements such as

r ≪ 0.01

or

dn

s

/d ln k ≈ 0

would

require a perturbation theory for the mode functions that is not constructed here.

21 End-to-end coherence and robustness (D)

The preceding mechanisms were derived piecewise; we test whether they cohere under a

single

xed rule. We run one Metropolis-kinetic dynamicspropose stitch and lift wher-

ever 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 always accepted, the thermal fac-

tor 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 is conrmed by sweeping the bath temperature over a factor of eight (

β =

k

B

T/ε

from

0.4

to

3.0

, i.e.

q

from

0.40

to

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 17).

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

9

(a nite, polycrystalline

crystallite with modal

K=12

interior, matching the Part I bulk mean

11.40

); reaching

single-crystal

⟨K⟩ → 12

is an annealing question, not a missing parameter.

40

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 17,

√

κm L

2

/ℏ > 28.6

,

rearranges to a lower bound on the node inertia,

m > 28.6

2

ℏ

2

/(κL

4

)

. Using the rigidity

requirement

κL

2

≳ 36 ε

and

v

lat

= Lε/ℏ

gives

m ≳ 22.7 ε/v

2

lat

; with the light cone

c = 4

√

2 v

lat

,

m c

2

≳ 7.3 ×10

2

ε.

(36)

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 17.

Critical rigidity temperature (D).

The melting condition

kT/(κL

2

) < 0.031

of Sec-

tion 17 sets a critical temperature

T

c

≈ 0.031 κL

2

/k

B

. Imposing the derived stiness

κL

2

≳ 36 ε

,

T

c

≳ 0.031 ×33

ε

k

B

≈

ε

k

B

= T

0

.

(37)

The melting threshold sits right at the bath scale

T

0

= ε/k

B

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,

ε

geom

≡

δ

2π

=

2π − 5 arccos

1

3

2π

≈ 2.0 ×10

−2

,

(38)

geometric rather than tuned. It controls the spectral tilt at

O(ε

geom

)

(Section 20), residual

anisotropy at

O(ε

2

geom

)

, and the foam instability strength at

O(ε

geom

)

.

Branching rate and exit time (C/S).

With the construction estimate

v

2D

≈ 0.40 v

lat

of Section 18, the branching rate of Section 19,

Γ ∼ (2πv

2

2D

σ)

1/3

with seeding density

σ = α

σ

P

lift

/(L

2

τ

0

)

, reduces to

Γτ

0

∼



2π(0.40)

2

α

σ

P

lift



1/3

≈



α

σ

P

lift



1/3

,

(39)

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,

t

exit

= Γ

−1

ln

V

foam

V

seed

,

(40)

separating

how much

expansion (the reservoir ratio, the

e

-fold count) from

how long

it

lasts (the seeding rate). Growth halts when

V

crystal

= V

foam

and

Γ → 0

, a graceful exit

requiring no added decay mechanism.

41

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

f

defect

∼ 1 −

⟨K⟩

12

≈ 0.25,

(41)

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

12

, 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

2

/ℏ

2

of Section 12. Both sharpen, rather than 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 17 (the dynamical matrix (28)) is a nearest-neighbor central-force model; in the

long-wavelength limit it is governed by the

rank-four

bond tensor [8],

D

αβ

(q) ≃

κL

2

2m

q

µ

q

ν

T

µναβ

, T

µναβ

=

12

X

j=1

ˆn

µ

j

ˆn

ν

j

ˆn

α

j

ˆn

β

j

= δ

µν

δ

αβ

+δ

µα

δ

νβ

+δ

µβ

δ

να

−∆

µναβ

,

(42)

with

∆

µναβ

= 1

only when all four indices coincide. Unlike the rank-two

S

µν

= 4δ

µν

that

governs the scalar channel,

T

carries the cubic term

∆

, so the three acoustic branches are

isotropic only in their trace. Diagonalizing along the symmetry directions gives, in units

of

L

p

κ/m

,

ˆ

q

longitudinal transverse transverse

[100] 1 1/

√

2 1/

√

2

[110]

√

5/2 1/

√

2 1/2

[111]

p

4/3 1/

√

3 1/

√

3

The longitudinal speed runs from

L

p

κ/m

along

[100]

to

p

4/3 L

p

κ/m

along

[111]

,

a parameter-free anisotropy of

p

4/3 − 1 ≈ 15.5%

set entirely by the FCC bond ge-

ometry. With

κ

xed by GW

=

light (Eq. (17)) these become

v

L

∈ [0.71, 0.82] c

and

v

T

∈ [0.35, 0.50] c

, the three branch speeds obeying the trace identity

v

2

L

+v

2

T

1

+v

2

T

2

= c

2

in

every direction (the content of Eq. (17)): the bare lattice carries

sub-luminal, anisotropic

sound, and the isotropic scalar speed

c

s

= L

p

2κ/m = c

of Section 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

42

the

p

4/3 − 1 ≈ 15.5%

level in its longitudinal branch and the

√

2 − 1 ≈ 41%

level in

its transverse branches (spread

[

1

2

,

1

√

2

] L

p

κ/m

across directions)the transverse sector

being the one relevant to tensor (gravitational-wave) modes; full continuum isotropy, with

a single

c

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 quantitative 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

12

= C

44

(Zener ratio

A = 2C

44

/(C

11

− C

12

) = 2

) characteristic of a

central-force lattice. The vacuum of Part I, however, is a polycrystal of randomly oriented

grains. VoigtReussHill averaging of the cubic tensor over orientationsthe standard

elastic-tensor average over the

SO(3)

rotation group, Voigt and Reuss giving the uniform-

strain and uniform-stress bounds and Hill their meanremoves the directional depen-

dence entirely, leaving an isotropic eective medium with

v

L

≃ 1.08 L

p

κ/m ≃ 0.76 c

and

v

T

≃ 0.62 L

p

κ/m ≃ 0.44 c

(Hill). Two consequences follow. Above the grain sizethe

misoriented-crystallite scale that produces the coordination scatter

σ

K

≈ 0.99

of Part I

the vacuum has no preferred direction, 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 longitudinal and transverse speeds remain split,

v

L

/v

T

≃ 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).

First,

nearest-neighbor pair

forces cannot produce a stable isotropic elastic medium

. For the general axially symmetric

nearest-neighbor force matrix

Φ = α ˆnˆn

T

+ β (⊮ − ˆnˆn

T

)

(radial stiness

α

, tangential

β

),

a direct computation of the FCC elastic constants gives

C

11

∝ α + β

,

C

12

∝ (α − 5β)/2

,

C

44

∝ (α + 3β)/2

, so the isotropy condition

C

11

− C

12

− 2C

44

∝ (β − α)/2

vanishes

only

at

β = α

where the force matrix degenerates to a scalar spring

Φ = α⊮

, all

three acoustic branches coincide, and the constants stand in the ratio

C

11

: C

12

: C

44

=

2 : −2 : 2

, i.e.

λ = −µ

and bulk modulus

K

B

= λ +

2

3

µ = −

µ

3

< 0

: mechanically

unstable. Second, and independently of any force model,

no stable elastic medium carries

all polarizations at one speed

:

v

L

= v

T

requires

λ+2µ = µ

, hence

λ = −µ

and

K

B

< 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

is

the unique equal-speed

conguration, and stability excludes it. The consequences are constructive: (i) restoring

isotropy

with

stability requires three-body angular terms,

H

bend

=

κ

θ

2

P

⟨ijk⟩

(ˆr

ij

· ˆr

ik

−

cos θ

0

)

2

, or couplings beyond nearest neighbors; solving for the ratio

κ

θ

/κ

that cancels

the cubic anisotropy is a dened target calculation. (ii) A shared single speed across

polarizations cannot be carried by the displacement eld of

any

stable medium: the

photon and gravitational sectors cannot both be elastic polarizations, and a distinct,

gauge-like excitation 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the relativistic tensor excitation, if the framework contains one, must be a gauge-

43

Table 7: Energy per node along the cascade, in units of the binding scale

ε

.

rung role

K

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

like, topological, or defect-based collective degree of freedom of the bond network rather

than the displacement eld itself.

The vacuum energy density (D/S).

The FCC bond density is

6

√

2/L

3

(six bonds

per node, node density

√

2/L

3

), so the binding term contributes

u

bind

= −ε

6

√

2

L

3

≈ −8.49

ε

L

3

,

(43)

negative, as a bound crystal requires. Phonon zero-point motion adds a positive

u

zp

∼

9

8

ℏω

D

(

√

2/L

3

)

with

ℏω

D

∼ 4 ε

from the xed

κ

, i.e.

u

zp

∼ +O(1) ε/L

3

, of the same order.

The net density is therefore

|ρ

vac

| ∼ ε/L

3

. With the binding oor

ε ≳ 2.3 ℏ

2

/(mL

2

)

and a Planck-scale lattice (

L ∼ ℓ

P

) this is of order the Planck density, roughly

120

orders of magnitude above the observed cosmological constant: the model reproduces the

cosmological-constant problem [22] in its standard, unsolved form. The single structural

feature it oers is that

u

bind

< 0

and

u

zp

> 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 7, Figure 18). The

K=6

sheet is the planar contact maximum; the

K=12

packing is the spatial contact

maximum, the kissing number [15] 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.

25 The CSS code on the close-packed lattice

The code term of Eq. (13) is the stabilizer Hamiltonian of the

[[192, 130, 3]]

CSS code

constructed in Part I (Section 4) [20]: vertex-star

Z

-checks and weight-12 octahedral

X

-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

44

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

assembly coordinate

7

6

5

4

3

2

1

energy per node (

ε

)

triangle

K

= 2

hex sheet

K

= 6

(2D max)

foam

K

= 4

(barrier)

close pack

K

= 12

(3D min)

+1

.

37

ε

Energy ladder (units of the binding scale

ε

)

registered path:

K

6

→

9

→

12

, downhill

unregistered: through foam barrier (glass)

Figure 18: The cascade as an energy ladder (

ε

): 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.

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 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 (

3

) 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. The selection rests primarily on the rst two of three selectors. (i) Bra-

vais structureFCC is a Bravais lattice (

Z

3

), HCP is not; (ii) rank-4 isotropythe HCP

fourth-moment bond tensor carries a

c

-axis term

T

zzzz

= 2.667 = T

xxxx

= 2

, forbidden for

an isotropic continuum, while FCC has only cubic anisotropy; (iii) 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

10

−3

k

B

per node, and convention-dependent, so we treat it as a consistent tiebreaker

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

The codeable cuboctahedral order is in turn chosen over the icosahedral glass that also

realizes

K=12

: monogamy marginally favors icosahedral (Thomson

49.165 < 49.342

), but

the code term closes only with the cuboctahedron's twelve square faces (the icosahedron

has none), and codeability overrides the sub-percent monogamy preference.

27 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

2

/ℏ

2

; the detailed-balance

q = 1/(1 + e)

45

and the dissolution ladder; the forced

K=12

terminus and

R

∗

=

p

2 −

√

2 L

; the lift rate's

freedom and the non-existence of a derivable value; the rigidity bounds Eqs. (30)(31); the

energy ladder; the code parameters; and the FCC/HCP selectors.

Established in Part I:

the kinematic cascade, the metric wall, the nite-size scaling, and the structure-tensor

isotropy.

Imported as established mathematics:

the kissing and Kepler theorems [15, 3] un-

derlying the contact maxima, and the Regge prescription of Section 20.

Open:

(i) the reg-

istration bias of the births is motivated by the code term lowering registered-conguration

energy but not proven quantitatively from BornMarkov rates; (ii) the rigidity factor

≳ 36

and the oor

14.8

follow from the central-force harmonic spectrum, and a fully anhar-

monic well would sharpen them; (iii) reaching single-crystal

⟨K⟩ → 12

rather than the

polycrystalline modal-

12

crystallite is an annealing/quench-rate question, not a missing

parameter; (iv) the branching phase (Section 19) is exponential and self-terminating but

its frozen perturbation spectrum is uncomputed and its raw clustering is too red to match

n

s

≈ 0.965

, so its identication with ination is unestablished; (v) the light cone is com-

puted 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; (vi) the construction is

positional, presupposing node positions and a metric exclusion, so a fully pre-positional

dynamics, and with it any genuine claim about metric expansion, remains the deepest

open problem.

28 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 propagates

an emergent light cone of speed

v

lat

= ℓ

lat

ε/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 Part I cascade. Of the apparent inputs to that cascade,

K=12

is a

forced terminusthe kissing bound forbids a thirteenth neighbor for any exclusion

R

ex

>

R

∗

=

p

2 −

√

2 L

the lift rate factorizes into a free geometric acceptance and a thermal

e

−3

xed by

T

0

= ε

, with no single derivable value, and the genuine parameter is a

rigidity scale, the binding curvature exceeding 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.

Time is the Schrödinger parameter and length is

ℓ

lat

; 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 rather than ne-tuned; and

the lift-seeded branching of sheets makes the foam-to-crystal conversion exponential and

self-terminatingan ination-shaped geometric conversion. The same Regge decit that

frustrates the foam imprints a primordial tilt: weighted by its dual face area (under the

stated normalization), it gives the Regge-limit estimate

n

s

= 1 −

√

3 (δ/2π) = 0.9646

,

46

numerically close to the Planck value, though deriving the perturbation transfer from the

build dynamics remains open. The assembled lattice thus carries the features of physical

spaceisotropy, a causal cone, a curvature that expands, a Regge-transfer estimate for

the scalar tilt that is numerically close to the observed Planck valuewhile the emergence

of the positional substrate itself, from which a fully rst-principles metric and spectral

index would follow, remains the frontier.

Data Availability Statement

The simulation algorithm is fully specied in Sections 2.12.4 (kinematic operators, codi-

mension suppression, QEC interpretation, compound-QEC argument) and 3.13.8 (pa-

rameter values, sweep ranges, statistical protocols), and any equivalent implementation

will reproduce the results within the reported statistical uncertainties. Reference Python

implementations are provided as a verication aid:



ssm_sim.py

 lattice generator and saturation analysis used to produce Tables 23

and the layer-structure data behind Figures 57:

https://github.com/raghu9130

2/ssmtheory/blob/main/ssm_sim.py

.



plot_rex_sweep.py

 exclusion-radius sweep used to produce Figure 8:

https:

//github.com/raghu91302/ssmtheory/blob/main/plot_rex_sweep.py

.



ssm_bulk_analysis.py

 bulk-interior diagnostic of Section 3.3, which re-analyzes

the simulation output of

ssm_sim.py

to extract bulk-restricted coordination statis-

tics under the strict 42-neighbor criterion:

https://github.com/raghu91302/ss

mtheory/blob/main/ssm_bulk_analysis.py

.

The results of Part II are analytic and are reproducible 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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