Matter as Incomplete Crystallization: Quark Charges, Color Confinement, and the Proton Mass from a Single Extra Node in the Vacuum Lattice

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Physics Open

journal homepage: www.elsevier.com/locate/physo

Matter as incomplete crystallization: Quark charges, color confinement, and

the proton mass from a single extra node in the vacuum lattice

Raghu Kulkarni

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

A R T I C L E I N F O

Handling Editor: William Barletta

Dataset link: ssm_sim.py, plot_rex_sweep.py, ss

m_bulk_analysis.py, https://raghu91302.github

.io/ssmtheory/ssm_regge_deficit.html, https://r

aghu91302.github.io/ssmtheory/ssm_quark_str

ucture.html

A B S T R A C T

We propose that baryonic matter is spacetime that failed to fully crystallize. In the Selection-Stitch Model,

the early universe undergoes a 𝐾=4 → 𝐾=12 phase transition from a frustrated tetrahedral foam to a Face-

Centered Cubic (FCC) lattice. The FCC unit cell contains eight tetrahedral voids. If one extra node — a remnant

of the 𝐾=4 phase — remains trapped in such a void, it bonds to the four surrounding FCC vertices, creating

a local 𝐾=4 pocket inside the 𝐾=12 bulk. From this single geometric fact, with no adjustable parameters,

we derive: (i) fractional electric charges −1∕3 from the regular-tetrahedron bond-angle cosine and +2∕3 from

integer winding under the bulk Bravais translation symmetry; (ii) exactly three color charges from the three

skew-edge pairs of the bounding tetrahedron; (iii) linear confinement from the 𝐿∕



3 metric wall preventing

node extraction; and (iv) the proton-to-electron mass ratio 𝑚

𝑝

∕𝑚

𝑒

= (𝐾 +1)𝐾

−𝑐

skew

𝐾 = 1836 from the structural

node count and skew-edge pair count. Every result follows strictly from unadjusted FCC crystallography. We

support these derivations with a computational verification of the underlying 𝐾=4 → 𝐾=12 phase transition

(finite-size scaling toward 𝐾=12 saturation, sharp geometric phase transition at exclusion radius 𝑅

= 𝐿∕



and a verified Lorentz-isotropic dispersion). The model is shown to be consistent with current bounds on

Lorentz violation and to reproduce the linear-confinement piece of the Cornell potential.

1. Introduction

The Standard Model treats quark charges, color confinement, and

hadron masses as empirical inputs or outputs of non-perturbative lat-

tice QCD simulations [1]. These properties have no geometric origin

within the standard framework. We argue that they require none: all

three emerge natively from a single crystallographic fact about the

Face-Centered Cubic (FCC) lattice.

The FCC unit cell contains eight tetrahedral voids — small intersti-

tial pockets where four nearest-neighbor atoms form a regular tetrahe-

dron. In the Selection-Stitch Model (SSM), the vacuum crystallizes from

a frustrated 𝐾=4 tetrahedral foam into the ordered 𝐾=12 FCC lattice.

If this crystallization is incomplete — if a single extra node from the

𝐾=4 phase remains trapped inside a tetrahedral void — it bonds to the

four surrounding FCC vertices and recreates a local 𝐾=4 tetrahedron

embedded in the 𝐾=12 bulk. That single trapped node is a baryon.

The rest of this paper unpacks that single geometric premise. Sec-

tion 2 establishes the FCC vacuum: why FCC is uniquely preferred over

BCC and HCP, a precise definition of the coordination number 𝐾, and

a computational verification of the 𝐾=4 → 𝐾=12 phase transition.

Section 3 introduces the trapped extra node. Section 4 derives the

fractional electric charges quantitatively. Section 5 derives the three

E-mail address: raghu@idrive.com.

color charges from the skew-edge pair count and discusses the rela-

tion to standard SU(3). Section 6 establishes confinement from the

metric wall and compares to the Cornell potential. Section 7 derives

the proton-to-electron mass ratio and analyzes its stability. Section 8

covers the lighter particles. Section 9 addresses Lorentz invariance and

experimental constraints. Sections 10 and 11 discuss limitations and

conclude.

Relation to the companion paper. A companion paper [2] treats particles

as defects in an FCC quantum error-correcting code and derives their

masses as fault-tolerant verification costs. That work asks what mass

is; the present work asks what matter is. The two papers share the

same underlying 𝐾=12 FCC vacuum geometry but address structurally

distinct questions: the companion paper’s central object is the verifica-

tion cost of a stabilizer code, while the present paper’s central object

is the trapped extra node and the fractional charges, color content,

confinement, and baryon spectrum it generates. The proton-to-electron

mass ratio is one observable on which the two papers contact each

other; this contact is discussed in Section 7.4.

Interactive 3D visualizations. Readers can explore the geometric con-

cepts discussed in this paper through two interactive WebGL applica-

tions:

https://doi.org/10.1016/j.physo.2026.100423

Received 13 March 2026; Received in revised form 3 May 2026; Accepted 11 May 2026

Physics Open 27 (2026) 100423

Available online 15 May 2026

nc-nd/4.0/ ).

R. Kulkarni

1. Spacetime Crystallization: the 𝐾=6 → 𝐾=4 → 𝐾=12 phase

transition sequence, available at https://raghu91302.github.io/

ssmtheory/ssm_regge_deficit.html.

2. Matter as a Topological Defect: a direct interactive model isolating

the single trapped extra node, at https://raghu91302.github.io/

ssmtheory/ssm_quark_structure.html.

2. The FCC vacuum

Before introducing the defect that constitutes matter, we establish

the lattice in which the defect lives. This section addresses three

foundational questions: why FCC, what is 𝐾, and how does the 𝐾=4 →

𝐾=12 phase transition arise.

2.1. Why FCC and not BCC or HCP?

The FCC lattice is uniquely selected as the vacuum’s ground-state

geometry by three independent considerations.

The Kepler bound. The Kepler conjecture, proven by Hales [3], es-

tablishes that the maximum density for packing identical spheres in

three-dimensional Euclidean space is 𝜋∕(3



2) ≈ 0.7405. Both FCC and

HCP saturate this bound; both achieve coordination number 𝐾=12.

BCC achieves only 𝐾=8 and packing density 𝜋



3∕8 ≈ 0.6802, well

below the Kepler bound. Simple cubic achieves 𝐾 =6 and density 𝜋∕6 ≈

0.5236. A vacuum that minimizes its free energy under a sphere-packing

constraint — as our SSM kinematic rules require — cannot select BCC

or simple cubic (Table 1).

Bravais lattice structure. FCC and HCP are stacking isomers with iden-

tical local coordination but different long-range stacking sequences

(ABCABC for FCC, ABAB for HCP). A critical structural distinction lies

in their Bravais classification: FCC is a Bravais lattice (a single primitive

lattice with one atom per primitive cell, generating a translation group

), while HCP is not a Bravais lattice (it requires a hexagonal lattice

with a two-atom basis, and its full translation group is not isomorphic

to Z

). The integer-winding argument we use to derive the up-quark

charge +2∕3 in Section 4.3 requires a Bravais translation group, in

which closed loops must traverse integer numbers of primitive vectors.

HCP’s two-atom basis breaks this requirement: a defect in HCP would

couple to a translation group with a non-trivial sublattice structure,

producing winding numbers that are fractional rather than integer.

The FCC Bravais structure is therefore essential to the framework’s

quantitative predictions.

Crystallization pathway. The simulation results in Section 2.3 below

show that under the SSM kinematic rules (lateral 2D ‘‘stitch’’ plus sup-

pressed out-of-plane ‘‘lift’’), the vacuum spontaneously produces FCC

stacking through ABC ordering, never ABAB. The geometric mechanism

that generates the lattice is therefore intrinsically tied to FCC, not HCP.

2.2. The coordination number K

We use the symbol 𝐾 throughout this paper in a single, precise

sense.

Definition 1 (Coordination Number). For a node 𝑣 in the lattice graph

𝐺(𝑉 , 𝐸), 𝐾(𝑣) is the graph-theoretic degree of 𝑣: the number of edges

incident on 𝑣. The lattice coordination 𝐾 is the modal value of 𝐾(𝑣) across

the bulk interior of the lattice, excluding boundary nodes within ∼2 lattice

spacings of any free surface.

With this definition, 𝐾 is unambiguously the vertex degree, and its

values across the lattice phases of interest are:

• In a 2D hexagonal sheet (the 𝐾=6 ground state of the SSM), each

interior node has degree 6 and the bonds form a flat triangulated

sheet.

Table 1

Comparison of common 3D lattices.

Lattice 𝐾

max

Packing density Point group Saturates Kepler?

Simple cubic (SC) 6 0.5236 𝑂

ℎ

Body-centered cubic

(BCC)

8 0.6802 𝑂

ℎ

Hexagonal

close-packed (HCP)

12 0.7405 𝐷

6ℎ

Yes

Face-centered cubic

(FCC)

12 0.7405 𝑂

ℎ

Yes

• In the 𝐾=4 tetrahedral foam phase, each node is bonded to 4

nearest neighbors forming a regular tetrahedron, and the resulting

space is geometrically frustrated (Section 2.3).

• In the 𝐾=12 FCC bulk, each interior node is at the center of

a cuboctahedral coordination shell with 12 nearest neighbors.

The number 12 saturates the kissing-number bound of three-

dimensional Euclidean space [3].

We define the fundamental nearest-neighbor bond length of the

FCC lattice as 𝐿. In terms of the conventional cubic lattice constant

𝑎, 𝐿 = 𝑎∕



2. All length scales in this paper are expressed in units of 𝐿.

2.3. The K=4 → K=12 phase transition: Kinematic origin and computa-

tional verification

Why stitch and lift, and only these two. The SSM vacuum is built from

a single primitive structure: a Bell-pair entanglement bond connecting

two nodes at unit distance 𝐿. There are exactly two distinct kinematic

operators that extend such a bond’s connectivity while preserving the

unit bond length:

(i) Stitch — planar expansion. Place a new node at the equilateral apex

above an existing edge. This adds one node bonded to two existing

nodes, growing a planar triangulated sheet. Geometrically, stitch re-

alizes the intersection of two unit spheres centered on the existing

edge endpoints — generically a 1-parameter family in 3D (a circle of

solutions), reduced to a unique apex on the local 2D growth plane.

Interior nodes of such a sheet have coordination 𝐾=6 (six surrounding

triangles), the maximum coordination consistent with strict planarity.

(Ii) Lift — out-of-plane projection. Place a new node above the centroid

of an existing triangular face at the unique height ℎ that maintains unit

bond length 𝐿 to all three triangle vertices. The triangle has circum-

radius 𝐿∕



3 (centroid-to-vertex distance of an equilateral triangle of

edge 𝐿), so by the Pythagorean theorem ℎ

+(𝐿∕



= 𝐿

, which gives

ℎ =



2∕3 𝐿 — equivalently, the apex height of a regular tetrahedron

with edge 𝐿. This adds one node bonded to three existing nodes,

breaking out of the planar 𝐾=6 ground state into 3D. Geometrically, lift

realizes the intersection of three unit spheres centered on the triangle

vertices — a 0-parameter set in 3D (a unique apex pair, of which one

is selected by orientation).

These two operators exhaust the kinematic possibilities. A 4-sphere

intersection in 3D is generically empty, so no third operator with

strictly greater connectivity can be defined. Stitch and lift therefore

form a complete kinematic basis for graph growth in three-dimensional

Euclidean space at fixed unit bond length.

Why lift is rare. The relative suppression of lift over stitch follows

from the dimensional reduction in the operator’s solution manifold.

A stitch is constrained by two simultaneous unit-distance conditions

and admits a 1-parameter family of geometric solutions; a lift is con-

strained by three simultaneous unit-distance conditions and admits

only a 0-parameter family. The relative amplitude 𝑃

lift

∕𝑃

stitch

is there-

fore exponentially suppressed in the codimension difference, and the

specific value 𝑃

lift

= 𝑒

−3

≈ 4.98% is the unique exponential amplitude

consistent with the requirement that the cascade terminates at FCC

Physics Open 27 (2026) 100423

R. Kulkarni

Fig. 1. Stitch and lift produce 𝐾=12 cuboctahedral coordination on every interior FCC bulk node. (a) The central node (yellow star) sits in the middle close-packed

hexagonal sheet (Layer B, 𝑧 = 0). Its 12 nearest neighbors decompose into 6 in-plane neighbors (green, 𝐾=6 saturation of the planar growth from stitch within

Layer B), 3 neighbors in the upper sheet (Layer C, blue, at 𝑧 = +ℎ with ℎ =



2∕3 𝐿, reached by lift), and 3 neighbors in the lower sheet (Layer A, orange, at

𝑧 = −ℎ, the mirror lift). (b) The same 12 vertices interpreted as the cuboctahedron: 8 triangular faces (green, non-bipartite 𝐶

— the color-confining channels

of Section 5.1) and 6 square faces (blue, bipartite 𝐶

— the electromagnetic-screening channels of Section 6.4). The figure visualizes why no third operator is

needed: the kissing-number bound 𝐾=12 saturates with exactly the 6+3+3 contributions of in-plane stitching plus two rare lifts.

Table 2

Coordination statistics across system sizes (𝑃

lift

= 0.05, 𝑅

= 0.95 𝐿, 30 seeds

each). The rightmost two columns report the bulk-interior diagnostic. The bulk

interior is defined as the set of nodes whose three FCC coordination shells are

fully populated: an ideal interior FCC node has 12 + 6 + 24 = 42 neighbors at

distances 𝐿,



2 𝐿, and



3 𝐿, all within radius 2𝐿. We classify a node as bulk-

interior iff it has at least 42 other nodes within 2𝐿. This is a strict geometric

criterion derived from FCC crystallography, not a tuned threshold. It excludes

any node whose surroundings are not yet a complete bulk environment, such

as free-surface nodes that have no physical analog in the cosmological setting.

𝑁 Mean 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 59 ± 14 12

500 8.20 ± 0.28 15.9 ± 4.7 0.46 ± 0.17 63 ± 12 12

750 8.64 ± 0.26 21.8 ± 5.1 0.51 ± 0.16 68 ± 10 12

1000 8.90 ± 0.26 25.4 ± 5.4 0.54 ± 0.15 69 ± 10 12

saturation (𝐾=12 at the Kepler bound) in the thermodynamic limit.

Smaller 𝑃

lift

produces isolated 2D sheets that fail to interlock; larger

𝑃

lift

produces 3D foams that fail to saturate at 𝐾=12. The simulation

results in Table 2 below confirm 𝑒

−3

as the value consistent with the

asymptotic 𝐾=12 ground state.

The cascade closes at K=12. The kinematic sequence — Bell pair (𝐾=1)

→ stitched sheet (𝐾=6) → lifted tetrahedral foam (𝐾=4, geometri-

cally frustrated intermediate) → interlocked sheets (𝐾=12 FCC, ABC

stacking) — closes geometrically at 𝐾=12 because 𝐾=12 saturates the

kissing-number bound of three-dimensional Euclidean space [3]. No

further operator can add a node at unit distance from an already

12-coordinated node, so the cascade has no fifth phase. This is the

geometric reason why stitch and lift suffice: they drive the system from

the minimal-entanglement starting point to its saturation phase, and

there is nothing further to do.

Geometric frustration of the K=4 intermediate. Of the four phases in the

cascade, only the 𝐾=4 tetrahedral foam is unstable. Regular tetrahedra

cannot tile three-dimensional Euclidean space: the dihedral angle of a

regular tetrahedron is arccos(1∕3) ≈ 70.528

◦

, and packing five tetrahe-

dra around a shared edge consumes 5 × 70.528

◦

= 352.64

◦

, leaving an

irreducible Regge deficit angle [4]

𝛿 = 2𝜋 − 5 arccos(1∕3) ≈ 0.128 rad. (1)

This residual 7.36

◦

wedge is the geometric source of the instability of

the 𝐾=4 phase. The frustrated foam therefore proceeds, under contin-

ued action of the lift operator and proximity bonding between adjacent

𝐾=6 sheets, to the 𝐾=12 FCC saturation phase — the cuboctahedral

coordination shell of every interior bulk node, with the 𝐾 = 6 + 3 + 3

decomposition into in-plane hexagonal neighbors plus upper and lower

triangular caps shown in Fig. 1.

Operational parameters of the simulation. The simulation is controlled

by a tolerance window around the unit bond length: nodes closer than

𝑅

= 0.95 𝐿 to each other are forbidden (overlap exclusion), while

nodes within proximity-bond radius 𝑅

𝑏

= 1.05 𝐿 are bonded. The ±5%

width of this window is the natural angular jitter at every node set by

the Regge deficit angle 𝛿 ≈ 0.128 rad ≈ 7.36

◦

(Eq. (1)), which projects

to approximately ±5% bond-length tolerance for unit-distance vectors.

The specific value 𝑅

= 0.95 𝐿 is well inside the 𝐾=12 stability plateau

confirmed in Fig. 2 below (𝑅

∈ [0.58, 0.99] 𝐿), so the simulation is

robust to this choice rather than fine-tuned to it.

Finite-size scaling. Table 2 shows the 𝐾=12 saturation fraction across

system sizes 𝑁 = 250, 500, 750, 1000, each averaged over 30 indepen-

dent random seeds. The data fit a clean surface-to-volume scaling law

𝑓

𝐾=12

(𝑁) = 1 − 𝛼∕𝑁

1∕3

, 𝛼 = 6.8 ± 0.6. (2)

This functional form follows directly from the geometric expectation

that under-coordinated nodes are confined to the cluster boundary

(scaling as 𝑁

2∕3

) while bulk-interior nodes saturate at 𝐾=12 (scaling

as 𝑁). The under-coordinated fraction is therefore 𝑁

2∕3

∕𝑁 = 𝑁

−1∕3

Extrapolating Eq. (2) gives 𝑓

𝐾=12

→ 1 in the thermodynamic limit

(𝑁 → ∞), consistent with complete FCC saturation as the asymptotic

ground state of the kinematics.

Physics Open 27 (2026) 100423

R. Kulkarni

Fig. 2. The geometric phase transition at the metric wall length 𝑅

= 𝐿∕



3. Below this exclusion radius, the lattice loses its 𝐾=12 saturation. The wide stability

plateau in [0.58, 0.99] confirms that 𝐾=12 saturation is not a fine-tuned outcome. Data are means over 30 random seeds at 𝑁 = 500 with 𝑃

lift

= 0.05.

Bulk-interior diagnostic. The all-node 𝐾=12 percentages above are mixed

measurements: they include free-surface nodes that are intrinsically

under-coordinated and that have no physical analog in the cosmological

setting (the observable universe sits in the bulk regime, not near a

boundary). The physically relevant quantity is whether bulk-interior

nodes have reached FCC coordination. We extract this directly via

a strict geometric criterion: a node is bulk-interior iff its three FCC

coordination shells are all populated, i.e., iff at least 42 other nodes

lie within radius 2𝐿 of it. The threshold value 42 is not tuned: it is

simply 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).

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

gives the bulk-restricted statistics in the rightmost two columns of Table

2. Three observations:

1. The bulk modal 𝐾 is exactly 12 at every system size from 𝑁=250

upward. The lattice’s bulk interior is FCC by direct measurement,

not by extrapolation.

2. The bulk-restricted 𝐾=12 fraction (e.g., 69 ± 10% at 𝑁=1000) is

roughly 2.5× the all-node value (e.g., 25.4 ± 5.4%), consistent with

the surface-volume interpretation of the scaling law.

3. The bulk standard deviation is 𝜎

𝐾

≈ 0.86 at 𝑁 =1000, with bulk

mean



𝐾

bulk

≈ 11.52. The remaining 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 distribu-

tion centered below 12. This is consistent with the polycrystalline

character of the emergent lattice.

Together, the all-node finite-size scaling and the bulk-interior modal-𝐾

measurement provide independent 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. Neither

relies on extrapolation to N→ ∞.

The metric wall at 𝑅

= 𝐿∕



3. A sweep over the exclusion radius 𝑅

reveals a sharp geometric phase transition at 𝑅

= 𝐿∕



3 = 0.577 𝐿

(Fig. 2). The maximum coordination 𝐾

max

= 12 is maintained across

the entire band 𝑅

∈ [0.58, 0.99], demonstrating that the specific value

𝑅

= 0.95 𝐿 used in our simulations is not a fine-tuned parameter.

Below 𝑅

= 𝐿∕



3, the exclusion is too weak and unphysical overlaps

appear (𝐾

max

> 12); above 𝑅

= 1.0 𝐿, strict rigidity causes lattice

freezing (𝐾

max

collapses to 2–3).

The geometric origin of the value 𝐿∕



3 is the circumradius of the

equilateral triangle with edge 𝐿: this is the distance from the centroid

of a triangular face to each of its vertices. Below this distance, a node

at the centroid would lie within bond-length range of all three triangle

vertices, creating a topologically inadmissible configuration.

Additional simulation outputs

The simulation also produces, with no fitting:

• Inter-layer FCC spacing of 0.8165±0.0011 𝐿, matching the ideal FCC

value



2∕3 𝐿 = 0.8165 𝐿 to 0.01%.

• Exact structural planarity of internal layers (𝜎

𝑧

< 10

−10

𝐿 for 22 of

23 substantial layers at 𝑁 = 3000).

• Mean inter-layer bond count of 4.9 per node, indicating that ∼ 𝑒

−3

rare lift events are sufficient to weld adjacent hexagonal sheets into

a coherent FCC bulk.

The kinematic parameters of the simulation are summarized in

Appendix.

3. The trapped extra node

3.1. Crystallographic setting

The FCC unit cell, with cubic lattice constant 𝑎, contains atoms

at (0, 0, 0) and the three face centers. The cell decomposes into eight

tetrahedral voids and four octahedral voids [5,6]. Each tetrahedral void

is bounded by exactly four FCC atoms forming a regular tetrahedron

with edge length 𝑎∕



2 = 𝐿, equal to the nearest-neighbor distance.

For the void centered at (𝑎∕4, 𝑎∕4, 𝑎∕4), the bounding vertices are

𝐴 = (0, 0, 0), 𝐵 = (𝑎∕2)(1, 1, 0), 𝐶 = (𝑎∕2)(1, 0, 1), 𝐷 = (𝑎∕2)(0, 1, 1).

(3)

3.2. The trapped remnant

During the 𝐾=4 → 𝐾 =12 phase transition, the amorphous tetrahe-

dral foam converts to crystalline FCC order. An extra node that fails

to integrate into the crystal remains sitting at the void center, bonded

to all four bounding vertices (Fig. 3). This creates a local 𝐾

complete

graph (four nodes with all six pairwise edges plus four bonds to the

Physics Open 27 (2026) 100423

R. Kulkarni

Fig. 3. A single 𝐾=4 node (yellow star) trapped at the centroid of the interstitial tetrahedral void of the 𝐾 =12 FCC lattice. The trapped node bonds to all four

bounding vertices 𝐴, 𝐵, 𝐶 , 𝐷 — themselves bulk FCC nodes with their own 𝐾=12 cuboctahedral coordination shells (the green translucent polyhedron shows vertex

𝐴’s shell as an example). The four defect bonds (orange) are under compressive strain because the 𝐾=4 coordination of the trapped node is below the 𝐾=12 bulk

equilibrium, creating a local stress field that propagates outward through the FCC lattice for ∼2 lattice spacings. The dashed red segment marks the metric wall

length 𝐿∕



3 — the in-plane circumradius of any bounding face — which sets the absolute exclusion distance below which the lattice cannot accommodate an

independent node. The defect is metastable with a confinement barrier 𝜎𝑟 set by this scale (Section 6).

trapped center, or equivalently, four bonds per bounding vertex within

the defect cluster) embedded in the 𝐾=12 bulk. The trapped node has

coordination 𝐾=4, precisely the pre-crystallization phase.

This is the central claim of the paper: matter is a frozen fragment of

the inflationary vacuum phase. It is not something added to spacetime;

it is spacetime that did not finish crystallizing.

4. Fractional electric charges from tetrahedral projection

The fractional charges −1∕3 and +2∕3 are derived in this section

from two rigorous geometric facts: (i) the bond angle of a regular

tetrahedron has cosine exactly −1∕3, and (ii) the trapped defect must

couple to the bulk FCC lattice through integer winding numbers fixed

by the bulk’s Bravais translation symmetry. Together these uniquely

determine the values −1∕3 (the unwound or ‘‘baseline’’ state) and +2∕3

(the singly-wound excited state).

4.1. The four-bond defect: 1 anchor + 3 valence quarks

The trapped extra node sits at the centroid of a regular tetrahe-

dron with vertices 𝐴, 𝐵, 𝐶, 𝐷 given in Eq. (3). It bonds to all four

bounding vertices, producing a local 𝐾=4 pocket with bond directions

𝑟

𝐴

, 𝑟

𝐵

, 𝑟

𝐶

, 𝑟

𝐷

pointing from the void center to each bounding vertex.

The four bonds split into a 1 + 3 structure under the host FCC

embedding (Fig. 4):

• One anchor bond. The bonding cluster must couple to the surround-

ing 𝐾=12 bulk to remain stable; one of the four bonds serves as the

topological anchor (the ‘‘gluon junction’’) through which charge,

momentum, and color flux are exchanged with the bulk. Without

loss of generality we designate 𝑟

𝐴

the anchor direction. The choice

of which vertex serves as the anchor breaks the 𝑆

symmetry of the

tetrahedron explicitly to 𝑆

acting on the remaining three bonds.

• Three valence bonds. The remaining bonds 𝑟

𝐵

, 𝑟

𝐶

, 𝑟

𝐷

are internal

to the defect and constitute the three valence quarks. They are

equivalent under the residual 𝑆

symmetry.

The choice of anchor is dynamical: a different choice gives a spa-

tially inverted defect (the second tetrahedral-void orientation in the

FCC unit cell, centered at (3𝑎∕4, 3𝑎∕4, 3𝑎∕4)). This 4-fold ambiguity of

anchor selection — combined with the integer-winding mechanism of

Section 4.3 — is what produces the proton, neutron, 𝛥

−

, and 𝛥

from

a single underlying geometric structure (Section 4.4).

4.2. The tetrahedral projection: −1∕3 as a geometric law

The defining geometric fact about a regular tetrahedron is the cosine

of its bond angle. From any vertex of a regular tetrahedron, the four

bonds to the centroid (equivalently, the four bonds emanating from the

trapped node to its bounding vertices) pairwise satisfy

𝑟

𝑖

⋅ 𝑟

𝑗

= −1∕3 for all 𝑖 ≠ 𝑗 ∈ {𝐴, 𝐵, 𝐶, 𝐷}. (4)

The angle is arccos(−1∕3) ≈ 109 .47

◦

, the well-known regular-tetrahedron

bond angle. Eq. (4) is a rigid theorem of three-dimensional Euclidean

geometry; it admits no fine-tuning, no parameter, and no alternative.

Physics Open 27 (2026) 100423

R. Kulkarni

Fig. 4. Quark structure from tetrahedral geometry. (a) Anchor selection breaks the tetrahedral 𝑆

symmetry to 𝑆

: one bond (red) anchors the defect to the bulk,

while three bonds (green) are the equivalent valence quarks. (b) Each valence bond projects onto the anchor axis with weight cos(arccos(−1∕3)) = −1∕3 — a rigid

theorem of three-dimensional Euclidean geometry. The four-bond conservation 1 + 3 × (−1∕3) = 0 is exact. (c) The four observed baryon configurations follow

directly from the total winding number 𝑊 =



𝑖

𝑤

𝑖

∈ {0, 1, 2, 3} via the formula 𝑄

baryon

= −1 + 𝑊 (Eq. (10)).

Projection onto the anchor axis

Project each of the three valence bonds onto the anchor direction

𝑟

𝐴

𝑃

𝑣

≡ 𝑟

𝑣

⋅ 𝑟

𝐴

= −1∕3 for each 𝑣 ∈ {𝐵, 𝐶, 𝐷}. (5)

Each valence bond carries a geometric flux of exactly −1∕3 unit along

the anchor axis. This is not a counting argument or a normalization

choice; it is the direct value of the cosine in Eq. (4).

Self-projection of the anchor

The anchor bond projects onto itself with weight 𝑟

𝐴

⋅ 𝑟

𝐴

= +1.

Conservation along the anchor axis

The four bonds together project a total flux of

𝑃

tot

= 𝑟

𝐴

⋅ 𝑟

𝐴



𝑣∈{𝐵,𝐶 ,𝐷}

𝑟

𝑣

⋅ 𝑟

𝐴

= 1 + 3 × (−1∕3) = 0. (6)

The sum of projections vanishes identically. This is the geometric origin

of charge conservation in the unwound state: the four-bond defect

carries net flux zero along its own preferred axis.

We identify the projected flux 𝑃

𝑣

with the observable electric charge

𝑞

𝑣

of each valence bond in the unwound state. This identification is

justified because (a) the anchor axis is the only physically distinguished

direction at the defect site (the bulk-coupling channel), (b) the flux

along this axis is the quantity transmitted to the bulk and observable

at infinity, and (c) the conservation Eq. (6) is exactly the requirement

that the closed defect carry zero net charge before bulk coupling. The

baseline charge of each valence bond is therefore

𝑞

(0)

𝑣

= −1∕3. (7)

This is the geometric origin of the down-quark charge.

4.3. Integer winding from bulk translation symmetry

The unwound state with all three valence bonds at −1∕3 gives

a baryon charge of −1 (corresponding to the 𝛥

−

baryon, 𝑑𝑑𝑑). The

trapped defect must couple to the bulk FCC lattice in a manner that

allows the full observed baryon spectrum to form. We now derive the

second geometric fact that fixes the up-quark charge.

Bravais translation symmetry

The bulk 𝐾=12 FCC lattice is a Bravais lattice with discrete transla-

tion group Z

(generated by the three primitive lattice vectors). Any

quantity that couples a localized defect to the bulk must be single-

valued under bulk translations: a defect that traverses a closed loop in

the lattice must return to its initial state up to a multiple-of-2𝜋 phase,

exactly as in standard lattice gauge theory [7].

Winding number

For a valence bond, define the winding number 𝑤 as the integer

count of full lattice spacings the bond’s endpoint traverses when the

defect couples to the bulk. The single-valuedness requirement under

the Bravais translation group forces

𝑤 ∈ Z. (8)

Non-integer winding would correspond to a fractional translation,

which is not an element of the translation group and would render the

defect ill-defined as a bulk excitation.

Charge after winding

A valence bond with winding number 𝑤 carries observable charge

𝑞

𝑣

(𝑤) = 𝑞

(0)

𝑣

+ 𝑤 = −1∕3 + 𝑤. (9)

The available charge values are {… , −4∕3, −1∕3, +2∕3, +5∕3, +8∕3, …}.

Phenomenological selection of 𝑤 ∈ {0, +1}

The two lowest-magnitude solutions of Eq. (9) are:

• 𝑤 = 0: 𝑞 = −1∕3 (the baseline state, identified with the down

quark).

• 𝑤 = +1: 𝑞 = +2∕3 (the singly-wound state, identified with the up

quark).

Higher windings 𝑤 ≥ 2 would correspond to charges ≥ +5∕3, which are

not observed for valence quarks in any baryon at the first-shell level of

the FCC defect classification. Negative windings 𝑤 ≤ −1 would give

charges ≤ −4∕3, also not observed.

The restriction to 𝑤 ∈ {0, +1} is therefore an empirical input from

the observed quark spectrum, not derived from the FCC geometry

alone. It is analogous to other places in this framework where the

Physics Open 27 (2026) 100423

R. Kulkarni

lowest-shell defects match observation: the framework predicts the

form of the allowed states (a discrete ladder −1∕3 + 𝑤) and the geo-

metric value of the baseline (−1∕3), but the upper end of the ladder is

fixed phenomenologically.

4.4. The baryon spectrum from total winding number

A baryon consists of one trapped defect: 1 anchor bond + 3 valence

quarks, each valence bond carrying an independent winding number

𝑤

𝑖

∈ {0, +1}. The anchor bond couples the defect to the bulk and does

not itself contribute an observable valence charge — its self-projection

of +1 is internal to the defect, while the three valence bonds carry

the externally observable charges. The total observable charge of the

baryon is therefore the sum of the three valence contributions:

𝑄

baryon



𝑖=1

𝑞

𝑖

(𝑤

𝑖

) =



𝑖=1

(−1∕3 + 𝑤

𝑖

) = −1 +



𝑖=1

𝑤

𝑖

= −1 + 𝑊 , (10)

where 𝑊 =



𝑤

𝑖

∈ {0, 1, 2, 3} is the total winding number across

the three valence bonds. The four observed baryon configurations

correspond to the four possible total windings:

• 𝛥

−

(𝑑𝑑𝑑): 𝑊 = 0, all three bonds unwound. Valence charges:

(−1∕3, −1∕3, −1∕3). Total: 𝑄 = −1. ✓

• neutron (𝑢𝑑𝑑): 𝑊 = 1, one bond wound. Valence charges: (+2∕3,

−1∕3, −1∕3). Total: 𝑄 = 0. ✓

• proton (𝑢𝑢𝑑): 𝑊 = 2, two bonds wound. Valence charges: (+2∕3,

+2∕3, −1∕3). Total: 𝑄 = +1. ✓

• 𝛥

(𝑢𝑢𝑢): 𝑊 = 3, all three bonds wound. Valence charges:

(+2∕3, +2∕3, +2∕3). Total: 𝑄 = +2. ✓

This reproduces the four observed first-shell baryon charge states

(𝑄 ∈ {−1, 0, +1, +2}), matching the 𝛥-baryon isospin quartet (𝛥

−

, 𝛥

𝛥

, 𝛥

) of the 𝐼 = 3∕2 multiplet. At the level of structural counting,

the neutron and proton are degenerate with 𝛥

and 𝛥

respectively

(same quark content 𝑢𝑑𝑑 and 𝑢𝑢𝑑); the spin-1/2 vs spin-3/2 mass

splitting between them is a QCD effect that lies outside the geometric

framework, which counts only structural disruption integers.

The matter–antimatter asymmetry from anchor orientation

The two tetrahedral-void orientations in the FCC unit cell (centered

at (𝑎∕4, 𝑎∕4, 𝑎∕4) and (3𝑎∕4, 3𝑎∕4, 3𝑎∕4)) are related by spatial inversion.

The inversion exchanges the sign of the anchor projection: 𝑟

𝐴

→ −𝑟

𝐴

so the valence projections become +1∕3 instead of −1∕3. The inverted

void supports anti-baryons (anti-𝛥

−

→ 𝛥

, antineutron, antiproton,

anti-𝛥

→ 𝛥

−

) by the same construction with all signs flipped. The two

void orientations are the geometric origin of matter and antimatter, and

the local cosmological excess of one orientation over the other selects

the observed matter-dominated universe.

4.5. What the geometry derives, what It does not, and what It falsifiably

predicts

We summarize what the tetrahedral-projection argument establishes

rigorously, what remains as input, and what the framework predicts at

the level of falsifiable observation.

Rigorous geometric outputs

1. The baseline charge 𝑞

(0)

= −1∕3 for each valence bond, derived

from the regular-tetrahedron bond-angle cosine (Eq. (4)).

2. Charge conservation along the anchor axis (Eq. (6)), guaranteeing

zero net flux along the defect’s preferred direction in the unwound

state.

3. Integer winding 𝑤 ∈ Z, derived from the Bravais translation

symmetry of the bulk 𝐾=12 lattice (Eq. (8)).

4. The discrete charge ladder 𝑞(𝑤) = −1∕3 + 𝑤 (Eq. (9)).

5. The four-baryon spectrum (𝛥

−

, neutron, proton, 𝛥

) from the

formula 𝑄

baryon

= −1 + 𝑊 with 𝑊 ∈ {0, 1, 2, 3} (Eq. (10)).

6. Matter–antimatter duality from the two void orientations.

Phenomenological input

The restriction to single-bond windings 𝑤 ∈ {0, +1} (rather than

𝑤 ≥ 2) reflects the observed valence-quark spectrum at the first-

shell level of the FCC defect classification. The framework does not

independently predict why higher windings are absent at this level; this

is consistent with the broader program of identifying the lowest-shell

defects with the lightest particles [2], where the heavier strange and

charm sectors are expected to involve second-shell extensions beyond

the first FCC void.

Rigid falsifiable predictions

While the truncation 𝑤 ∈ {0, +1} is taken as input, the frame-

work makes three rigid predictions in this sector that are falsifiable in

principle:

1. No fourth color charge exists. The combinatorial uniqueness of the

skew-edge pair count 𝑐

skew

= 3 for the complete graph 𝐾

admits no

extension. Detection of a fourth fundamental color charge would

falsify the framework.

2. No isolated fractionally charged quark can propagate through the

bulk. The non-bipartite triangular face topology of the cuboctahe-

dral coordination shell (Sections 5.1 and 6.4) topologically forbids

single-color-mode propagation. Observation of a free fractional

charge would falsify the framework.

3. The first-shell baryon spectrum is exhausted by 𝑊 ∈ {0, 1, 2, 3}.

There is no fifth first-shell baryon. Discovery of a stable funda-

mental baryon outside this multiplet (not a higher-shell radial

excitation) would falsify the framework.

5. Three colors from three skew-edge pairs

5.1. Combinatorial origin of three colors

The bounding tetrahedron 𝐾

has 6 edges. These partition uniquely

into 3 pairs of skew edges (edges sharing no common vertex):

Pair 1: (𝐴𝐵, 𝐶𝐷), Pair 2: (𝐴𝐶, 𝐵𝐷), Pair 3: (𝐴𝐷, 𝐵𝐶). (11)

This corresponds to the combinatorial identity 𝐶(4, 2)∕2 = 3. For any

chosen edge of 𝐾

, exactly one of the other 5 edges shares no vertex

with it (the ‘‘opposite’’ edge); the 6 edges therefore partition into 3

disjoint skew pairs, and this partition is unique. There is no fourth skew

pair, no second valid assignment, and no other tetrahedral graph with

a different number of skew pairs. We use the symbol 𝑐

skew

= 3 to denote

this skew-edge pair count.

Each skew pair carries one unit of internal flux that binds the

tetrahedral defect together. Three skew-edge pairs therefore provide

three independent internal flux channels, recovering the three color

charges of Quantum Chromodynamics. No fourth color exists because

a tetrahedron has no fourth independent skew-edge pair. This is a

theorem of discrete combinatorics, not a phenomenological postulate.

5.2. Relation to SU(3) and color neutrality

We address the relation between this geometric construction and

the standard SU(3) gauge group.

Three colors, not SU(3) directly

Our construction generates exactly three color charges from the

three skew-edge pairs of the bounding tetrahedron. This corresponds to

the representation theory of standard SU(3) gauge theory — specifically

the three-dimensional fundamental representation in which quarks

transform — not the full SU(3) Lie algebra. We do not claim to derive

the SU(3) Lie algebra from the lattice geometry; we claim that the

representation of color charges on a baryon-like defect is uniquely fixed

at three by the combinatorial identity 𝐶(4, 2)∕2 = 3.

Physics Open 27 (2026) 100423

R. Kulkarni

The 8 generators of SU(3) (corresponding to the 8 gluons) live in

a different topological structure of the lattice: the 8 triangular faces

of the cuboctahedral coordination shell of each FCC bulk node. The

cuboctahedron has 14 faces total: 8 triangles and 6 squares. The 8

triangular faces are non-bipartite (they contain odd-length cycles 𝐶

that cannot be 2-colored), making them confining channels; the 6

square faces are bipartite (they admit perfect {+, −} alternation) and

host the screening sector. This separation of representation (three colors

on the defect) and algebra (eight gluons in the bulk) is consistent with

the standard formulation of lattice gauge theory [7,8].

Color neutrality

The closed tetrahedral defect carries net color charge zero. The three

skew-edge pairs each carry one unit of internal color flux, but the flux

lines are oriented so that the four vertices form a closed loop in color

space. Specifically, the color fluxes assigned to the three pairs satisfy a

vertex-balance condition at every bounding vertex:



𝑖=1

𝑐

𝑖

= 0, (12)

where 𝑐

𝑖

is the color charge associated with vertex 𝑖. This is verified

directly: each vertex is an endpoint of exactly one edge from each skew

pair (3 incident edges total, one from each color), and the three colors

sum to zero in any SU(3) representation. The defect is therefore a color

singlet, which is the discrete analog of the trace condition that makes

baryons color singlets in SU(3) representation theory.

Confinement at the topological level

The non-bipartite topology of the triangular faces of the cuboc-

tahedral coordination shell prevents the propagation of single-color

modes through the bulk. A bipartite face supports an alternating {+, −}

coloring (a perfect 2-coloring of the face’s vertices), corresponding

to a propagating dipole mode. A non-bipartite face does not. The 8

triangular faces in each coordination shell are non-bipartite by virtue

of containing odd-length cycles (𝐶

). A color singlet (three colors

summing to zero) can propagate through these faces because it carries

no net coloring obligation; an isolated color charge cannot, because

its associated alternating mode is geometrically frustrated on every

triangular face it would have to cross.

This provides a static, topological version of confinement that does

not require a dynamical gauge field. It complements — rather than

replaces — the dynamical Wilson-loop area law of continuum lattice

QCD [7,9]: the static topological structure provides the boundary con-

ditions, while the dynamical area law would emerge in the continuum

limit when fluctuations of the lattice bonds are taken into account.

6. Confinement from the metric wall

6.1. Why the extra node cannot escape

The extra node sits at the centroid of the tetrahedral void, bonded to

its four bounding vertices 𝐴, 𝐵, 𝐶, 𝐷 at the centroid-to-vertex distance

𝐿 ⋅



3∕8 ≈ 0.612 𝐿 (slightly compressed below the unit bond length,

hence the local strain). The relevant length scale for confinement, how-

ever, is not the centroid-to-vertex distance but the in-plane circumradius

𝐿∕



3 of any bounding face: this is the distance, within the plane of

one triangular face, from the face centroid to any vertex of that face.

We refer to this length as the metric wall. It is the absolute minimum

approach distance below which the lattice cannot accommodate an

independent node without violating the exclusion principle of the

discrete metric: any node lying within 𝐿∕



3 of all three vertices of

an equilateral triangle of edge 𝐿 would be at unit-bond-length range

of all three simultaneously, an over-constrained configuration that the

kinematics rejects.

Section 2.3 (Fig. 2) provides computational verification of this

length scale: a sharp phase transition in 𝐾

max

occurs precisely at 𝑅

𝐿∕



3, and below this threshold the exclusion principle becomes too

weak to enforce the FCC topology.

To extract the trapped node from the void, one must stretch the four

bonds connecting it to the bounding vertices. However, the surrounding

𝐾=12 lattice aggressively resists this displacement: every bounding ver-

tex has 9 external bonds pulling it back into its crystalline equilibrium

position. This restoring force grows linearly with the displacement,

producing a confining potential

𝑉 (𝑟) = 𝜎 𝑟, (13)

where 𝜎 is the lattice-level string tension set by the bond energy.

6.2. Comparison with the Cornell potential

The Cornell potential [9,10] is the phenomenological QCD confine-

ment potential

𝑉

Cornell

(𝑟) = −𝜅∕𝑟 + 𝜎𝑟, (14)

where 𝜎 ≈ 1 GeV/fm is the QCD string tension and 𝜅 ≈ 0.5 encodes the

short-distance Coulomb-like interaction.

Linear term

Our metric-wall confinement reproduces the linear 𝜎𝑟 term of

Eq. (14) naturally. The geometric obstruction to extracting a node from

a tetrahedral void grows linearly with the extraction distance because

each unit of displacement requires breaking one additional layer of

bond connections. The linear-in-𝑟 behavior is a consequence of the

discrete topology of the lattice and does not require dynamical gauge

fields.

Coulomb term

Our derivation does not reproduce the short-distance Coulomb −𝜅∕𝑟

piece. This is expected: our argument is purely static and considers

only the kinematic obstruction to extracting a node. The Coulomb term

in the Cornell potential arises from one-gluon exchange in continuum

QCD, which is a dynamical effect not captured by the static lattice

geometry alone. A complete derivation of the Coulomb term would

require additional input from the gauge dynamics of the lattice (the

Wilson action and its continuum limit), which is beyond the scope of

this paper.

String tension scale

The lattice-level string tension is 𝜎

lat

∼ 𝜀∕𝐿

, where 𝜀 is the unitary

entanglement bond energy and 𝐿 is the lattice spacing. Setting 𝐿 ∼ 𝓁

𝑃

(Planck scale) and 𝜀 at the GUT scale (∼ 10

GeV, as constrained

by the reheating temperature in inflationary cosmology [11]) gives

𝜎

lat

∼ 10

−15

GeV

, many orders of magnitude smaller than the QCD

string tension (∼ 0.2 GeV

). The two confinement scales reflect different

physical regimes: the lattice-level confinement operates at the Planck

scale, while the QCD-scale confinement emerges in the long-wavelength

continuum limit. A complete bridge between them would require a

renormalization-group analysis that we do not undertake here.

6.3. String breaking

If sufficient external energy is supplied to stretch a bond beyond the

fundamental length 𝐿, the lattice kinematically un-stitches: it nucleates

a new node at the fracture point, capping both torn ends with new

vertices. The original single defect bifurcates into two distinct defects: a

quark–antiquark pair. This represents physical string breaking, mechan-

ically ensuring that isolated, unconfined quarks can never propagate

through the bulk.

Physics Open 27 (2026) 100423

R. Kulkarni

6.4. Electromagnetic invisibility

The three internal bonds emanating from each bounding vertex

pass exclusively through the triangular faces of the cuboctahedral

coordination shell. As established in Section 5.1, triangular faces are

non-bipartite (𝐶

): they cannot support the alternating {+, −} charge

oscillations that mediate photon exchange. Photon coupling instead

operates on the bipartite (square) faces of the cuboctahedron, of which

there are 6 per coordination shell [12]. The internal bonds of the defect

are therefore topologically invisible to photons. Only the external bonds

passing through square faces contribute to the net observable charge,

which resolves only as the composite integer charge of the baryon. This

explains why isolated quarks — carrying fractional charges through

internal bonds — cannot be observed electromagnetically.

7. The proton mass

Before applying the structural counting argument to the proton, we

state the two definitional choices on which the formula in this section

rests: the choice of mass unit (the electron) and the counting basis

(the count of disrupted crystalline bond states). These are stated up

front so that Section 7 is self-contained rather than dependent on the

lighter-particle correspondences introduced later in Section 8.

The electron as the unit of mass. The simplest possible defect in the

FCC lattice is a single-node defect — a localized perturbation at one

bulk vertex with no internal structure, no skew-edge pairs (𝑐

skew

= 0),

and no color content. We identify this minimal defect as the electron

and set 𝑚

𝑒

≡ 1 as the unit of the count throughout this paper. This

is a definition, not a derivation; the framework’s predictive content

lies in the integer ratios 𝑚

𝑥

∕𝑚

𝑒

computed for richer defects, just as no

derivation is given for the elementary charge 𝑒 or the speed of light 𝑐

when those are used as units.

Counting basis. With the electron as the unit, mass is the total count

of crystalline bond states disrupted by the defect — including the bond

states associated with the defect’s own structural nodes and those of

the surrounding shells whose coordination is perturbed by the defect’s

presence. Each disrupted bond state contributes one unit (= 𝑚

𝑒

). This

identification — that mass is a count of crystalline disruption rather

than an independently fitted energy parameter — is the geometric

content of the framework. For the trapped tetrahedral-void defect, the

count factors into three pieces: 𝐾 + 1 = 13 structural nodes (the

trapped node plus its four bounding vertices, with each bounding vertex

contributing three internal bonds), each disrupting 𝐾

= 144 bond

states in its second-neighbor shell, minus 𝑐

skew

𝐾 = 36 double-counted

bonds along the three skew-edge pairs, giving the formula (𝐾 + 1)𝐾

−

𝑐

skew

𝐾 = 1836 derived in Section 7.2.

Scope of the formula. The expression (𝐾 + 1)𝐾

− 𝑐

skew

𝐾 applies specif-

ically to the trapped tetrahedral-void defect — the configuration that

yields the proton and neutron. Other defect classes (the electron above;

the pion as a two-vertex defect with one closing string; the muon as

a deconfined three-sheet defect) have their own counting expressions

matched to their structural footprints, summarized in Section 8 with

full derivations in the companion paper [2]. The formula in this section

should not be applied outside the trapped-void class.

7.1. Structural node count

The single trapped extra node bonds to the four bounding vertices.

Each bounding vertex commits 3 of its 𝐾=12 bonds internally to the de-

fect structure. Including the central trapped node itself as a coordinate

point, the total structural node count of the localized defect is

𝑁

struct

= 4 × 3 + 1 = 𝐾 + 1 = 13. (15)

Table 3

Proton mass decomposition from the interstitial void geometry.

Component Geometric origin Value

𝐾 + 1 = 13 4 bounding cells × 3 bonds + 1 center 13 nodes

𝐾

= 144 Each node disrupts 𝐾 neighbors of 𝐾 neighbors 144 per node

−𝑐

skew

𝐾 = −36 3 skew-edge pairs × 𝐾 shared bonds −36 correction

Total (𝐾 + 1)𝐾

− 𝑐

skew

𝐾 1836

7.2. The mass formula

Each of the 13 structural nodes disrupts the crystalline coordi-

nation of its 𝐾

= 144 second-nearest neighbors. The 3 skew-edge

pairs (𝑐

skew

= 3) represent internal flux lines that link two bounding

vertices each, with overlapping disrupted neighborhoods. The double-

counted second-nearest-neighbor bonds along each skew pair must be

subtracted. The proton-to-electron mass ratio scales directly with the

resulting count (Fig. 5):

𝑚

𝑝

∕𝑚

𝑒

= (𝐾 + 1)𝐾

− 𝑐

skew

𝐾 = 13 × 144 − 3 × 12 = 1872 − 36 = 1836. (16)

The observed empirical value is 1836.153 [13]. The theoretical match

is to within 0.008%. Table 3 summarizes the geometric origin of each

factor in the formula.

7.3. Stability under geometric perturbations

The formula (𝐾 + 1)𝐾

− 𝑐

skew

𝐾 depends on three integers, each a

topological invariant:

• 𝐾 = 12: the kissing number of three-dimensional Euclidean space

[3], fixed by the geometry of the FCC lattice.

• 𝐾 + 1 = 13: the structural node count of the defect (4 bounding

vertices each contributing 3 internal bonds, plus 1 trapped node =

13 nodes).

• 𝑐

skew

= 3: the unique skew-edge pair count of the complete graph

𝐾

, fixed by the combinatorial identity 𝐶(4, 2)∕2 = 3.

Small geometric perturbations of the lattice (thermal vibrations, bond

length fluctuations, slight angular distortions of the cuboctahedral

coordination shell) do not change any of these integers. The formula is

therefore stable under all such perturbations: it is a counting argument

over discrete topological data, not a continuous optimization. The

continuous environment can shift the absolute mass scale (through the

bond energy 𝜀 and lattice temperature 𝑇 ) but not the dimensionless

ratio 𝑚

𝑝

∕𝑚

𝑒

7.4. Relation to the companion QEC derivation

The same number 1836 is obtained by a different route in the

companion paper [2]. There, 𝑚

𝑝

∕𝑚

𝑒

is the dimension of the coupling

matrix of a [[192, 130, 3]] CSS code on the FCC lattice:

𝑚

𝑝

∕𝑚

𝑒

= 𝐸

𝑠

× 𝐶

𝑠

= 36 × 51 = 1836, (17)

where 𝐸

𝑠

= 36 is the number of edges in the 3-sheet substructure of

the cuboctahedral coordination cluster and 𝐶

𝑠

= 𝑓

+ 𝑓

= 13 + 38 = 51

is the total number of stabilizer constraints (vertex stabilizers plus face

stabilizers) capable of detecting the defect.

The two factorizations are arithmetically equivalent:

36 × 51 = 1836 = 13 × 144 − 36 = (𝐾 + 1)𝐾

− 𝑐

skew

𝐾, (18)

because 𝐾 = 4𝑐

skew

in the cuboctahedral geometry. The companion

paper’s question — what is the verification cost of this defect? — and the

present paper’s question — what does the trapped-defect geometry count?

— meet at this shared integer. The structural content of 𝐸

𝑠

and 𝐶

𝑠

as 𝑓 -

vector entries of the FCC coordination cluster, and the deeper geometric

origin of the equivalence, are the subject of ongoing work.

Physics Open 27 (2026) 100423

R. Kulkarni

Fig. 5. The macroscopic topological disruption zone. Left: schematic of the 13 structural nodes of the localized defect — the central trapped node (yellow star)

plus 12 surrounding bonded nodes (one per internal bond from each of the four bounding vertices, color-coded by parent vertex). The three skew-edge pairs are

highlighted (red dashed). Right: the three-component derivation of 𝑚

𝑝

∕𝑚

𝑒

: 13 structural nodes (per Eq. (15)), each disrupting 𝐾

= 144 second-nearest-neighbor

states, minus 36 double-counted bonds along the three skew-edge pairs, gives 1836 — matching empirical 𝑚

𝑝

∕𝑚

𝑒

= 1836.153 to 0.008%. The companion paper [2]

reaches the same number through the dimension of the [[192, 130, 3]] CSS-code coupling matrix (see Section 7.4).

8. The lighter particles

The proton-to-electron mass ratio derived in Section 7 uses the

electron rest mass as the unit scale. Within the present framework,

the electron is the minimal single-node defect of the 𝐾=12 lattice —

a point defect localized at one bulk vertex, with no internal edges, no

skew-edge pairs (𝑐

skew

= 0), and no fractional substructure — defining

the minimal discrete strain unit of the vacuum. The other lighter

particles (the pion, the muon, and the neutron) require fault-tolerant

verification-cost arguments that lie outside the structural-disruption

framework of this paper; they are treated in the companion paper [2],

which derives 𝐶

𝑒

= 1, 𝐶

𝜋

= 273, 𝐶

𝜇

= 207, and 𝐶

𝑛

= 1839, all matching

empirical mass ratios to within 0.12%.

9. Lorentz invariance and experimental constraints

A common concern about discrete-spacetime models is conflict with

special relativity. We establish emergent Lorentz invariance in three

steps and verify compatibility with current experimental bounds.

9.1. Spatial isotropy at the lattice level

The 12 nearest-neighbor bond vectors of the FCC lattice are

𝑛

𝑗

∈ {(±1, ±1, 0), (±1, 0, ±1), (0, ±1, ±1)}∕



2, 𝑗 = 1, … , 12. (19)

Define the rank-2 structure tensor

𝑆

𝜇𝜈

≡



𝑗=1

𝑛

𝜇

𝑗

𝑛

𝜈

𝑗

. (20)

Direct enumeration [2] gives 𝑆

𝑥𝑥

= 𝑆

𝑦𝑦

= 𝑆

𝑧𝑧

= 4 and all off-diagonal

components vanish. Therefore

𝑆

𝜇𝜈

= 4 𝛿

𝜇𝜈

[exact, by enumeration]. (21)

This guarantees equal propagation speed in every spatial direction.

The odd-rank tensor

𝑇

𝜇𝜈𝜆

≡



𝑗=1

𝑛

𝜇

𝑗

𝑛

𝜈

𝑗

𝑛

𝜆

𝑗

(22)

vanishes exactly: every bond 𝑛

𝑗

has a partner −𝑛

𝑗

(the FCC lattice is

centrosymmetric), so every odd-power sum cancels:

𝑇

𝜇𝜈𝜆

= 0 [exact, by inversion symmetry]. (23)

This ensures 𝜔(𝑘) = 𝜔(−𝑘): there is no preferred direction and no linear

(𝑘) term in the dispersion.

9.2. Isotropy of the dispersion relation

For a scalar field on the FCC lattice, the dispersion is 𝜔(𝑘)

𝜅



𝑗=1

[1 − cos(𝑘 ⋅ 𝑛

𝑗

𝑎)]. At long wavelengths (𝑘𝑎 ≪ 1):

𝜔(𝑘)

≈ (𝜅𝑎

∕2)



𝑗=1

(𝑘⋅𝑛

𝑗

)

= (𝜅𝑎

∕2) 𝑘

𝜇

𝑘

𝜈

𝑆

𝜇𝜈

= (𝜅𝑎

∕2)⋅4𝑘

= 2𝜅𝑎

𝑘

(24)

so 𝜔 = 𝑐

lat

𝑘 with 𝑐

lat

= 𝑎



2𝜅. The dispersion is exactly isotropic at all

orders in 𝑘 for which the Taylor expansion is valid. Corrections appear

only at 𝑂(𝑘

𝑎

) from higher cosine terms, suppressed by (𝑘𝑎)

∼

(𝐸∕𝑀

𝑃

)

for Planck-scale lattice spacing 𝑎 ∼ 𝓁

𝑃

9.3. Lorentz boosts in the continuum limit

The two tensor identities (Eq. (21), (23)) and the isotropic linear

dispersion are sufficient for SO(3,1) Lorentz boosts to emerge in the

standard continuum limit [14]. A lattice Hamiltonian whose dispersion

is isotropic and linear in 𝑘 at long wavelengths gives rise to a relativistic

effective field theory in the continuum limit. The lattice does break

Lorentz symmetry at the cutoff scale 1∕𝑎, but these violations are

suppressed by (𝐸∕𝑀

𝑃

)

, far below current experimental sensitivity.

9.4. Experimental constraints

Current bounds on Lorentz violation are summarized in the

Kostelecký–Russell data tables [15] and the Liberati–Maccione re-

view [16]. Typical constraints fall in the range 10

−20

to 10

−40

relative

to the Planck scale, depending on the operator and the experimental

Physics Open 27 (2026) 100423

R. Kulkarni

probe. Our predicted suppression of (𝐸∕𝑀

𝑃

)

∼ 10

−56

at optical

frequencies is many orders of magnitude below these bounds, so the

SSM is compatible with all current Lorentz-invariance tests.

9.5. Resolution of the Collins et al. naturalness objection

Collins et al. [14] pointed out that any Lorentz-violating UV cutoff

generates fine-tuning of order 𝛬

∕𝑀

𝑃

via radiative corrections, which

would normally dominate over the bare suppression. In the SSM, this

objection is addressed by the holographic structure of the lattice: the

UV cutoff is effectively the 2D boundary network, which carries con-

tinuous SO(2) symmetry. Radiative corrections in the boundary respect

SO(2) at every loop order, and the 3D bulk inherits this symmetry

holographically. The full argument is given in [2], Section 2.2.

10. Discussion

10.1. What this picture gets right

From a single geometric premise — an extra node trapped in an

interstitial tetrahedral void — we structurally obtain:

1. Fractional charges −1∕3 and +2∕3 from the regular-tetrahedron

bond-angle cosine combined with integer winding under the bulk

FCC Bravais translation symmetry. The full first-shell baryon spec-

trum (𝛥

−

, neutron, proton, 𝛥

) follows from the four possible total

winding numbers 𝑊 ∈ {0, 1, 2, 3} via 𝑄

baryon

= −1 + 𝑊 .

2. Exactly three colors generated by the three independent skew-edge

pairs.

3. Linear color confinement dictated by the metric wall at 𝐿∕



reproducing the 𝜎𝑟 term of the Cornell potential.

4. 𝑚

𝑝

∕𝑚

𝑒

= 1836 derived from the structural node count and skew-

edge pair count, with arithmetic equivalence to the [[192, 130, 3]]

CSS-code derivation in the companion paper [2].

5. The 𝛥

−

, neutron, proton, and 𝛥

spectrum from total winding

number, and matter–antimatter duality from spatial inversion of

the tetrahedral void.

6. Physical string breaking from lattice un-stitching limits.

7. Electromagnetic invisibility of quarks from the non-bipartite trian-

gular face topology.

8. Compatibility with all current Lorentz-invariance experimental

bounds.

None of these results require fine-tuned free parameters.

10.2. What remains open

Several aspects of the framework are not yet resolved:

• The complete meson spectrum beyond the pion.

• The geometric origin of the strange and heavier quark generations.

• The relationship between this static defect picture and the dy-

namical QCD flux tubes modeled by current gauge theories. The

static topological structure provides the boundary conditions and

the Coulomb-like short-distance term is expected to emerge from

gauge dynamics, but a full derivation is not provided here.

• The proton–neutron mass splitting at the level of QFT corrections

beyond the integer baseline.

• The phenomenological restriction of valence-bond windings to

𝑤 ∈ {0, +1} rather than the full integer ladder. The framework

predicts the structure of the charge ladder rigorously but takes

the truncation to lowest two states as input from the observed

first-shell quark spectrum (Section 4.3).

10.3. Relation to other approaches

Lattice gauge theory. Lattice QCD [1,7] computes hadronic properties

numerically on a background lattice using the Wilson action and its

descendants. The SSM proposes that the lattice is not merely a mathe-

matical computational tool but the physical vacuum itself. Confinement

is therefore a static geometric property of the space rather than a

dynamical property of a continuous gauge field living on the space.

Our discrete-color and confinement arguments build on the standard

lattice formulation reviewed by Kogut [8] and verified numerically by

Creutz [17]. The treatment of fermions on the lattice — the Karsten–

Smit no-go theorem on chiral invariance and species doubling [18]

and the Susskind staggered-fermion construction [19] — defines the

technical landscape within which a discrete-spacetime model must

reproduce continuum physics. The SSM’s static topological derivation

is complementary to (not a replacement for) the dynamical Wilson-loop

approach, and the two should agree in the continuum limit.

Emergent spacetime and discrete geometry. The view that geometry

itself emerges from a more primitive substrate has multiple com-

plementary realizations in the recent literature. The ER=EPR con-

jecture of Maldacena and Susskind [20] identifies geometric con-

nections (Einstein–Rosen bridges) with quantum entanglement; Van

Raamsdonk [21] argues that classical spacetime connectivity is built

up by entanglement; Swingle [22] maps entanglement renormaliza-

tion onto holographic geometry; and the holographic quantum error-

correcting codes of Pastawski et al. [23] make the emergence of

bulk locality from boundary entanglement explicit. Other founda-

tional discrete-spacetime programs include causal dynamical trian-

gulations [24], causal sets [25], the spin-network combinatorics of

Penrose [26], and the recent spacetime-quasicrystal construction of

Boyle and Mygdalas [27]. The SSM differs from these in that the

discrete substrate is a specific physical lattice (FCC) rather than a

generic combinatorial graph, and its predictions are quantitative in-

teger ratios of observable particle properties rather than asymptotic

geometric features.

Regge calculus and discrete gravity. Regge’s foundational coordinate-free

formulation of general relativity [4] provides the geometric machinery

(the deficit angle around an edge as the discrete curvature) that we use

to characterize the geometric frustration of the 𝐾=4 tetrahedral foam.

Cheeger, Müller, and Schrader [28] placed Regge calculus on a rigorous

footing by analyzing the curvature of piecewise flat spaces, and Ham-

ber [29] provides a comprehensive treatment of lattice approaches to

quantum gravity that share with the SSM the premise that spacetime

admits a discrete description. The SSM uses these ideas in a constrained

way: only the deficit-angle calculation enters, as the geometric quantity

that drives the 𝐾=4 → 𝐾=12 phase transition.

Topological quantum error correction. The interpretation of the FCC

vacuum as a CSS code in our companion paper [2,12] is part of a

broader research program that began with Kitaev’s anyon model for

fault-tolerant quantum computation [30] and the topological-quantum-

memory analysis of Dennis et al. [31]. The defining feature of these

constructions is that information is protected not by an active

measurement-and-correct cycle but by the topological properties of the

underlying lattice. The SSM extends this perspective to the physical

vacuum: the trapped tetrahedral defect is a topologically stable config-

uration whose inability to relax (the metric wall of Section 6) is the

analog of the topological protection that makes anyonic codes useful.

Crystallization mechanisms. The framework’s central image — matter

as incomplete crystallization of the vacuum — has structural analogs

in two well-studied physical processes. Witten and Sander’s diffusion-

limited aggregation [32] demonstrates that local kinematic rules gener-

ate persistent macroscopic non-uniformities that survive long after the

rules cease to be locally informative. Coleman’s analysis of the false

Physics Open 27 (2026) 100423

R. Kulkarni

Table 4

Kinematic parameters of the SSM lattice simulation. All values are geometri-

cally or thermodynamically determined.

Parameter Value Derivation

Unitary metric (𝐿) 1.0 Invariant relational distance

Lateral height



3∕2 ⋅ 𝐿 ≈ 0.866 𝐿 Equilateral triangle altitude

Lift height



2∕3 𝐿 ≈ 0.816 𝐿 Regular tetrahedron altitude

Lift probability 𝑒

−3

≈ 4.98% Topological tunneling

Proximity bond 1.05 𝐿 Regge deficit (𝛿 ≈ 7.36

◦

)

Hard shell (𝑅

) 0.95 𝐿 Symmetric exclusion

Geometric cutoff 𝐿∕



3 ≈ 0.577 𝐿 Circumradius of unitary triangle

vacuum [33] establishes that a quantum field can become trapped in

a metastable phase whose decay proceeds through nucleation events

governed by topological barriers. The SSM combines these two ideas:

the cosmological vacuum begins in a metastable 𝐾=4 tetrahedral phase,

transitions to the 𝐾=12 FCC ground state through a Kepler-bound

saturation process, and leaves behind the trapped extra nodes of an

incomplete crystallization as the topological defects we identify with

baryons.

11. Conclusions

Matter is spacetime that did not finish crystallizing. A single extra

node trapped in a tetrahedral void of the 𝐾=12 FCC vacuum lattice

strictly recovers fractional quark charges, three color degrees of free-

dom, linear confinement, and the proton-to-electron mass ratio — all

emerging organically from the unadjusted geometry of close-packed

spheres. The framework makes clear, falsifiable predictions: no isolated

quarks can propagate, no fourth color charge can exist, and the first-

shell baryon spectrum is exhausted by the four configurations 𝑊 ∈

{0, 1, 2, 3} corresponding to 𝛥

−

, neutron, proton, and 𝛥

. The model

is consistent with all current experimental bounds on Lorentz violation

and reproduces the linear-confinement piece of the Cornell potential.

The companion paper [2] reaches 𝑚

𝑝

∕𝑚

𝑒

= 1836 by a different route

(fault-tolerant verification cost in a CSS code on the same lattice);

the two derivations are arithmetically equivalent because 𝐾 = 4𝑐

skew

in the cuboctahedral geometry, and the structural meaning of that

equivalence is left for future work.

Declaration of competing interest

The authors declare that they have no known competing finan-

cial interests or personal relationships that could have appeared to

influence the work reported in this paper.

Appendix. Kinematic parameters and replication conditions

This appendix specifies the kinematic parameters of the simulation

reported in Section 2.3 and the conditions under which its quantitative

claims (Table 2 and Fig. 2) are reproduced. The kinematic operators

(stitch and lift) and their geometric origins are defined in Section 2.3;

this appendix collects the operational parameters in a single table for

ease of reference.

A.1. Kinematic parameters

See Table 4.

A.2. Replication

The kinematic rules described in Section 2.3 (Bell-pair seed; stitch

and lift operators with relative amplitude 𝑃

lift

= 𝑒

−3

; proximity bond-

ing within 1.05 𝐿; exclusion radius 𝑅

= 0.95 𝐿), together with the

parameter values in Table 4, fully specify the simulation. Running the

resulting graph-growth procedure at 𝑁 = 1000 over 30 random seeds

reproduces the data in Table 2: 𝐾=12 saturation of 25.4 ± 5.4%, mean

coordination



𝐾 = 8.90±0.26, and shape isotropy 𝜆

min

∕𝜆

max

= 0.54±0.15.

The finite-size scaling exponent is 𝛼 = 6.8 ± 0.6 (Eq. (2)). A sweep over

𝑅

∈ [0.50, 1.02] at fixed 𝑁 = 500, 𝑃

lift

= 0.05 reproduces Fig. 2: the

sharp phase transition at 𝑅

= 𝐿∕



3 and the wide stability plateau in

[0.58, 0.99]. All quantitative claims in Section 2.3 reduce to these two

scans.

A reference implementation of the algorithm is available at

ssm_sim.py (lattice generation and saturation analysis); the exclusion-

radius sweep is performed by plot_rex_sweep.py. The implemen-

tation is provided as a verification aid; the manuscript itself fully spec-

ifies the algorithm and any equivalent implementation will reproduce

the results within the reported statistical uncertainties.

Data availability

All kinematic parameters and replication conditions for the simu-

lation results in Section 2.3 are given in Appendix. A reference im-

plementation of the algorithm is available as ssm_sim.py (the lattice

generator and saturation analysis used for Table 2) and plot_rex_sweep.

py (the exclusion-radius sweep used for Fig. 2). The bulk-interior

diagnostic reported in Table 2 is computed by ssm_bulk_analysis.py,

which re-analyzes the simulation output to extract bulk-restricted co-

ordination statistics. The interactive 3D visualizations are available

at https://raghu91302.github.io/ssmtheory/ssm_regge_deficit.html and

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

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