Emergent Face-Centered Cubic Vacuum from
Discrete Entanglement Networks
Forced
K=12
Saturation and an Emergent Light Cone
Raghu Kulkarni
SSMTheory Group, IDrive Inc., Calabasas, CA 91302, USA
raghu@idrive.com
Abstract
We ask what macroscopic geometry follows if the vacuum is a discrete entangle-
ment network rather than a continuum, and show that a close-packed (
K=12
, face-
centered-cubic) structure, an emergent light cone, and approximate Lorentz invari-
ance arise from the network's own growth and dynamics instead of being imposed.
The treatment has two parts.
Part I (kinematic construction; consolidates and extends the growth simulation in-
troduced in Ref. [21]).
Local probabilistic assembly rules for discrete space tend to
jam into porous, geometrically frustrated graphs rather than saturated volumetric
bulks [1]. We show computationally that a two-operator growth lawa planar stitch
and a rare out-of-plane lift, suppressed by a factor
e
−3
(motivated kinematically by
the codimension dierence between the operators, and obtained in Part II from a
transition-state ansatza
3ε
exposed-bond barrieras the thermal factor of the
lift rate, with a free geometric prefactor)drives a discrete entanglement network
through the cascade
K=1 → 6 → 4 → 12
, past the frustrated
K=4
tetrahedral
foam (Regge decit
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
updirect, non-extrapolated evidence of a close-packed, predom-
inantly twelve-coordinated bulk consistent with FCC crystalliteswith a leading-
order surface-to-volume trend
f ≈ 1−αN
−1/3
(eective amplitude
α = 6.8±0.6
over
the measured range
N=250
1000
) and a wide exclusion-radius plateau conrming
the operating values are not ne-tuned. An ideal (defect-free, periodic) bulk patch
of the emergent edge lattice carries an explicit
[[192, 130, 3]]
CSS code at
67.7%
rate [20].
Part II (dynamical foundations).
We develop the open-system dynamics beneath
the growth: a single bond Hamiltonian whose dissipative limit assembles the lat-
tice at the detailed-balance rate
q = 1/(1 + e)
at the operating point
k
B
T
0
= ε
,
and whose coherent limit propagates a ballistic signal cone with front speed
c =
4
√
2 v
lat
. We prove
K=12
is a
forced
terminus (largest cuboctahedral vacancy at
R
∗
=
p
2 −
√
2 L
), factorize the lift rate into a xed thermal factor
e
−3
and a free ge-
ometric amplitude, and derive a rigidity scale
κL
2
≳ 36 ε
that a single-scale vacuum
cannot meet. Requiring the light cone to coincide with the elastic trace speedthe
closest lattice statement of the observed equality of gravitational-wave and photon
1
speedsxes the stiness,
κ = 16 mε
2
/ℏ
2
, and with rigidity oors the binding scale
at
ε ≳ 2.3 ℏ
2
/(mL
2
)
. A conditional Regge-curvature estimate of the scalar spectral
tilt closes the paper.
Part I
Kinematic Construction
1 Introduction
The constructive generation of three-dimensional macroscopic spaces from discrete, fun-
damental components is a central problem in statistical mechanics, network theory, and
discrete models of quantum gravity [1, 4]. A persistent obstacle is geometric frustration.
When discrete nodes are assembled via purely randomized three-dimensional probabilis-
tic rules, the resulting structures typically resemble diusion-limited aggregation [2] or
kinematically jammed, fractal-like foams. These porous congurations 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 dened 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 signicant computational
challenge. In this paper, we explore a resolution to this geometric bottleneck by altering
the dimensional probabilities of the generative kinematics. Motivated by holographic mod-
els of boundary-volume correspondence [4, 5], we hypothesize that saturated 3D volumes
can be deterministically generated if the underlying assembly process is overwhelmingly
two-dimensional. We introduce a discrete simulation based on the Selection-Stitch Model
(SSM), seeded by a Bell-pair triangle (three unit-distance bonds joining three nodes, the
minimal closed simplicial unit) and grown through two primary topological operators: a
2D lateral expansion (the Stitch) and a rare, out-of-plane 3D projection (the Lift).
The relative amplitude of these operators is taken as
P
lift
= e
−3
≈ 4.98%
, motivated by
the codimension dierence between their solution manifolds (Section 2.2) and consistent
with the QEC survival ansatz of the
[[192, 130, 3]]
CSS code that the emergent lattice
supports (Sections 2.3 and 4); Part II derives this value as the thermal factor of the lift
rate, the geometric prefactor remaining free (Section 16). Under these constraints, the
system traces a kinematic cascade
K = 1 → K = 6 → K = 4 → K = 12
, escaping
the geometrically frustrated
K = 4
tetrahedral foam [7] and crystallizing into a polycrys-
talline FCC lattice that saturates the Kepler bound. We computationally verify that this
kinematic regime naturally drives the network through the cascade, bypassing geometric
frustration to yield a highly ordered, polycrystalline FCC geometry.
Relation to Ref. [21].
The growth simulation at the center of Part I was introduced in
Ref. [21], where it served as the supporting substrate for that paper's primary subject
the identication of matter with defects trapped in the close-packed vacuumand was
2
accordingly compressed. The simulation and its derivatives warrant a consolidated treat-
ment of their own, which this two-part paper provides. Part I therefore deliberately
reproduces, with expanded methodology, the kinematic denitions, the
e
−3
codimension
argument, the 30-seed dataset with its nite-size scaling, and the sensitivity sweeps of
Ref. [21], so that the dynamical derivations of Part II rest on a self-contained foundation
rather than a citation chain. All previously reported material is reused by the author with
attribution; Table 3 states exactly which columns are reproduced unchanged and which
are re-analyzed here (the revised bulk-interior columns supersede the published values).
The quantum error-correction reading of the lift amplitude (Sections 2.3 and 2.4) is new
to this paper, and Part II is entirely new.
Interactive 3D Visualization.
A 31-frame animated WebGL application
showing step-by-step lattice growthfrom the seed triangle through 2D sheet
expansion to
e
−3
out-of-plane lift eventsis 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 36. Controls: play/pause,
frame stepping, drag to orbit, scroll to zoom.
2 Methodology
The simulation algorithm is fully specied in this section; a reference implementation
is available at the URLs in the Data Availability statement. All simulations use
open
boundary conditions
: the lattice grows freely from the seed triangle without periodic
wrapping, xed walls, or any imposed simulation cell. The cluster terminates at a free
surface determined dynamically by the kinematics, and under-coordinated nodes therefore
reside at this free surfacea 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 23) report means
±
standard deviations over 30 independent seeds. Illustrative
single-run gures use seed 42.
2.1 Generative operators and proximity bonding
The Bell-pair primitive and the triangular seed.
We treat the unit Bell-pair
bond(two nodes joined at distance
L
, the discrete structural analog of a maximally entan-
gled pair)as the primitive entanglement element of the network. The simulation begins
from three connected Bell-pair bonds forming an equilateral triangle of side
L
; this Bell-
pair triangle is the minimal closed simplicial unit that the kinematic operators below can
act on, and the FCC lattice grows from it. We use Bell pair here in a structural rather
than dynamical sense: the simulation is a classical graph model and does not implement
gauge elds, a Wilson action, or a Hamiltonian. The connection to lattice gauge theory
is purely geometricthe triangle is the minimal closed loop on a simplicial complex, and
we adopt this terminology to motivate the structural operators.
The vacuum lattice grows from this seed via two kinematic operators.
3
Stitch (2D Expansion):
In gauge theory, a single open link is not a physical observable;
only closed loops are gauge-invariant [6]. The minimal physical entanglement structure on
a simplicial complex is the triangle. The Stitch operator generates these minimal closed
gauge-invariant loops by placing a new node at the equilateral apex of an existing edge,
growing planar
K = 6
hexagonal sheets. Geometrically, the Stitch realizes the intersection
of two unit spheres centered on the existing edge endpointsa 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 verticesa 0-parameter
(0D) solution manifold consisting of exactly two points (one selected by orientation).
Minimal operator basis.
Under the stated assumptionsinsertion of a single node at
a generic intersection of unit spheres, at xed unit bond lengthStitch and Lift form the
minimal local operator basis that strictly increases connectivity. Cooperative multi-node
insertions, bond rearrangements and swaps, non-generic four-anchor sites, and collective
registration moves lie outside this basis; no claim of exhaustiveness is made for them. The
Stitch (2-sphere intersection) yields a 1D family; the Lift (3-sphere intersection) yields a
0D pair; a four-sphere intersection in 3D is generically empty, so no third operator with
strictly greater connectivity can be dened. Together, Stitch and Lift form a complete
kinematic basis for
generic single-node insertion
in three-dimensional Euclidean space at
xed bond lengthgrowth through two- or three-anchor sphere intersections; the multi-
node and rearrangement moves excluded above lie outside this basis.
Proximity Bonding:
As the network expands, any two nodes within a threshold radius
(
1.05 L
) automatically bind. This mechanism allows independent, adjacent 2D layers
to geometrically interlock, producing the
6(
in-plane
) + 3(
above
) + 3(
below
) = K = 12
cuboctahedral coordination of the FCC lattice (Figure 4).
The
K = 4 → K = 12
cascade and its closure.
Acting on the Bell-pair triangle,
the two operators drive a kinematic cascade through the coordination phases
K = 1
(the
elementary Bell-pair entanglement bond, the per-node degree of an isolated pair)
→ K = 6
(stitched hexagonal sheet)
→ K = 4
(tetrahedral foam, the conguration produced by
lifts before proximity bonding closes the layers)
→ K = 12
(interlocked ABC-stacked
FCC). The
K = 4
tetrahedral foam is geometrically frustrated: regular tetrahedra cannot
tile three-dimensional Euclidean space because the dihedral angle of a regular tetrahedron
is
arccos(1/3) ≈ 70.528
◦
, and packing ve tetrahedra around a shared edge consumes only
5 × 70.528
◦
= 352.64
◦
, leaving an irreducible Regge decit [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 decit.
The exclusion radius
R
ex
= 0.95 L
and
4
proximity bond radius
R
b
= 1.05 L
are not independent free parameters: they form a
symmetric
±5%
tolerance window around the unit bond length, set directly by the Regge
decit. The wedge angle
δ ≈ 0.128
rad projects to a fractional bond-length jitter of
|sin(δ/2)| ≈ 0.064
at each node; the adopted symmetric window of half-width
5%
is of
this natural size (slightly tighter than the full jitter), and the wide plateau of Section 3.8
shows that nothing depends on the precise width. The robustness of
K = 12
saturation
across the entire band
R
ex
∈ [0.58, 0.99] L
(Section 3.8, Figure 8) further conrms that
the specic 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 dier 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
dierence:
P
lift
= e
−S
= e
−3
≈ 0.04978,
(2)
where
S = 3
counts the independent codimensional constraints lifting the operator from
a continuous (1D) family to an isolated (0D) pair. The terminus itself does not select
this value:
K=12
is forced for any lift rate by the kissing bound (Part II, Section 16).
What the lift rate controls is the growth
morphology
: smaller
P
lift
produces at, poorly
interlocked sheet structures, while larger
P
lift
produces foams with poor crystalline order
(Table 2). We treat equation (2) as a structural/kinematic motivation rather than a
derivation from a Euclidean tunneling action; the simulation's value of
p
lift
= 0.05
is
taken from this codimension argument and tested against the volumetric-yield analysis of
Section 3.1.
2.3 Quantum error-correction interpretation of
e
−3
The codimension-suppressed amplitude (2) receives a second, independent motivation
from the quantum error-correcting code that the emergent lattice supports. The FCC
lattice at the smallest system supporting this code (a
4×4×4
unit-cell arrangement, 192
edges) carries a
[[192, 130, 3]]
CSS quantum error-correcting code [10, 20], with 192 phys-
ical qubits (edges), 130 logical qubits, encoding rate
k/n = 67.7%
, and code distance
d = 3
. The X-stabilizers act on octahedral voids (each touching the 12 edges of one oc-
tahedral cell) and the Z-stabilizers act on vertices (each touching the 12 incident edges);
both families have uniform weight 12. The minimal closed gauge-invariant loops on the
simplicial complex, the triangular faces of the local coordination cluster [6], underlie the
compound error-detection argument that follows. A node participating in
t
triangular
stabilizer checks is protected by
t
independent error-detection circuits. Below the correc-
tion threshold, the error suppression scales exponentially with the decit. This motivates
the survival ansatz (a heuristic, not a consequence of code distance:
d = 3
guarantees
correction of arbitrary weight-one errors, and no decoding model, noise channel, syndrome
5
dynamics, or threshold calculation is claimed to produce Eq. (3)):
P (
survive
|t) =
(
1
if
t ≥ d + 1
e
−(d+1−t)
if
t < d + 1
(3)
The exponential form is qualitatively consistent with threshold behavior in topological
codes [11], where logical errors below threshold are suppressed exponentially in the syn-
drome decit by direct analogy with thermal activation across a free-energy barrier in
the underlying random-bond Ising mapping of the toric code. The specic 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 decit
d+ 1−t
, and (iii)
recovers the textbook below-threshold scaling
P ∝ e
−∆
in the large-decit 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 specic functional form, depending only on the monotonic increase
of eective protection with neighborhood depth. A node produced by a tetrahedral Lift
participates in exactly
t = 1
triangle. At code distance
d = 3
:
P
lift
= e
−(d+1−1)
= e
−d
= e
−3
(4)
The same value as the codimension argument (2). The two routes are consistent rather
than independent: both exponents count the same three simultaneous unit-distance con-
straints (the three anchoring bonds of the tetrahedral lift), read kinematically in one case
and as missing stabilizer protection in the other.
2.4 Compound QEC: why at growth dominates
The QEC framework explains why 2D growth dominates without being externally pre-
scribed. Throughout this argument the triangular plaquettes serve as
heuristic
local
checks: they are the closed gauge-invariant loops of the lattice, not stabilizer generators
of the
[[192, 130, 3]]
code itself, whose checks are the weight-12 vertex and octahedral op-
erators (Section 4). What follows is a motivation for the lift suppression, not a property
of the stated code.
The in-plane node.
A node in the interior of a hexagonal sheet has
K = 6
neighbors and
participates in 6 triangles. Each triangle shares 2 edges with neighboring triangles. An
error at the central node triggers a syndrome at all 6 surrounding stabilizer checks. The
error is also independently detectable through the triangles of the node's second-nearest
neighbors: each of the 6 neighbors participates in 5 additional triangles beyond the one
shared with the central node, giving an additional set of compound detection paths within
2 hops. The eective protection is well above the
d + 1 = 4
threshold, and sheet-interior
nodes survive at eectively 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 eective triangle count is
t
eff
≈ 1
.
The asymmetry.
The ratio of eective protection is much larger than the raw triangle-
count ratio of
6:1
, because compound detection paths grow with distance from the bound-
6
ary. A node
n
hops into the sheet interior accumulates compound detection paths from all
triangles within
n
-hop neighborhoods, while the out-of-plane peninsula remains capped
at 1. This makes any threshold-based selection rule, not just the specic formula (3)
preferentially destroy out-of-plane protrusions while preserving the sheet. The compound
QEC argument is structurally stable: it does not depend on the exact functional form of
the survival probability.
2.5 Summary of kinematic parameters
Every variable in the simulation is derived from foundational geometry: the lateral and
lift heights are unique altitudes of regular triangles and tetrahedra, the lift probability
is motivated by codimension suppression and by the QEC distance (and is derived in
Part II as a thermal factor with a free geometric prefactor), and the proximity-bond
and exclusion radii together form the symmetric
±5%
Regge-decit tolerance window
derived in Section 2.1. Table 1 provides the complete kinematic parameter space. The
robustness of the operating values is conrmed 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 specic value
R
ex
= 0.95 L
is not a
ne-tuned parameter.
Table 1: Kinematic parameters. All values are geometrically or thermodynamically de-
termined; the proximity bond and exclusion radii together form the symmetric
±5%
Regge-decit 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 2.22.3)
Proximity Bond (
R
b
)
1.05 L
Regge decit (
δ ≈ 7.36
◦
,
Eq. (1))
Hard Shell (
R
ex
)
0.95 L
Regge decit (Section 2.1);
plateau veried in Section 3.8
Observed kinetic cuto
1
√
3
L ≈ 0.577 L
Circumradius of unit trian-
gle; operator-kinetic wall (Sec-
tion 3.8)
3 Results: Kinematic Saturation and FCC Registry
3.1 Morphological dependence on lift probability
We conducted a parameter sweep across the probability of the 3D Lift operator. As shown
in Table 2 and Figure 1, all entries are means
±
standard deviations over 30 independent
7
random seeds.
Table 2: Lattice saturation, volumetric yield, and cluster shape as a function of lift prob-
ability (
N = 1000
, 30 seeds).
Φ = f
K=12
×(z
ext
/xy
ext
)
penalizes both poor crystallization
and failure to percolate into a 3D bulk.
Regime
K=12
(%) Layers Aspect
z/xy Φ
1% Lift
24.5 ± 5.8 13.4 ± 2.1 0.59 ± 0.12 0.145 ± 0.045
3% Lift
27.3 ± 6.1 15.4 ± 3.2 0.76 ± 0.16 0.207 ± 0.064
5% Lift (
e
−3
)
25.4 ± 5.4 19.2 ± 6.7 0.83 ± 0.14 0.211 ± 0.057
10% Lift
20.7 ± 6.2 25.2 ± 9.8 0.84 ± 0.10 0.174 ± 0.056
15% Lift
18.0 ± 4.5 33.3 ± 9.3 0.90 ± 0.11 0.162 ± 0.045
30% Lift
7.9 ± 3.1 48.3 ±5.7 0.92 ± 0.10 0.073 ± 0.030
50% Lift
1.5 ± 0.7 59.4 ±4.2 0.96 ± 0.09 0.014 ± 0.007
85% Lift
1.5 ± 0.4 64.8 ±4.6 0.99 ± 0.11 0.015 ± 0.004
Raw
K=12
percentage alone does not capture the quality of the 3D structure. At 1%
lift, the lattice achieves
24.5 ±5.8% K=12
(comparable to 5%) but only 13 layers with an
aspect ratio
z/xy = 0.59
: the structure is a at pancake that has failed to percolate into a
volumetric bulk. At 85% lift, the structure is nearly spherical (
z/xy = 0.99
) with 65 lay-
ers, but poorly crystallized (
1.5% K=12
). The volumetric yield
Φ = f
K=12
× (z
ext
/xy
ext
)
captures both requirements: high crystallization
and
3D volumetric extent. The prod-
uct (rather than sum) form is essential.
Φ
vanishes whenever
either
factor vanishes: a
perfectly crystallized but at sheet (high
f
K=12
, aspect
→ 0
, the low-lift regime) and a
spherical but uncrystallized foam (aspect
→ 1
, low
f
K=12
, the high-lift regime) are both
correctly identied as failures. A sum
f
K=12
+(z
ext
/xy
ext
)
would reward either achievement
independently and could rank a at sheet of perfect crystallinity above a moderately crys-
tallized 3D bulk, missing the physical point that both criteria are required simultaneously.
The multiplicative form is the simplest functional that vanishes whenever either require-
ment fails and grows monotonically when both improve, making it a natural composite
gure of merit for crystallized 3D bulk. The peak in
Φ
falls in the band
p ∈ [3%, 5%]
(with
Φ = 0.207 ± 0.064
at 3% and
0.211 ± 0.057
at 5%, statistically indistinguishable
within the 30-seed error bars), and the codimension-suppressed value
e
−3
≈ 4.98%
sits
squarely inside this optimal band.
Φ
degrades clearly below this band (
0.145 ± 0.045
at
1%, dominated by at-pancake structure) and above it (
0.073 ±0.030
at 30%, dominated
by poor crystallization), establishing
e
−3
as a value consistent with the structural opti-
mum: the lift rate at which the system simultaneously achieves crystalline order and 3D
bulk percolation. With only a coarse grid of tested lift rates, the data locate the optimum
only within a broad band of roughly
2
7%
; a denser sweep with condence intervals on the
optimum's position, alternative objective functions, and larger
N
is left to future work.
3.2 Finite-size scaling and the thermodynamic limit
The simulation was executed at four system sizes (
N = 250
1000
) with identical parame-
ters, each repeated over 30 independent random seeds. All results in Table 3 report means
±
standard deviations.
8
0 10 20 30 40 50 60 70 80 90
Lift Probability (%)
0
10
20
30
40
50
Fraction of Nodes (%)
(a) Lattice saturation (N=1000, 30 seeds)
e
3
5.0%
K = 12 (%)
K 10 (%)
0 10 20 30 40 50 60 70 80 90
Lift Probability (%)
0.00
0.05
0.10
0.15
0.20
0.25
Volumetric Yield
(b) Volumetric yield peaks at
e
3
e
3
(Vol. Yield)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Aspect ratio
z
/
xy
Aspect
z
/
xy
Figure 1: (a)
K=12
saturation vs lift probability (30 seeds,
1σ
error bars). Raw
K=12%
is similar for
p ≤ 5%
. (b) Volumetric yield
Φ
(green diamonds) reaches its optimal band
at
p ∈ [3%, 5%]
encompassing
e
−3
; aspect ratio
z/xy
(purple triangles) reveals that low-
lift structures are at pancakes.
At leading order the data follow the surface-to-volume form [21]
f(K=12) = 1 −
α
N
1/3
, α = 6.8 ± 0.6
(5)
as shown in Figure 2. This functional form has a direct geometric interpretation: under-
coordinated nodes (
K < 12
) reside exclusively at the cluster boundary, whose node count
scales as
N
2/3
. The bulk interior, scaling as
N
, achieves full
K = 12
saturation. The
under-coordinated fraction therefore scales as
N
2/3
/N = N
−1/3
.
Equation (5) is a leading-order description over a factor of four in
N
rather than a con-
verged asymptotic law: the eective amplitude
α
eff
(N) = (1−f) N
1/3
rises monotonically
from
5.8
at
N=250
to
7.5
at
N=1000
, and the one-term form would give
f < 0
below
N ≈ α
3
≈ 310
; a subleading
N
−2/3
(edge) correction brings the t within the quoted
errors. We therefore quote
α = 6.8 ± 0.6
as the mean eective amplitude across the
measured range; larger systems (
N
up to
∼ 1.6 × 10
4
) would materially strengthen the
extrapolation and are the natural next step. Extrapolating the leading-order form gives
f → 1
in the thermodynamic limit (
N → ∞
), consistent with complete FCC saturation
as the asymptotic ground state of the kinematics. Power-law ts over so narrow a range
are more easily believed when the functional form is geometrically motivated, as it is
here, but the central evidence that the bulk is close packed at modal coordination
K=12
comes not from extrapolation but from the bulk-interior diagnostic introduced next. The
last column of the all-node summary in Table 3 shows the macroscopic shape isotropy
λ
min
/λ
max
of the spatial covariance matrix, rising steadily toward 1.0 with increasing sys-
tem sizeconsistent with the expectation that larger polycrystalline clusters average over
more grain orientations.
9
Table 3: Coordination statistics across system sizes (
p
lift
= 0.05
,
R
ex
= 0.95 L
, 30 seeds
each). All columns are computed from the same simulation runs: the bulk-interior
columns are obtained by re-analyzing the lattice output under the criterion of Sec-
tion 3.3a node is classied as bulk-interior i at least 42 other nodes lie within radius
2L
(the count of an ideal FCC interior node's neighbors in its rst three coordination
shells,
12 + 6 + 24
, at distances
L
,
√
2 L
, and
√
3 L
). The threshold 42 is geometrically
derived from FCC crystallography, not tuned. The bulk-restricted columns supersede the
corresponding columns of Table 2 in Ref. [21]: the all-node columns are identical (com-
puted from the same 30-seed simulation output), while the bulk-interior analysis script
was revised after that publication.
N
¯
K K=12
(all, %)
λ
min
/λ
max
K=12
(bulk, %) Bulk modal
K
250
7.41 ± 0.33 8.2 ± 3.7 0.39 ± 0.17 49.3 ± 13.0
12
500
8.20 ± 0.28 15.9 ± 4.7 0.46 ± 0.17 56.0 ± 11.1
12
750
8.64 ± 0.26 21.8 ± 5.1 0.51 ± 0.16 61.7 ± 8.9
12
1,000
8.90 ± 0.26 25.4 ± 5.4 0.54 ± 0.15 64.2 ± 8.8
12
3.3 Bulk-interior diagnostic: direct evidence of close-packed bulk
The all-node
K = 12
percentages in Table 3 are mixed measurements: they include free-
surface nodes that are intrinsically under-coordinated and that have no physical analog
in the cosmological setting, where the observable universe sits in a bulk regime far from
any boundary. The physically relevant question is not what fraction of
all
nodes have
reached
K = 12
, but whether the
bulk interior
has done so.
We extract this directly via a strict geometric criterion: a node is bulk-interior i at least
42 other nodes lie within radius
2L
of it, the population of an ideal FCC interior node's rst
three coordination shells. This is a population criterion, not a crystallographic classier:
it matches the shell count of ideal FCC but does not by itself verify shell radii, angular
registry, or ABC (rather than ABAB) stacking, and coordination
K=12
is shared by FCC
and HCP. The results below should accordingly be read as establishing a close-packed,
predominantly twelve-coordinated bulk; the crystallographic FCC assignment rests on the
interlocking construction and awaits conrmation by a standard classier. The threshold
42 is not tuned: it is exactly
12+6+24
, the count of an ideal FCC interior node's neighbors
in its rst three coordination shells (at distances
L
,
√
2 L
, and
√
3 L
, respectively). This
is a strict crystallographic criterion derived from FCC geometry; it excludes any node
whose surroundings are not yet a complete bulk environment.
Re-analyzing the same 30-seed simulation output under this criterion gives the rightmost
two columns of Table 3. Three observations:
1. The bulk modal
K
is exactly 12 at every system size from
N = 250
upward: the
close-packed character of the bulk is a direct measurement, not an extrapolation.
The crystallographic FCC label awaits the classier follow-up (Section 3.3).
2. The bulk-restricted
K = 12
fraction (e.g.,
64.2±8.8%
at
N = 1000
) is roughly
2.5×
the all-node value (e.g.,
25.4 ± 5.4%
), consistent with the surface-volume interpre-
tation of the scaling law: the under-coordinated population that drags the all-node
mean down is concentrated at the boundary.
10
200 400 600 800 1000 1200 1400
N
20
10
0
10
20
30
40
50
K=12 (%)
(a) Scaling: = 6.8 ± 0.6
f
= 1 6.8/
N
1/3
K=12 (%)
300 400 500 600 700 800 900 1000
N
5
6
7
8
9
10
11
12
13
K
(b)
K
Kepler bound
K=12
K=6
K
Figure 2: Finite-size scaling (30 seeds,
1σ
error bars, shaded band). (a)
K=12
fraction vs
system size with the leading-order form
f = 1 −6.8/N
1/3
. (b)
¯
K
approaching the Kepler
bound.
3. The bulk standard deviation is
σ
K
≈ 0.99
at
N = 1000
with bulk mean
¯
K
bulk
≈
11.40
. The bulk-interior nodes that fall short of
K = 12
are concentrated at
K =
10
,
11
rather than spread broadlya ngerprint of grain-boundary nodes between
FCC crystallites of dierent orientations, not of a coordination distribution centered
below 12. This is consistent with the polycrystalline character of the emergent
lattice.
The all-node nite-size scaling and the bulk-interior modal-
K
measurement therefore
provide independent and complementary evidence: the rst establishes that under-
coordinated nodes are conned to the surface (where they decay as
N
−1/3
), the second
establishes that the bulk itself is close packed with modal coordination exactly
K=12
at
every system size measured (the crystallographic FCC label awaiting the classier follow-
up above). Neither relies on extrapolation to
N → ∞
.
3.4 Three-dimensional lattice structure
Figure 3 shows the emergent
N = 3000
cluster. The
K = 12
saturated core (green) is
surrounded by a thin under-coordinated shell (red), consistent with the surface-to-volume
scaling law. The central hexagonal layer (
z ≈ 0
) displays the emergent hex bond topology
with perfect structural planarity.
Figure 4 illustrates the cuboctahedral coordination shell: panel (a) shows the ABC stack-
ing of three hexagonal sheets at
z = 0
,
z = +h
,
z = −h
with the focal node and its
6 + 3 + 3 = 12
neighbors, and panel (b) shows the same neighbor set as the cuboctahe-
dron polyhedron with its 8 triangular and 6 square faces.
11
10.0
7.5
5.0
2.5
0.0
2.5
5.0
7.5
10.0
x
10.0
7.5
5.0
2.5
0.0
2.5
5.0
7.5
10.0
y
10.0
7.5
5.0
2.5
0.0
2.5
5.0
7.5
10.0
z
(a) N=3000 cluster, color by K
7.5 5.0 2.5 0.0 2.5 5.0 7.5 10.0
x
7.5
5.0
2.5
0.0
2.5
5.0
7.5
y
(b) Central layer z 0 (257 nodes)
Figure 3: (a) Full
N = 3000
cluster colored by coordination: green (
K = 12
), orange
(
K = 10
11), red (
K < 10
). (b) Central hexagonal layer at
z ≈ 0
.
3.5 Layer planarity and FCC registry
A
z
-clustering analysis of the
N = 3000
lattice identies 28 well-dened planar layers
(Figure 5).
Exact structural atness.
Of the 23 layers containing
≥ 10
nodes, 22
exhibit
σ
z
< 10
−10
L
(Figure 5c). The stitch operator places each node at the exact
equilateral apex, yielding perfect planarity by construction.
Ideal FCC layer spacing.
The measured inter-layer spacing is
0.8165 ± 0.0011 L
, matching the ideal FCC value
p
2/3 L = 0.8165 L
to 0.01% (Figure 5b).
Surface shell thickness.
All
K < 12
nodes
are conned to a boundary shell of constant thickness
t
shell
≈ 1.6 L
, corresponding to
approximately 2 FCC layer spacings.
3.6 Coordination distribution
Figure 6 shows the coordination distribution at
N = 3000
. The peak at
K = 12
contains
1,141 nodes (38.0%). Under-coordinated nodes reside exclusively at the cluster surface.
3.7 Inter-layer bonding: welds vs. rivets
The 5% lift events act as topological rivets, but proximity bonding ensures rapid percola-
tion. At
N = 3000
, 99.0% of layer nodes possess inter-layer bonds, averaging 4.9 per node
(Figure 7). This far exceeds the
∼ 20%
threshold for rigidity percolation. The layers are
structurally welded, approaching the theoretical maximum of 6 inter-layer bonds for an
ideal
K = 12
site.
12
1.5
1.0
0.5
0.0
0.5
1.0
1.5
x (L)
1.5
1.0
0.5
0.0
0.5
1.0
1.5
y (L)
0.8
0.6
0.4
0.2
0.0
0.2
0.4
0.6
0.8
z (L)
Layer A (z = h)
Layer B (z = 0)
Layer C (z = +h)
(a) ABC stacking of 3 hexagonal sheets:
6 in-plane (Layer B) + 3 above (Layer C) + 3 below (Layer A) = 12 NN
Focal node (Layer B)
6 NN in plane (Layer B)
3 NN above (Layer C)
3 NN below (Layer A)
1.00
0.75
0.50
0.25
0.00
0.25
0.50
0.75
1.00
x (L)
0.75
0.50
0.25
0.00
0.25
0.50
0.75
y (L)
0.8
0.6
0.4
0.2
0.0
0.2
0.4
0.6
0.8
z (L)
(b) Cuboctahedral coordination shell:
8 triangular + 6 square faces
Triangular faces (8)
Square faces (6)
Focal node (interior)
Figure 4: The
K = 12
cuboctahedral coordination shell of an interior FCC bulk node.
(a) ABC stacking: the focal node sits in the central hexagonal sheet (Layer B,
z = 0
),
with 6 in-plane neighbors (green) plus 3 in the upper sheet (Layer C, blue,
z = +h
with
h =
p
2/3 L
) and 3 in the lower sheet (Layer A, orange,
z = −h
), reached by the rare
e
−3
Lift events. The
6+3+3 = 12
decomposition saturates the kissing-number bound. (b) The
same 12 vertices viewed as the cuboctahedron, with its 8 triangular faces (green) and 6
square faces (blue). The triangular faces are the closed gauge-invariant loops underlying
the compound-QEC argument of Section 2.4; the square faces, representing the absence
of a diagonal bond, are topologically distinct.
3.8 Sensitivity sweep and the
R
ex
= 1/
√
3
geometric cuto
To ensure
K
max
= 12
is not an artifact of a tuned exclusion radius, a ne-grain sweep
was conducted across
R
ex
(reference implementation:
plot_rex_sweep.py
in the Data
Availability statement). As shown in Figure 8, the maximum coordination of 12 is main-
tained across the entire band
R
ex
∈ [0.58, 0.99]
. The observed breakdown at
R
ex
≈ 0.58
corresponds to the circumradius of the equilateral triangle (
1/
√
3 ≈ 0.577 L
); below it the
exclusion is too weak and unit-radius packing violations appear (
K > 12
). This observed
wall is an
operator-kinetic
threshold, not the absolute geometric one: Part II proves that
a thirteenth-neighbor site exists geometrically for all
R
ex
< R
∗
=
p
2 −
√
2 L ≈ 0.765 L
(Section 15). In the intermediate band
R
ex
∈ (0.577, 0.765) L
the persistence of
K
max
= 12
is therefore kineticthe stitch and lift operators never propose the thirteenth-neighbor
site, which sits at unit distance from only one existing node. At 1.00 and above, strict
rigidity causes lattice freezing. The wide stability plateau conrms that the specic value
R
ex
= 0.95
is not ne-tuned.
13
0 5 10 15 20 25
Layer index
0
50
100
150
200
250
Nodes
(a) 28 layers
0.813 0.814 0.815 0.816 0.817 0.818 0.819 0.820
Spacing (L)
0.0
2.5
5.0
7.5
10.0
12.5
15.0
17.5
20.0
Count
(b) 0.8165 ± 0.0011 L
FCC = 0.8165
50 100 150 200 250
Layer pop.
10
14
10
12
10
10
10
8
10
6
10
4
10
2
z
(L)
(c) 22/23 exactly flat
10
10
Figure 5: Layer structure (
N = 3000
). (a) Nodes per layer. (b) Inter-layer spacings match
the FCC ideal to 0.01%. (c) 22 of 23 substantial layers have
σ
z
< 10
−10
L
: exact atness.
2 4 6 8 10 12
Coordination K
0
5
10
15
20
25
30
35
Fraction (%)
K=12: 1140
(38.0%)
Coordination distribution (N = 3000)
Figure 6: Coordination distribution (
N = 3000
). Green:
K = 12
(38.0%). Orange:
K = 10
11. Red:
K < 10
.
4 The
[[192, 130, 3]]
CSS Code
An ideal bulk patch of the emergent FCC lattice carries a mathematically veriable quan-
tum error-correcting code. The code is dened on a defect-free
4×4×4
unit-cell arrange-
ment with periodic boundariesthe single-crystal idealization of the polycrystalline clus-
ters grown here, whose grain boundaries and free surfaces host unclosed stabilizerswith
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, veried computationally in [20], are:
14
0 2 4 6 8
Inter-layer bonds per node
0
200
400
600
800
1000
1200
1400
Count
(a) Mean=4.9, 99% bonded
0 1 2 3 4 5 6 7
Inter-layer bonds
2
4
6
8
10
12
K
(b) K=12 requires ~6 IL bonds
Figure 7: Inter-layer bonding (
N = 3000
). (a) Distribution of inter-layer bonds per node;
modal value = 6 (FCC maximum). (b)
K = 12
requires
∼ 6
inter-layer bonds.
Physical qubits (
n
) 192 (edges of the FCC lattice)
Logical qubits (
k
) 130
Code distance (
d
) 3
Encoding rate (
k/n
) 67.7%
Vertices (
V
) 13 per coordination cluster
Edges (
E
) 36 per coordination cluster
Faces (
F
) 38 (
= 32
triangles
+6
squares)
The 13-node coordination cluster (the central node together with its 12 cuboctahedrally
arranged neighbors of Figure 4) decomposes as a simplicial complex into 32 triangular 2-
cells and 6 square 2-cells: 8 of the triangles are the cuboctahedron's surface faces (visible
in Figure 4b), and the remaining 24 are spoke triangles formed by each cuboctahedral edge
with the central node. The 32 triangles are the minimal closed gauge-invariant loops [6]
that underlie the compound-QEC argument of Section 2.4; the 6 squares, representing
the absence of a diagonal bond, are topologically distinct. The encoding rate
k/n =
67.7%
means two-thirds of the lattice degrees of freedom carry logical information. By
comparison, 2D topological codes on planar or toroidal geometries encode a xed number
of logical qubits regardless of system size, so their rate
k/n → 0
as
n → ∞
. The 3D
projection, rare as it is at
∼ 5%
per frontier site, is what enables an extensive encoding
rate. Three quantities coincide at the value 3: the code distance
d = 3
, the number of
independent distance constraints
S = 3
for the tetrahedral lift, and the ambient spatial
dimension. Whether this triple coincidence reects a structural identity or is accidental
remains an open question that may be addressable through a generalized construction at
other values of
d
.
15
0.5 0.6 0.7 0.8 0.9 1.0
Exclusion radius
R
ex
(units of
L
)
0
4
6
8
12
16
20
24
Maximum coordination
K
max
Unphysical
overlap
(
K >
12
)
Lattice
freezing
FCC saturation
(Kepler bound)
Geometric phase transition at the metric wall
R
ex
=
L/
p
3
K = 12 plateau
R
ex
=
L/
p
3
≈
0
.
577
L
K
max
(simulation)
Figure 8: Observed (operator-kinetic) breakdown at
R
ex
= L/
√
3
(
N = 500
, 30 seeds,
p
lift
= 0.05
). The maximum coordination
K
max
= 12
is maintained across the entire stabil-
ity plateau
R
ex
∈ [0.58, 0.99] L
. Below
L/
√
3 ≈ 0.577 L
(the equilateral-triangle circum-
radius), the exclusion is too weak and unit-radius-packing violations appear (
K
max
> 12
);
above
1.0 L
, strict rigidity causes lattice freezing. The absolute geometric threshold for
a thirteenth neighbor is
R
∗
≈ 0.765 L
(Part II); the plateau segment between
0.577 L
and
0.765 L
reects kinetic protection by the operator basis (see text). The wide plateau
conrms that the operating value
R
ex
= 0.95 L
is not ne-tuned.
16
5 Discussion
5.1 Exact rank-two spatial isotropy and conditions for an emer-
gent Lorentz limit
A persistent objection to discrete spacetime models is the apparent incompatibility be-
tween lattice regularity and continuous Lorentz invariance. If the 3D FCC lattice were
a foundational background, it would possess preferred directions (the crystallographic
axes), leaving the framework vulnerable to the Collins et al. naturalness objection [12],
which highlights that radiative corrections amplify even small tree-level Lorentz violations
into macroscopic, experimentally falsiable anomalies. We address this in three stepsan
exact algebraic isotropy of the FCC bond set at the lattice level, the resulting isotropy
of the scalar lattice dispersion at long wavelengths, and the conditions under which a
Lorentz-invariant continuum limit follows, confronted with experimental bounds.
Step 1: Exact spatial isotropy of the FCC bond set.
The
K = 12
FCC nearest-
neighbor bond vectors are
n
j
∈
(±1, ±1, 0), (±1, 0, ±1), (0, ±1, ±1)
/
√
2, j = 1, . . . , 12.
(6)
Dene the rank-2 structure tensor
S
µν
≡
P
12
j=1
n
µ
j
n
ν
j
. By direct enumeration:
S
xx
= 4 ×
1
2
|{z}
(±1,±1,0)
+ 4 ×
1
2
|{z}
(±1,0,±1)
+ 0
|{z}
(0,±1,±1)
= 4,
(7)
S
xy
=
1
2
(+1)(+1) + (+1)(−1) + (−1)(+1) + (−1)(−1)
+ 0 + 0 = 0,
(8)
and the three-fold permutation symmetry of the bond set gives
S
yy
= S
zz
= 4
and
S
xz
= S
yz
= 0
. Therefore
S
µν
= 4 δ
µν
(exact, by enumeration)
.
(9)
This algebraic identity guarantees equal propagation speed in every spatial direction for
the scalar sector at leading order in
k
. The odd-rank tensor
T
µνλ
≡
P
j
n
µ
j
n
ν
j
n
λ
j
vanishes
exactly because the FCC bond set is centrosymmetric: every
n
j
has a partner
−n
j
, so all
odd-power sums cancel,
T
µνλ
= 0
(exact, by inversion symmetry)
.
(10)
Equation (10) forbids any preferred direction and any linear-in-
k
term in the lattice dis-
persion.
Step 2: Isotropy of the dispersion relation.
For a scalar eld on the FCC lattice, the
dispersion relation is
ω(k)
2
= κ
P
12
j=1
1 − cos(k · n
j
a)
. At long wavelengths (
|k|a ≪ 1
),
expanding the cosine,
ω(k)
2
≈
κa
2
2
12
X
j=1
(k · n
j
)
2
=
κa
2
2
k
µ
k
ν
S
µν
= 2κa
2
|k|
2
,
(11)
so
ω = c
s
|k|
with
c
s
= a
p
2κ/m
, where
κ
is the bond stiness that xes rigidity in
Part II and
m
the node inertia. This scalar speed is the
trace
channel of the displacement
17