K=12 Lattice Saturation and the [[192,130,3]] CSS Code from a Bell-Pair Triangle: A Kinematic Growth Model

K = 12 Lattice Saturation and the

[[192, 130, 3]] CSS Code from a Bell-Pair Triangle:

A Kinematic Growth Model

Raghu Kulkarni

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

raghu@idrive.com

Abstract

Discrete kinematic models of spacetime, such as tensor networks and causal dy-

namical triangulations, frequently encounter the problem of geometric frustration:

local probabilistic assembly rules tend to yield disordered, porous graphs rather than

structurally saturated volumetric bulks [1]. In this study, we investigate the kine-

matic growth of a discrete spatial network governed by a strict exclusion radius and

proximity entanglement rules. We demonstrate computationally that imposing a

strongly 2D-dominant lateral growth algorithm—where out-of-plane nodal projec-

tions are exponentially suppressed by the codimension diﬀerence between the 2D

Stitch (1-parameter solution manifold) and 3D Lift (0-parameter pair) operators,

yielding P

lift

= e

−3

—resolves this geometric frustration. This suppression ampli-

tude receives a second, independent motivation from quantum error correction: the

emergent FCC lattice supports a [[192, 130, 3]] CSS code [15] whose distance d = 3

sets the survival threshold for out-of-plane nodes at exactly e

−d

= e

−3

. Under

these constraints, the three-dimensional network undergoes the kinematic cascade

K = 1 → K = 6 → K = 4 → K = 12, escaping the geometrically frustrated K = 4

tetrahedral foam (Regge deﬁcit δ ≈ 7.36

◦

) and crystallizing into a polycrystalline

Face-Centered Cubic (FCC) lattice that saturates the Kepler kissing-number bound

(K = 12) [3]. A bulk-interior diagnostic—based on the strict crystallographic cri-

terion that a node has at least 42 neighbors within radius 2L (its three full FCC

coordination shells)—establishes that the bulk modal coordination is exactly K = 12

at every system size from N = 250 upward, providing direct (non-extrapolated) evi-

dence of FCC bulk crystallization. The all-node K = 12 fraction follows the surface-

to-volume scaling f = 1−(6.8±0.6) N

−1/3

, conﬁning under-coordinated nodes to the

cluster surface (30 seeds per size). A parameter sweep over lift probability identiﬁes

an optimal band p ∈ [3%, 5%] for the volumetric yield Φ = f

K=12

× (z

ext

/xy

ext

with e

−3

sitting inside this band; both lower and higher lift rates are clearly de-

graded. The simulation demonstrates exact layer planarity (σ

< 10

−10

L) and

uniform FCC inter-layer spacing (0.8165 ±0.0011 L). We analyze the network’s sen-

sitivity to the exclusion radius, identifying a strict geometric failure threshold at

= 1/

√

3 L that induces structural jamming, with a wide stability plateau across

∈ [0.58, 0.99] L conﬁrming that operating values are not ﬁne-tuned.

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 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 entangled

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 =

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.

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

= 0.95 L and

proximity bond radius R

= 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

∈ [0.58, 0.99] L

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

= 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

stitch

is exponentially suppressed in the codimension

diﬀerence:

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.

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 [15, 10], 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 [15]. 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

−(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:

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

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

∈ [0.58, 0.99] L,

far wider than any plausible tuning window, so the speciﬁc value R

= 0.95 L is not a

ﬁne-tuned parameter.

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

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

Regge-deﬁcit tolerance window of Section 2.1.

Parameter Value Derivation

Unitary Metric (L) 1.0 Invariant relational distance

Lateral Height

√

L ≈ 0.866 L Equilateral triangle altitude

Lift Height

L ≈ 0.816 L Regular tetrahedron altitude

Lift Probability e

−3

≈ 4.98% Codimension suppression /

QEC (Sections 2.2–2.3)

Proximity Bond (R

) 1.05 L Regge deﬁcit (δ ≈ 7.36

◦

Eq. (1))

Hard Shell (R

) 0.95 L Regge deﬁcit (Section 2.1);

plateau veriﬁed in Section 3.8

Geometric Cutoﬀ

√

L ≈ 0.577 L Circumradius of unitary trian-

gle

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.

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

ability (N = 1000, 30 seeds). Φ = f

K=12

×(z

ext

/xy

ext

) penalizes both poor crystallization

and failure to percolate into a 3D bulk.

Regime K=12 (%) Layers Aspect z/xy Φ

1% Lift 24.5 ± 5.8 13.4 ± 2.1 0.59 ± 0.12 0.145 ± 0.045

3% Lift 27.3 ± 6.1 15.4 ± 3.2 0.76 ± 0.16 0.207 ± 0.064

5% Lift (e

−3

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

10% Lift 20.7 ± 6.2 25.2 ± 9.8 0.84 ± 0.10 0.174 ± 0.056

15% Lift 18.0 ± 4.5 33.3 ± 9.3 0.90 ± 0.11 0.162 ± 0.045

30% Lift 7.9 ± 3.1 48.3 ± 5.7 0.92 ± 0.10 0.073 ± 0.030

50% Lift 1.5 ± 0.7 59.4 ± 4.2 0.96 ± 0.09 0.014 ± 0.007

85% Lift 1.5 ± 0.4 64.8 ± 4.6 0.99 ± 0.11 0.015 ± 0.004

Raw K=12 percentage alone does not capture the quality of the 3D structure. At 1% lift,

the lattice achieves 24.5 ± 5.8% K=12—comparable to 5%—but only 13 layers with an

aspect ratio z/xy = 0.59: the structure is a 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

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.

0 10 20 30 40 50 60 70 80 90

Lift Probability (%)

Fraction of Nodes (%)

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

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

(Vol. Yield)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Aspect ratio

Aspect

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

= 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.

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

f(K=12) = 1 −

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

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-

200 400 600 800 1000 1200 1400

K=12 (%)

(a) Scaling: = 6.8 ± 0.6

= 1 6.8/

1/3

K=12 (%)

300 400 500 600 700 800 900 1000

(b)

Kepler bound

K=12

K=6

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.

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

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

√

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.

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-

10.0

7.5

5.0

2.5

0.0

2.5

5.0

7.5

10.0

7.5

5.0

2.5

0.0

2.5

5.0

7.5

10.0

7.5

5.0

2.5

0.0

2.5

5.0

7.5

10.0

(a) N=3000 cluster, color by K

7.5 5.0 2.5 0.0 2.5 5.0 7.5 10.0

7.5

5.0

2.5

0.0

2.5

5.0

7.5

(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.

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

≈ 0.99 at N = 1000 with bulk mean

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.

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 =

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

< 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

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.

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

0 5 10 15 20 25

Layer index

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.

(L)

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

< 10

−10

L: exact ﬂatness.

2 4 6 8 10 12

Coordination K

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.

(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

= 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

(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

∈ [0.58, 0.99]. The breakdown at R

≈ 0.58 corresponds to

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

= 0.95 is not ﬁne-tuned.

0 2 4 6 8

Inter-layer bonds per node

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

(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.

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 [15, 10],

X-stabilizers act on octahedral voids (each touching the 12 edges connecting its 6 sur-

rounding 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 [15], 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

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 same combinatorial counts

(V = 13, E = 36, F = 38) appear as inputs to the companion mass-spectrum framework

of [16]. The encoding rate k/n = 67.7% means two-thirds of the lattice degrees of free-

dom 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

0.5 0.6 0.7 0.8 0.9 1.0

Exclusion radius

(units of

)

Maximum coordination

max

Unphysical

overlap

(

K >

)

Lattice

freezing

FCC saturation

(Kepler bound)

Geometric phase transition at the metric wall

K = 12 plateau

≈

577

max

(simulation)

Figure 8: Geometric phase transition at the metric wall R

= L/

√

3 (N = 500, 30

seeds, p

lift

= 0.05). The maximum coordination K

max

= 12 is maintained across the

entire stability plateau R

∈ [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

max

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

that the operating value R

= 0.95 L is not ﬁne-tuned.

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 struc-

tural identity or is accidental remains an open question that may be addressable through

a generalized construction at other values of d.

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—

following the construction laid out in [16].

Step 1: Exact spatial isotropy of the FCC bond set. The K = 12 FCC nearest-

neighbor bond vectors are

∈



(±1, ±1, 0), (±1, 0, ±1), (0, ±1, ±1)



√

2, j = 1, . . . , 12. (6)

Deﬁne the rank-2 structure tensor S

µν

≡

j=1

. By direct enumeration:

= 4 ×

|{z}

(±1,±1,0)

+ 4 ×

|{z}

(±1,0,±1)

+ 0

|{z}

(0,±1,±1)

= 4, (7)



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



+ 0 + 0 = 0, (8)

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

= S

= 4 and

= S

= 0. Therefore

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

≡

vanishes exactly because the FCC bond set is

centrosymmetric: every n

has a partner −n

, so all odd-power sums cancel,

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)

= κ

j=1



1 − cos(k · n



. At long wavelengths (|k|a ≪ 1),

expanding the cosine,

ω(k)

≈

κa

j=1

(k · n

)

κa

µν

= 2κa

|k|

, (11)

so ω = c

lat

|k| with c

lat

= a

√

2κ. The dispersion is exactly isotropic at leading order: this

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

corrections appear only at O(k

), suppressed by (|k|a)

∼ (E/M

)

for Planck-scale

lattice spacing a ∼ ℓ

Step 3: Emergence of Lorentz boosts. A lattice Hamiltonian whose long-wavelength

dispersion is isotropic and linear in |k| 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

)

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

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

= ⟨(x

−¯x

)(x

−¯x

)⟩. 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 [15] provides

the error-correction framework whose distance d = 3 independently determines the same

suppression amplitude.

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

< 10

−10

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

(4) a sharp geometric phase transition at R

= 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 Ta-

bles 2–3 and the layer-structure data behind Figures 5–7: https://github.com/

raghu91302/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/

ssmtheory/blob/main/ssm_bulk_analysis.py.

Declarations

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

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