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). Local probabilistic assembly rules for discrete space

tend to jam into porous, geometrically frustrated graphs rather than saturated vol-

umetric bulks [1]. We show computationally that a two-operator growth law—a

planar stitch and a rare out-of-plane lift, suppressed to P

lift

= e

−3

by the codi-

mension difference between the operators—drives a discrete entanglement network

through the cascade K=1 → 6 → 4 → 12, past the frustrated K=4 tetrahedral

foam (Regge deficit 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 FCC bulk—with surface-to-

volume scaling f = 1 − (6.8 ± 0.6)N

−1/3

and a wide exclusion-radius plateau con-

firming the operating values are not fine-tuned. 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 lattice

at the parameter-free detailed-balance rate q = 1/(1 + e), and whose coherent limit

propagates an isotropic signal cone c = 2v

lat

. We prove K=12 is a forced terminus

(largest cuboctahedral vacancy at R

∗

=

p

2 −

√

2 L), factorize the lift rate into

a fixed thermal factor e

−3

and a free geometric 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 (metric-perturbation) speed—the observed equality of

gravitational-wave and photon speeds—fixes the stiffness, κ = 2mε

2

/ℏ

2

, and with

rigidity floors the binding scale at ε ≳ 18 ℏ

2

/(mL

2

). A conditional Regge-curvature

estimate of the scalar spectral tilt closes the paper.

1

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 diffusion-limited aggregation [2] or

kinematically jammed, fractal-like foams. These porous configurations 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 defined 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 significant 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 fixed at P

lift

= e

−3

≈ 4.98% by the codi-

mension difference between their solution manifolds (Section 2.2), and is independently

determined to the same value by the survival threshold of the [[192, 130, 3]] CSS code that

the emergent lattice supports (Sections 2.3 and 4). Under these constraints, the system

traces a kinematic cascade K = 1 → K = 6 → K = 4 → K = 12, escaping the geometri-

cally frustrated K = 4 tetrahedral foam [7] and crystallizing into a polycrystalline 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.

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

2 Methodology

The simulation algorithm is fully specified 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, fixed 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 figures 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 fields, 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.

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).

Kinematic completeness. Stitch and Lift exhaust the operators that strictly increase

connectivity at fixed unit bond length. 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

defined. Together, Stitch and Lift form a complete kinematic basis for graph growth in

three-dimensional Euclidean space at fixed bond length.

3

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 configuration 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 five tetrahedra around a shared edge consumes only

5 × 70.528

◦

= 352.64

◦

, leaving an irreducible Regge deficit [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 deficit. The exclusion radius R

ex

= 0.95 L and

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

deficit. The wedge angle δ ≈ 0.128 rad projects to a fractional bond-length jitter of

|sin(δ/2)| ≈ 0.064 at each node, so a tolerance window of half-width ∼ 5% accommodates

the residual angular distortion at every interior site without admitting unintended coin-

cidences. The robustness of K = 12 saturation across the entire band R

ex

∈ [0.58, 0.99] L

(Section 3.8, Figure 8) further confirms that the specific 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 differ 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

difference:

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. This is the unique exponential amplitude

consistent with the kinematic requirement that the cascade terminates at K = 12 FCC

saturation in the thermodynamic limit: smaller P

lift

produces isolated 2D sheets that fail

to interlock, while larger P

lift

produces 3D foams that fail to saturate at the Kepler bound.

4

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 deficit. This motivates

the survival ansatz:

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 deficit by direct analogy with thermal activation across a free-energy barrier in

the underlying random-bond Ising mapping of the toric code. The specific 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 deficit d+1−t, and (iii)

recovers the textbook below-threshold scaling P ∝ e

−∆

in the large-deficit 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 specific functional form, depending only on the monotonic increase

of effective 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 obtained from the codimension argument (2), arrived at from a completely

independent direction.

2.4 Compound QEC: why flat growth dominates

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

scribed. 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

5

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 detec-

tion paths within 2 hops. The effective protection is well above the d + 1 = 4 threshold,

and sheet-interior nodes survive at effectively 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 effective

triangle count is t

eff

≈ 1.

The asymmetry. The ratio of effective protection is much larger than the raw triangle-

count ratio of 6:1, because compound detection paths grow with distance from the bound-

ary. A node 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 specific 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

follows from codimension suppression and from the QEC distance, and the proximity-bond

and exclusion radii together form the symmetric ±5% Regge-deficit tolerance window

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

robustness of the operating values is confirmed 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 specific value R

ex

= 0.95 L is not a

fine-tuned parameter.

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

random seeds.

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 flat 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

6

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

termined; the proximity bond and exclusion radii together form the symmetric ±5%

Regge-deficit 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 deficit (δ ≈ 7.36

◦

,

Eq. (1))

Hard Shell (R

ex

) 0.95 L Regge deficit (Section 2.1);

plateau verified in Section 3.8

Geometric Cutoff

1

√

3

L ≈ 0.577 L Circumradius of unitary trian-

gle

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

perfectly crystallized but flat 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 identified as failures. A sum f

K=12

+(z

ext

/xy

ext

) would reward either achievement

independently and could rank a flat 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

figure 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 flat-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.

7

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 flat pancakes.

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.

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 classified as bulk-interior iff at least 42 other nodes lie within radius

2L (the count of an ideal FCC interior node’s neighbors in its first 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.

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

The data are described by the finite-size scaling law [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

.

8

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 fit f = 1 − 6.8/N

1/3

. (b)

¯

K approaching the Kepler bound.

Extrapolating equation (5) gives f → 1 in the thermodynamic limit (N → ∞), consistent

with complete FCC saturation as the asymptotic ground state of the kinematics. Single-

decade fits to power laws are more easily believed when the functional form is geometrically

motivated, as it is here, but the central evidence that the bulk is FCC 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 system size—

consistent with the expectation that larger polycrystalline clusters average over more

grain orientations.

3.3 Bulk-interior diagnostic: direct evidence of FCC 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 iff at least

42 other nodes lie within radius 2L of it, equivalently, iff its three FCC coordination shells

are fully populated. The threshold 42 is not tuned: it is exactly 12 + 6 + 24, the count of

an ideal FCC interior node’s neighbors in its first 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

bulk interior is FCC by direct measurement, not by extrapolation.

9

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.

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.

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 fingerprint of grain-boundary nodes between

FCC crystallites of different orientations, not of a coordination distribution centered

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

lattice.

The all-node finite-size scaling and the bulk-interior modal-K measurement therefore

provide independent and complementary evidence: the first establishes that under-

coordinated nodes are confined to the surface (where they decay as N

−1/3

), the second

establishes that the bulk itself is FCC at every system size measured. 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.

10

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.

3.5 Layer planarity and FCC registry

A z-clustering analysis of the N = 3000 lattice identifies 28 well-defined planar layers

(Figure 5). Exact structural flatness. 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 confined 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.

11

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 flatness.

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.

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.

3.8 Sensitivity sweep and the R

ex

= 1/

√

3 geometric cutoff

To ensure K

max

= 12 is not an artifact of a tuned exclusion radius, a fine-grain sweep was

conducted across R

ex

(reference implementation: plot_rex_sweep.py in the Data Avail-

ability statement). As shown in Figure 8, the maximum coordination of 12 is maintained

across the entire band R

ex

∈ [0.58, 0.99]. The breakdown at R

ex

≈ 0.58 corresponds to

12

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.

the circumradius of the equilateral triangle (1/

√

3 ≈ 0.577 L). Below this threshold, the

exclusion is too weak, allowing unit-radius packing violations (K > 12). At 1.00 and

above, strict rigidity causes lattice freezing. The wide stability plateau confirms that the

specific value R

ex

= 0.95 is not fine-tuned.

4 The [[192, 130, 3]] CSS Code

The emergent FCC lattice carries a mathematically verifiable quantum error-correcting

code. At the smallest system supporting this code (a 4×4×4 unit-cell arrangement), the

lattice has 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, verified computationally in [20],

are:

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

13

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: Geometric phase transition at the metric wall R

ex

= L/

√

3 (N = 500, 30

seeds, p

lift

= 0.05). The maximum coordination K

max

= 12 is maintained across the

entire stability plateau R

ex

∈ [0.58, 0.99] L. Below L/

√

3 ≈ 0.577 L (the equilateral-

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

(K

max

> 12); above 1.0 L, strict rigidity causes lattice freezing. The wide plateau confirms

that the operating value R

ex

= 0.95 L is not fine-tuned.

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 fixed 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 reflects a structural identity or is accidental

remains an open question that may be addressable through a generalized construction at

other values of d.

14

5 Discussion

5.1 Exact spatial isotropy and emergent Lorentz invariance

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 falsifiable anomalies. We resolve this in three steps—an

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

of the lattice dispersion relation at long wavelengths, and the standard continuum-limit

emergence of full Lorentz invariance with confrontation against 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)

Define 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.

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 field 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—the elastic (metric-perturbation) speed of node dis-

placements, set by the same FCC phonon spectrum and stiffness κ that fix rigidity in

Part II, with m the node inertia. The dispersion is exactly isotropic at leading order: this

15

isotropy is an algebraic consequence of equation (9), not an approximation. Anisotropic

corrections appear only at O(k

4

a

4

), suppressed by (|k|a)

4

∼ (E/M

P

)

4

for Planck-scale

lattice spacing a ∼ ℓ

P

.

Step 3: Emergence of Lorentz boosts. The isotropy (9) guarantees equal propagation

speed in every direction for any excitation on the bond set—the geometric prerequisite

for emergent Lorentz invariance. 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 equality is not a

tuning but an observed fact—gravitational-wave and photon speeds agree to one part in

10

15

, and Part II shows it fixes the stiffness κ. With one cone, an isotropic linear dispersion

gives rise to a relativistic effective field theory in the standard continuum limit [12]. Lattice

corrections at the cutoff 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. Spatial SO(3) isotropy combined with time-reversal

symmetry then implies SO(3, 1) Poincaré invariance for all dimension-4 operators in the

effective theory. Any residual O

h

anisotropy computed from raw 3D lattice positions is

an artifact of the coordinate description, not a physical effect.

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), confirming 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. 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 difference 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 independently determines the same

suppression amplitude.

16

6 Conclusion

We have verified 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

codimension-suppressed lift probability, independently moti-

vated by QEC survival at code distance d = 3—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, providing direct (non-extrapolated)

evidence of FCC bulk crystallization, complemented by an all-node finite-size scaling law

f = 1−(6.8±0.6) N

−1/3

that confines 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 indistinguishable from p = 3% given

the 30-seed error bars, but clearly degraded at p ≤ 1% due to flat-pancake structures

with aspect ratio z/xy = 0.59, and at p ≥ 10% due to poor crystallization); (3) exact

structural flatness (σ

z

< 10

−10

L) of internal layers with spacings matching FCC to 0.01%;

(4) a sharp geometric phase transition at R

ex

= 1/

√

3 L, with a wide stability plateau

across [0.58, 0.99] L confirming that operating values are not fine-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-

firming polycrystalline averaging that complements the exact 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 sets the same barrier that controls

the lift probability: the difficulty of creating 3D volume is the strength of the code’s error

protection.

Data Availability Statement

The simulation algorithm is fully specified 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 verification 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.

17

Declarations

Conflict of interest: The author declares that they have no conflict of interest.

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

geometric phase transition at the metric wall, finite-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 fixes 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 recovered as one stochastic unraveling. 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

fixed 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

18

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.

Status of the claims. The results below sit at different confidence 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 confirm reaches bulk K=12 generically across temperature (Section 21). (C)

Consistent but separately grounded: the light cone, the relation c = 2v

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 as-

sumptions: 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 deficit) and its

coefficient

√

3 is selected by the Regge area-weighting, numerically consistent with Planck,

but this is an estimate within Regge calculus on a presupposed metric, and identification

with cosmic inflation 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 =

2v

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 fixed 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].

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 configurations already carry a length L

(Section 14, Eq. (19))—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 fixed nucleation rate that sets the unit of time and a

19

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 dynamics, two limits: the Hamiltonian (13) coupled to a

thermal bath at T

0

= ε/k

B

. Coherent part → emergent light cone; dissipative part →

constructive assembly with derived rate q = 1/(1 + e).

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

pressed in these units; the model fixes 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 configuration 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

satisfiable 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 deficit 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 deficit lives during crystallization. Sheets arrive

one at a time: a sheet is seeded by a lift and grows laterally, and the next sheet is indepen-

dently seeded and grows, so there is always a finite delay between the arrival of consecutive

20

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

first (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 satisfied. Only the

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

registration tetrahedra faced fraction edge irregularity stabilizers satisfiable

regular (FCC) 32 0.73 0.000 maximal

eclipsed (AA) 0 0.00 0.000 frustrated

relaxed (irregular) 0 0.00 0.068 frustrated

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 flat K=6 sheets at the close-packed

offset 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 tetrahedron (all six edges

equal to L; 100% of interior nodes in a finite patch). The frustrated tetrahedral bond-

ing here is the same that defines 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 deficit is fixed, not in-

cidental. 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 pla-

quettes that are genuine cell faces, i.e. the fraction of Z-stabilizers that can be consistently

satisfied. Only the regular registration forms tetrahedral cells at all (32 versus none) and

closes its plaquettes (73% faced versus 0%); the alternatives are stabilizer-frustrated. The

regular registration therefore uniquely maximizes the satisfiable stabilizers, which is what

selection by the code means operationally. (This is a stabilizer-satisfiability comparison

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 deficit δ = 2π − 5 arccos

1

3

= 7.36

◦

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

instead distribute a much larger and variable deficit. Because the seeding-plus-code con-

straint enforces the same regular registration everywhere along the contact, the deficit

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

converts K=9 → 12 and the deficit vanishes, the cuboctahedral shell tiling space without

frustration. The crystallization therefore passes through a genuine, spatially extended,

21

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 fixed value of the deficit. 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 deficit), 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 deficit is

therefore spatially homogeneous and fixed at δ, and vanishes when the third sheet closes

the cuboctahedral shell (K=9 → 12).

finite-duration tetrahedral foam:

flat 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 deficit 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.

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)

22

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/D three exposed bonds at T

0

; geomet-

ric prefactor free

κL

2

≳ 36 ε rigidity C/D FCC harmonic phonon estimate

n

s

= 0.9646 C/S Regge dual-area transfer assump-

tion

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)

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 off, 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).

23

12 Coherent limit: a single-excitation signal cone and

lattice isotropy

In the single-excitation sector the kinetic term is tight binding; a localized single ex-

citation spreads ballistically (Figure 11) with front velocity v

front

= 2ℓ

lat

ε/ℏ = 2v

lat

,

verified as 2.01 ℓ

lat

ε/ℏ on 401 bonds. A local Hamiltonian has a finite 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 field 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 dispersion is isotropic

and linear with anisotropy only at O((|k|ℓ

lat

)

4

). The causal cone is the signal front itself:

c ≡ v

front

= 2v

lat

= 2ℓ

lat

ε/ℏ, the fastest speed at which any influence 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 mea-

sured equality of the gravitational-wave and photon speeds to one part in 10

15

[19] requires

c

s

= c, which fixes the otherwise-free stiffness,

κ =

2mε

2

ℏ

2

(GW = light : c

s

= c). (16)

This is sharpened by the rigidity requirement κL

2

≳ 36 ε of Section 17: the two to-

gether impose a floor on the binding scale, ε ≳ 18 ℏ

2

/(mL

2

)—the bond energy cannot

lie parametrically below the lattice kinetic scale ℏ

2

/(mL

2

). Equal cones thus turn a free

parameter into a prediction.

Mode speeds and the single cone. The cone c = 2v

lat

used here is the maximum

group velocity (Lieb–Robinson front) of the scalar bond excitation. 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 latter the long-wavelength sound speed of

the displacement field. Emergent SO(3, 1) invariance of the long-wavelength theory was

established in Section 5.1 from the exact bond-set isotropy S

µν

= 4δ

µν

and centrosym-

metry T

µνλ

= 0, given a common propagation speed; the role of Part II is to supply that

input dynamically. The GW=light identification (Eq. (16)) forces the signal and elastic

branches to coincide, c

s

= c, on exactly the pair that is observationally constrained, turn-

ing the assumed equal speed of Section 5.1 into a consequence of the stiffness. 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 ≡ 2v

lat

as the causal cone of the

signal sector and note the multi-speed structure explicitly rather than hiding it in a single

symbol.

24

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 spreads with front velocity v = 2ℓ

lat

ε/ℏ, an

emergent light cone reproducing the isotropic linear dispersion.

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

= e, and the single-bond rate equation has stationary failure probability

q =

γ

↓

γ

↑

+ γ

↓

=

1

1 + e

≈ 0.2689, (17)

with no free parameter: 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}.

(18)

Stability rises steeply with coordination: the foam turns over while the K=12 core is

effectively immortal. This relative stability is the detailed-balance content of Eq. (15),

not an input.

14 The assembly as a quantum-trajectory unraveling

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

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. The unraveling deserves a precise state-

ment, because the Lindblad jumps act on bond states while the moves are node place-

ments. A σ

+

b

jump sets a bond b from |0⟩ to |1⟩. In the unraveling, a node is born when

25

Algorithm 1 Quantum-trajectory assembly (one unraveling of Eq. (15))

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

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), weighted by min(1, e

β |A(x)|

) from detailed balance on the bonds it forms; a σ

−

b

jump lowers a bond, and a node dissolves when its surviving bonds fall below rigidity,

reproducing P

dissolve

(K) of Eq. (18). The continuous bond-flip rates γ

↑

, γ

↓

thus generate

the discrete moves through the admissible anchor sets: the geometry enters by restricting

which coincident σ

+

patterns correspond to a unit-distance registered site.

The configuration space, made precise. To state the positional conditioning exactly

rather than informally, define the configuration space

C =

n

(V, E, x) : |x

i

− x

j

| ≥ R

ex

∀i = j, E ⊆



ij : |x

i

− x

j

| ≤ R

b



o

, (19)

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

satisfies the exclusion constraint in Eq. (19); 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 define 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 specific coincident-jump pattern with the detailed-

balance weight—but the conditioning itself is the positional input flagged as the deepest

open problem (Section 27).

26

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.

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, (20)

a closed-form constant confirmed by a global maximin search (Figure 13). Therefore

R

∗

< R

ex

< L =⇒ no thirteenth neighbor, all twelve admitted, (21)

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; the

lower edge R

∗

is where a thirteenth neighbor intrudes—both confirmed by our simulation

(Section 16). K=12 is thus not a value the kinematics must be tuned to reach.

16 The lift rate factorizes

Because Eq. (21) makes the terminus history-independent, the lift rate cannot affect 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 reflects 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

27

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.

one extra power of the bond-shell width w,

P

lift

P

stitch

≃ C w, C = O(1), (22)

verified numerically (the ratio falls monotonically 0.137 → 0.027 as w : 0.15 → 0.03, no

plateau). This geometric acceptance fixes the form, never the value.

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

factor. A lift places an apex with three out-of-plane anchoring bonds that are exposed—

unstabilized by any in-plane neighbor—until the cooperative three-apex seed of the next

sheet closes around them; a stitch, lying in the plane, exposes none. At the detailed-

balance temperature kT

0

= ε that fixes 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

, (23)

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. (22), not derivable) times the thermal factor e

−3

(Eq. (23), fixed 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 predicts the thermal suppression

e

−3

, not 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 fixed in its thermal factor; the two statements refer to the two factors and do not

conflict.

28

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.

17 The parameter the model must supply: a rigidity

scale

If the lift rate factorizes and K=12 is forced, what must the model fix for the lattice to

exist? Here the “one energy scale” of the bond-state Hamiltonian is not sufficient: the

positional realization of the bond requires a curvature scale κ in addition to the binding

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

, (24)

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

, (25)

averaged over the Brillouin zone. The forced lattice is rigid only if bond fluctuations 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 floor,

√

κm L

2

ℏ

> 14.8, (26)

so even at T = 0 the lattice must be stiff and heavy enough that quantum motion stays

inside the window; and a thermal separation,

kT

κL

2

< 0.028 (a factor ≳ 36). (27)

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

29

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 fluctuating

separation stays above the K=12 wall R

∗

(red), and melts past t

c

; rigidity fixes the bound

Eq. (27).

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 stiff in this sense.

18 The construction velocity

The dynamics fixes 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

. 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 parameter-free speed

v

2D

= (1 − 2q)

√

3

2

v

lat

≈ 0.40 v

lat

≈ 0.20 c. (28)

The out-of-plane lift is rarer (one fewer solution dimension, and the binding term favors

the flat 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. (28), 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

= 2v

lat

(Section 12), the construction front runs at

v

2D

= 0.20 c: local assembly proceeds at one-fifth of the signal speed it builds toward.

There are two physical speeds, the slow construction front and the single light cone, not

three.

30

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 inflation-shaped, self-

terminating conversion. The rate scales as P

1/3

lift

and so inherits the lift-rate freedom.

19 Exponential branching growth: an inflation-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

, (29)

an exponential growth of the crystallized volume (Figure 16), confirmed 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 fixed number; the number

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 inflation-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 inflationary 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 inflation

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

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

not established.

31

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 macro-

scopic geometric content is a curvature, and that curvature makes contact with observa-

tion. During crystallization the structure passes through the two-sheet tetrahedral foam

established in Section 9 (Eq. (12)): in the finite 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 deficit δ = 2π −5 arccos

1

3

≈ 0.1284 homogeneously across

the contact region (Eq. (1)). The value is fixed by the seeding-equidistance constraint

and the code stabilizers, which hold the tetrahedra regular rather than letting them relax:

among candidate stackings only the regular registration maximizes the satisfiable stabi-

lizers (Table 4, Section 9), and the frustration of five regular tetrahedra about a hinge

is exactly δ = 7.36

◦

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

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

cuboctahedral shell. A constant positive deficit is 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

, indepen-

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

the geometric signature associated with a de Sitter-like expansion, so the two-sheet foam

drives an 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 deficit vanishes: a geometric end to the deficit-driven phase with

no added mechanism.

The spectral tilt. The same deficit imprints a curvature perturbation. The dimen-

sionless fractional curvature per hinge is

ζ

hinge

=

δ

2π

=

2π − 5 arccos

1

3

2π

≈ 0.02043, (30)

the deficit measured against the full 2π of flat space. The primordial spectrum is defined

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

the Regge dual-area transfer assumption, the coefficient is fixed by the geometry of the

simplex.

The dual-area weighting is definitional, 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 deficit 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 deficit by the dual-face area is the

definition of scalar curvature in discrete general relativity, not an option exercised here.

Why the coefficient is exactly

√

3. The building blocks of the foam are regular tetrahedra,

so the hinges are edges of length L and their duals are equilateral triangular faces of

area A

h

=

√

3

4

L

2

. The

√

3 is the unavoidable signature of equilateral geometry: when

the edge length is normalized out to obtain the dimensionless coefficient connecting the

per-hinge curvature to the boundary measure, the

√

3

4

of the simplex face leaves the factor

√

3. This is the same regular tetrahedron that fixes the deficit δ = 2π −5 arccos

1

3

, so the

32

perturbation amplitude and its transfer coefficient share a single geometric origin. With

this weighting,

n

s

− 1 = −

√

3

δ

2π

=⇒ n

s

= 1 −

√

3

2π − 5 arccos

1

3

2π

= 0.9646, (31)

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 identification below, not on a completed perturbation calculation.

Why no other value is available. There is no parameter to tune. Standard cosmology

adjusts the slope of an inflaton potential V (ϕ) to set n

s

[17, 18]; here every quantity is a

fixed Euclidean constant: arccos

1

3

(the rigid dihedral angle of a regular tetrahedron), 5

(the maximal integer packing around a hinge, ⌊2π/ arccos

1

3

⌋), 2π (flat space), and

√

3 (the

equilateral dual-face area). Because the rigid foam phase admits no squished or stretched

tetrahedra, the angles and areas are locked to the exact mathematics of regular simplices,

and the coefficient cannot take another value.

The one assumption that remains. Regge’s theorem fixes 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 identification,

not a corollary of the action theorem: it assumes the perturbation propagates with the

weighting that defines the background curvature. This is the Regge-limit transfer esti-

mate, parameter-free within Regge calculus. We note one sharpening: the coefficient

√

3

does not depend on the front velocity. If the crystallization front releases each hinge’s

deficit onto the boundary as it advances, the front speed enters the release rate and the

boundary normalization identically and cancels in the dimensionless coefficient, which

reduces to the static equilateral face-area Jacobian

√

3 irrespective of how fast the front

moves. The transfer coefficient 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 identification 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 deficit already load-bearing in the model, so the tilt has the right mag-

nitude for the right reason (a small geometric deficit makes a small red tilt). Coefficient

from a standard prescription: the coefficient

√

3 follows from the Regge action’s dual-area

weighting (curvature as the area-weighted deficit on the dual face), so n

s

is parameter-free

within Regge calculus. Geometric, not free: the

√

3 is the static equilateral face-area Ja-

cobian of the close packing, a fixed property of the geometry rather than a tunable input

or an external import, and it is independent of the front speed (which cancels). One

identification 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 specific open step that

would make n

s

parameter-free from first principles. Presupposed: the De Sitter reading

uses the Friedmann relation H

2

∝ R

eff

and so a metric, the same positional input flagged

throughout. Two falsifiable consequences follow that distinguish this from tuned inflation:

no tensor modes at leading order (r ≪ 0.01, the deficit being scalar) and zero running

(dn

s

/d ln k ≈ 0, n

s

being a geometric constant), both consistent with current bounds [16].

33

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 cliff.

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

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

a single fixed rule. We run one Metropolis-kinetic dynamics—propose stitch and lift

wherever geometrically available, accept a birth with probability min(1, e

β ∆n

) (∆n the

bonds it forms), 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 fight at the seams.

Robustness is confirmed 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 fine-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 finite, 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.

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 floor of Section 17,

√

κm L

2

/ℏ > 14.8,

rearranges to a lower bound on the node inertia, m > 14.8

2

ℏ

2

/(κL

4

). Using the rigidity

34

requirement κL

2

≳ 36 ε and v

lat

= Lε/ℏ gives m ≳ 6.1 ε/v

2

lat

; with the light cone c = 2v

lat

,

m c

2

≳ 24 ε. (32)

A stable close-packed node must carry an effective inertia of order one hundred times the

binding scale—a derived mass–energy hierarchy, conditional on the c = 2v

lat

light cone.

Critical rigidity temperature (D). The melting condition kT/(κL

2

) < 0.028 of Sec-

tion 17 sets a critical temperature T

c

≈ 0.028 κL

2

/k

B

. Imposing the derived stiffness

κL

2

≳ 36 ε,

T

c

≳ 0.028 × 36

ε

k

B

≈

ε

k

B

= T

0

. (33)

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 deficit supplies a genuine small di-

mensionless number,

ε

geom

≡

δ

2π

=

2π − 5 arccos

1

3

2π

≈ 2.0 × 10

−2

, (34)

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

, (35)

the O(1) prefactor collapsing to near unity, so the exponential rate is essentially the cube

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

then has a definite exit time,

t

exit

= Γ

−1

ln

V

foam

V

seed

, (36)

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.

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, (37)

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)

effect: 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.

35

23 Further consequences: the acoustic spectrum and

the vacuum energy

Two quantities follow directly from the geometry and the GW-fixed stiffness κ = 2mε

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 (24)) 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

,

(38)

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 fixed by GW=light (Eq. (16)) these become v

L

∈ [0.71, 0.82] c and

v

T

∈ [0.41, 0.71] c: 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 in its tensor sector at the 15%

level at the lattice scale; full continuum isotropy, with a single c shared by all polariza-

tions, requires either angular (bond-bending) couplings beyond nearest-neighbor central

forces or a renormalization-group flow to an isotropic fixed point. We record the bare

anisotropy as a sharp quantitative target, not a solved problem.

Polycrystalline isotropization (D). The 15% directional anisotropy above 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 removes the directional dependence entirely, leaving an

isotropic effective 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

36

Table 6: 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

preferred direction, so any directional gravitational-wave birefringence is confined to sub-

grain scales, a falsifiable 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 still requires bond-bending couplings or fixed-point flow to close.

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

, (39)

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

∼

√

2 ε from the fixed 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 floor ε ≳ 18 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 offers 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 6, 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

37

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.

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

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 verified, 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 first 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 confirm 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.

38

27 Scope and open problems

Derived here: the emergent light cone c = 2v

lat

and, from the GW=light constraint,

the fixed stiffness κ = 2mε

2

/ℏ

2

; the detailed-balance q = 1/(1 + e) 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. (26)–(27); the energy ladder;

the code parameters; and the FCC/HCP selectors. Established in Part I: the kinematic

cascade, the metric wall, the finite-size scaling, and the structure-tensor isotropy. Im-

ported as established mathematics: the kissing and Kepler theorems [15, 3] underlying the

contact maxima, and the Regge prescription of Section 20. Open: (i) the registration bias

of the births is motivated by the code term lowering registered-configuration energy but

not proven quantitatively from Born–Markov rates; (ii) the rigidity factor ≳ 36 and the

floor 14.8 follow from the central-force harmonic spectrum, and a fully anharmonic 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 per-

turbation spectrum is uncomputed and its raw clustering is too red to match n

s

≈ 0.965,

so its identification with inflation is unestablished; (v) the light cone is computed in the

single-excitation sector; (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 and c =

2v

lat

; its dissipative limit assembles the structure, its rate q = 1/(1+e) derived by detailed

balance and its quantum-trajectory unraveling 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

fixed 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 floor 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, identified 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 dynamics, 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

fine-tuned; and the lift-seeded branching of sheets makes the foam-to-crystal conversion

exponential and self-terminating—an inflation-shaped geometric conversion. The same

Regge deficit that frustrates the foam imprints a primordial tilt: weighted by its dual face

area, it gives the Regge-limit estimate n

s

= 1 −

√

3 (δ/2π) = 0.9646, numerically close

to the Planck value, though deriving the perturbation transfer from the build dynamics

39

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 first-principles metric and spectral index would follow,

remains the frontier.

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