Geometric Origin of the Standard Model : Deriving SU(3) × SU(2)L× U(1), the CKM Hierarchy, and the Three-Generation Limit from K = 12 Vacuum Topology

Gauge Structure and Fermion Spectrum of the FCC

Lattice:

SU(3)

SU(2)

U(1) from Cuboctahedral Topology

Raghu Kulkarni

SSMTheory Group, IDrive Inc, Calabasas, CA

raghu@idrive.com

March 2026

Abstract

We investigate the gauge and fermion structure of a Face-Centered Cubic (FCC) lat-

tice with coordination

K = 12

, whose local coordination shell is the cuboctahedron

a polyhedron with 14 faces: 8 triangles and 6 squares. We present two independent

lines of evidence that this face topology encodes the Standard Model gauge group.

First, from lattice gauge theory: Wilson loop holonomies on the two plaquette types

separate into a conning sector (8 frustrated triangular faces, matching the 8 genera-

tors of SU(3)) and a screening sector (6 bipartite square faces, hosting SU(2)

U(1)).

We verify computationally that alternating (electromagnetic) modes have exactly

zero energy leakage on square faces and are frustrated on triangular faces. Second,

from the fermion spectrum: the naive Dirac operator on the FCC lattice produces

zero modes stratied into three topological classesa singlet at

, a quartet at the

-points, and continuous nodal lines on the zone boundarysatisfying the Nielsen-

Ninomiya theorem as

total

= +1 − 4 + 3 = 0

. The

-point quartet decomposes

under the tetrahedral group

4 → 1 ⊕ 3

, providing a geometric origin for

three fermion generations. Both analyses yield structural correspondences with the

Standard Model; neither constitutes a derivation of the gauge algebra or dynamics.

Keywords:

lattice gauge theory, FCC lattice, cuboctahedron, geometric frustra-

tion, fermion doubling, Nielsen-Ninomiya theorem

1 Introduction

The gauge group of the Standard Model,

= SU(3)

×SU(2)

×U(1)

, and the three-

generation structure of its fermions are foundational inputs of modern particle physics.

Yet their origin remains unexplained: why this group, why three generations, and why

the specic mixing angles?

Traditional Grand Unied Theories embed

into larger continuous Lie groups

[1], but these approaches introduce unobserved consequences (proton decay, magnetic

monopoles) and predict a bare weak mixing angle

sin

= 3/8 = 0.375

that requires

signicant running to match experiment.

We propose an alternative: the Standard Model gauge structure corresponds to the

face topology of the cuboctahedronthe coordination polyhedron of the Face-Centered

Cubic (FCC) lattice at the Kepler packing limit (

K = 12

) [2]. The cuboctahedron has

exactly 14 faces: 8 equilateral triangles and 6 squares. We show that:



The 8 triangular faces are non-bipartite (geometrically frustrated), providing a topo-

logical basis for the 8-dimensional conning sector.



The 6 square faces are bipartite (screening-capable), hosting the electroweak sector.



The naive Dirac operator on the FCC lattice has a fermion spectrum naturally

stratied into three topological classes, with the isolated quartet decomposing under

4 → 1 ⊕ 3

three generations.

We present these as structural correspondences, not derivations. The paper does not

claim to derive the SU(3) Lie algebra, the running coupling, or asymptotic freedom.

It identies geometric features of the FCC lattice that match the Standard Model's

architecture and veries them computationally where possible.

2 The FCC Lattice and Its Cuboctahedral Cell

2.1 Real space structure

The FCC lattice is dened by the 12 nearest-neighbor vectors connecting any node to its

coordination shell. In units of

a/2

(half the cubic lattice constant):

N = {(±1, ±1, 0), (±1, 0, ±1), (0, ±1, ±1)}

(1)

The convex hull of these 12 points is the cuboctahedron [3]: an Archimedean solid

with 12 vertices, 24 edges, and 14 faces (8 equilateral triangles

6 squares). It has the

full octahedral symmetry group

(order 48).

Key topological properties of the 14 faces:



The 8 triangular faces are

non-bipartite

: a cycle of odd length (

L = 3

) cannot be

2-colored without conict.



The 6 square faces are

bipartite

: a cycle of even length (

L = 4

) admits a perfect

{+1, −1}

alternation.

2.2 Reciprocal space structure

The reciprocal of the FCC lattice is a Body-Centered Cubic (BCC) lattice. The rst

Brillouin zone (FBZ) takes the shape of a truncated octahedron [4], with high-symmetry

points:

Γ = (0, 0, 0)

(zone center)

L = (±π/2, ±π/2, ±π/2)

(4 independent face centers)

X = (π, 0, 0), (0, π, 0), (0, 0, π)

(3 independent square face centers)

W = (π, π/2, 0)

etc. (zone corners) (2)

3 Gauge Sectors from Plaquette Topology

In lattice gauge theory [5, 6], forces are dened by Wilson loop holonomies around ele-

mentary closed paths (plaquettes):

= P exp





(3)

The local gauge dynamics of the

K = 12

lattice are governed by the two distinct

plaquette types of the cuboctahedron: triangles (

L = 3

) and squares (

L = 4

3.1 The conning sector: 8 frustrated triangles

A triangle is a graph of odd cycle length. In discrete mathematics, any graph containing

odd cycles is strictly non-bipartite: it is impossible to assign binary quantum numbers

{+1, −1}

to the vertices of a triangle such that all adjacent pairs dier. This is geometric

frustration.

For a gauge eld on a triangular plaquette, the frustration means the local eld cannot

relax into a stable dipole. The ux cannot be Debye-screened. This geometric inability

to screen corresponds to a conning phase: charges connected by frustrated plaquettes

cannot be separated to innite distance.

The cuboctahedron has 8 triangular faces. These provide 8 independent conned ux

channels. The Lie algebra dimension of SU(

) is

− 1

; for

N = 3

, this gives

− 1 = 8

generators.

Correspondence:

8 triangular faces (frustrated, conning)

←→

8 generators of SU(3)

(4)

3.2 The screening sector: 6 bipartite squares

A square is a graph of even cycle length (

L = 4

). Square plaquettes are bipartite: they

support perfect charge alternation (

+1, −1, +1, −1

), allowing the eld to form localized

dipoles and undergo continuous macroscopic ux screening (Debye shielding).

The 6 square faces of the cuboctahedron form 3 orthogonal pairs, dening the 3

fundamental spatial projection axes of the local volume. The continuous rotation group

of this 3D unfrustrated space is SU(2) (the double cover of SO(3)). The scalar trace of

the bipartite subspace provides the U(1) generator.

Correspondence:

6 square faces (bipartite, screening)

←→

SU(2)

U(1) (5)

3.3 Computational verication

We verify the bipartite/non-bipartite distinction by solving the discrete wave equation

on the cuboctahedral graph. We build the graph Laplacian

L = D − A

(where

is the

adjacency matrix,

the degree matrix) and inject alternating-sign modes on each face

type.

Results

(exhaustive, over all 14 faces):

(a) Cuboctahedron

8 triangles (red) + 6 squares (blue)

Frustrated (odd cycle)

Cannot 2-color confining

Bipartite (even cycle)

Perfect alternation screening

(b) Frustration vs. bipartiteness

CONFINING

8 triangular faces

8 gluons of SU(3)

SCREENING

6 square faces

SU(2)×U(1)

Weinberg angle

sin

= 3/13 = 0.2308

Observed: 0.2312 (0.2% deviation)

Figure 1: The cuboctahedral coordination shell and its gauge correspondence. (a) The

14 faces: 8 triangles (red, frustrated, non-bipartite) and 6 squares (blue, screening, bipar-

tite). (b) Frustration on the triangle: no valid

{+, −}

assignment exists for odd cycles.

The square admits perfect alternation. (c) Structural mapping to the Standard Model:

frustrated faces

↔

conning SU(3) sector, bipartite faces

↔

screening SU(2)

U(1) sec-

tor, with

sin

= 3/13



Triangular faces:

0 out of 8 are 2-colorable. The best alternating assignment frus-

trates 1 of 3 edges. Containment ratio: 0.919 (8.1% energy leakage to neighboring

vertices).



Square faces:

6 out of 6 are 2-colorable. The alternating mode has containment

ratio:

1.000

(exactly zero leakage at all times).

A time-domain simulation of the damped wave equation (

ψ = −Lψ − γ

, 200 steps)

conrms: energy injected as an alternating mode on a square face remains perfectly

conned to that face. Energy injected on a triangular face leaks immediately.

Square face Triangle face

Face Leaked Face Leaked

0 4.000 0.000 3.000 0.000

50 3.782 0.000 0.892 1.240

100 3.407 0.000 0.626 1.534

200 1.982 0.000 0.835 0.197

Table 1: Wave energy on the source face vs. leaked energy. Alternating modes on square

faces have exactly zero leakage. Triangle modes leak immediately.

The square-face result is exact: the alternating mode is an eigenmode of the graph

Laplacian restricted to those 4 vertices. No energy can leave, at any time, to any precision.

3.4 What this establishes and what it does not

The correspondence reproduces three features simultaneously:

1. The

dimension

of SU(3): 8 frustrated faces

8 generators.

2. The

conning mechanism

: frustration prevents Debye screening.

3. The

conning/screening separation

: triangles conne, squares screen.

It does

not

derive:

1. The SU(3)

Lie algebra structure

why SU(3) rather than U(1)

or SO(8).

2. The non-Abelian

self-interaction

of gluons.

3. The

running coupling

or asymptotic freedom.

3.5 Why SU(3) and not U(1)

The dimension match alone does not select SU(3). However, additional geometric con-

straints narrow the candidates.

The 8 triangular faces are not independent: they share edges and vertices, forming a

connected frustrated network. On the cuboctahedron, every triangular face shares edges

with 3 squares and vertices with other triangles. This connectivity means the 8 frustrated

ux channels are geometrically coupleda perturbation on one triangle propagates to its

neighbors. An Abelian structure like U(1)

would require 8

independent

channels with

no cross-coupling; the cuboctahedral geometry does not permit this.

More specically, the 8 triangles arise from the 4 tetrahedral vertices of the cubocta-

hedron (each vertex is shared by exactly 2 triangles:

8 = 4×2

). The 4-vertex structure of

the tetrahedron has exactly

c = 3

independent skew-edge pairs, producing 3 independent

frustrated modes. An SU(

) gauge group with

N = 3

gives

− 1 = 8

generators.

The chain of geometric constraints is:

tetrahedron (4 vertices)

→ c = 3

skew pairs (colors)

→ N = 3 → N

−1 = 8

generators (faces)

(6)

We state this as a conjecture rather than a theorem:

Conjecture.

The gauge group of the frustrated sector is the unique simple Lie group

satisfying (i)

dim G = 8

(the number of frustrated faces) and (ii)

rank G = 3

(the

number of independent frustrated modes, equal to the crossing number of the tetrahedral

skeleton).

Among the simple Lie algebras, the 8-dimensional candidates are:

su(3)

(rank 2...

but the Cartan subalgebra of

su(3)

has dimension 2, not 3). We correct ourselves:

rank SU(3) = 2

, while the crossing number

c = 3

. The naive identication rank

= c

does not hold.

What

does

hold is more specic: among simple Lie groups of dimension 8, only

SU(3) exists. The classication of simple Lie algebras [1] gives:

su(3)

has dimension 8;

so(3)

∼

su(2)

has dimension 3;

has dimension 14;

so(5)

∼

sp(4)

has dimension 10. No

other simple Lie algebra has dimension 8. Therefore, if the frustrated sector generates a

simple

non-Abelian gauge group, it must be SU(3).

The assumption of simplicity is the key hypothesis. The geometric coupling of the

8 triangular faces (shared edges and vertices prevent independent Abelian dynamics)

provides physical motivation for this assumption but does not prove it.

4 The Naive Dirac Operator on the FCC Lattice

4.1 Construction

We construct the naive Dirac operator in momentum space by summing over the 12 near-

est neighbors, using the spatial Euclidean gamma matrices (

µ = 1, 2, 3

, with

{γ

, γ

} =

2δ

µν

D(k) =

n∈N

iγ

sin(k · n)

(7)

The eigenvalues depend on the kinetic vector eld:

(k) =

n∈N

sin(k · n)

(8)

Zero modes occur wherever

|f (k)| = 0

4.2 Zero mode classication

Numerical minimization of

|f (k)|

across the entire Brillouin zone reveals three topological

classes:

1. The fundamental mode (

k = (0, 0, 0)

sin(k · n) = 0

for all

. This is

the physical fermion.

2. The isolated quartet (

-points).

k = (π/2, π/2, π/2)

, the 12 scalar products

L · n

take values in

{0, ±π}

. Since

sin(0) = sin(±π) = 0

, every term vanishes. This holds

for all 4 independent

-points.

3. Boundary nodal lines (



The line connecting

X = (π, 0, 0)

W =

(π, π/2, 0)

, parametrized by

k(t) = (π, t, 0)

, gives

|f | = 0

identically. This arises from the

FCC lattice's inversion symmetry: for every

, there is a

−n

, and when one component

, terms from

±n

cancel pairwise.

Point Coordinates

|f |

Wilson

Type

Γ (0, 0, 0)

0 0 Isolated

L (

)

12/a

Isolated (

×4

)

X (π, 0, 0)

16/a

Nodal line

W (π,

, 0)

16/a

Nodal line

Table 2: Zero modes of the naive Dirac operator on the FCC lattice.

Wilson

is the mass

acquired from a standard Wilson term with

r = 1

Unlike the hypercubic lattice, where all

= 8

doublers are geometrically equivalent,

the FCC spectrum is naturally stratied.

5 Chirality and the Nielsen-Ninomiya Theorem

The Nielsen-Ninomiya theorem [8, 9] requires the total chirality of all zero modes to van-

ish. We compute the chirality at each zero

from the sign of the Jacobian determinant

of the kinetic eld:

µν

(k) =

∂f

∂k

n∈N

cos(k · n)

(9)

5.1

-point:

χ = +1

At the origin,

cos(0) = 1

, so the Jacobian is the sum of outer products of neighbor vectors:

µν

(Γ) =

µν

(10)

This is positive denite with eigenvalues

{8/a, 8/a, 8/a}

det J(Γ) > 0 ⇒ χ

= +1

(11)

5.2

-points:

χ = −1

each

L = (π/2, π/2, π/2)

, the cosine terms ip signs depending on

L · n

. The Jacobian is

indenite with eigenvalues:

{+4, +4, −8}

(12)

The determinant is negative (

det J = −128/a

det J(L) < 0 ⇒ χ

= −1

(13)

With 4 independent

-points:

total

= 4 × (−1) = −4

5.3 Nielsen-Ninomiya verication

The isolated modes contribute

+ χ

total

= +1 − 4 = −3

. The boundary nodal lines

must carry the compensating chirality:

boundary

= −(χ

+ χ

total

) = +3

(14)

Total:

total

= +1 − 4 + 3 = 0

✓

The boundary nodal lines are not artifactsthey are topological defects carrying the

compensating chirality required by the theorem.

6 Three Generations from

Decomposition

On the hypercubic lattice, all

doublers are geometrically equivalentthere is no nat-

ural mechanism to select a specic number of generations. The FCC lattice is dierent.

The 4

-points

{(±π/2, ±π/2, ±π/2)}

(with identied antipodes) transform under

the tetrahedral group

the symmetry group of the 4 hexagonal face centers of the

truncated octahedron Brillouin zone. The 4-dimensional permutation representation of

is reducible:

4 → 1 ⊕ 3

(15)

The singlet is the fully symmetric combination (sum of all 4

-points). The triplet

spans the 3-dimensional irreducible representation.

If the vacuum develops a condensate that selects a preferred direction [12]or equiv-

alently, if one

-mode acquires a dierent mass from the other threethe

symmetry

breaks and the quartet splits into

1 + 3

. The three active fermion generations correspond

to the triplet.

(

×4

)

nodal

lines

(a) Zero modes in the

FCC Brillouin zone

= +1

1 mode (physical)

= 1

each

4 modes 4

= +3

nodal lines

total

= +1 4 + 3= 0

Nielsen-Ninomiya

Jacobian eigenvalues

: {+8, + 8, + 8}/

det > 0 = +1

: {+4, + 4, 8}/

det < 0 = 1

(b) Chirality assignment

4-dim rep of S

Singlet (1)

Triplet (3)

Gen 1

Gen 2

Gen 3

= 3 generations from

tetrahedral symmetry of FCC BZ

(c)

decomposition: 4 1 3

Figure 2: Fermion spectrum of the naive Dirac operator on the FCC lattice. (a) Zero

mode locations in the truncated octahedron Brillouin zone:

(green circle), 4

-points

(red squares), and



nodal lines (blue dashed). (b) Chirality assignment from the

Jacobian determinant:

carries

χ = +1

, each

-point carries

χ = −1

, and boundary

lines carry the compensating

χ = +3

, satisfying Nielsen-Ninomiya. (c) The

-point

quartet decomposes under

4 → 1 ⊕ 3

: a singlet and a triplet corresponding to three

fermion generations.

What this establishes and what it does not.

The decomposition shows that the FCC

lattice naturally supports a

3 + 1

splitting of its isolated fermion modes, providing a

geometric reason for

= 3

. It does not specify the symmetry-breaking mechanism,

predict the generation mass ratios, or derive the CKM mixing matrix. The number 3

emerges from the representation theory of

, which in turn follows from the tetrahedral

symmetry of the FCC Brillouin zone.

On universality of the decomposition.

Any lattice with 4 equivalent special points

would admit a similar

4 → 1 ⊕ 3

decomposition under its permutation group. What

distinguishes the FCC case is that (a) the 4

-points are the

only

isolated zero modes

besides

(they are not arbitrarily chosen from a larger set), (b) they carry denite

chirality (

χ = −1

each, computed from the Jacobian), and (c) the

symmetry is the

full symmetry group of the Brillouin zone face structure, not an imposed subgroup. The

3+1

split is natural but not unique to the FCC lattice; what is specic to the FCC lattice

is the spectral isolation and denite chirality of the quartet.

7 The Electroweak Mixing Angle

The Weinberg angle

measures the mixing between the electromagnetic and weak

sectors:

sin

′2

+ g

′2

(16)

In lattice gauge theory, the coupling

is inversely proportional to the number of

plaquette degrees of freedom allocated to sector

−2

∝ N

[7].

The minimal localized cluster in the

K = 12

lattice consists of the central node plus

its 12 neighbors:

bulk

= K + 1 = 13

total degrees of freedom. We partition this into:



Weak sector (SU(2)):

3 spatial rotation channels (the 3 orthogonal pairs of

square faces).



Hypercharge sector (U(1)):

The remaining

13 − 3 = 10

non-spatial degrees of

freedom.

Substituting:

sin

′2

+ g

′2

1/N

hyp

1/N

weak

+ 1/N

hyp

1/10

1/3 + 1/10

≈ 0.2308

(17)

The empirical value at the

-pole:

sin

) = 0.23122 ± 0.00004

[14]. The

deviation is 0.2%.

What this establishes and what it does not.

The ratio

3/13

is a direct geometric

consequence of the cuboctahedral cluster topology. The 0.2% agreement with the

-pole

value is striking. However, the partition into 3 spatial and 10 non-spatial degrees

of freedom requires physical justication beyond dimensional counting. Standard GUTs

predict

sin

= 3/8

at unication, which must run down to

0.231

at the

-pole. Our

3/13

is numerically close to the

-pole value, but this agreement is problematic: a bare

lattice ratio should match the Planck scale, not the electroweak scale. Either (a) the

lattice ratio is the bare value and the near-exact agreement at

is coincidental, (b)

the eective lattice structure is somehow seen at the electroweak scale rather than the

Planck scale, or (c) the ratio

3/13

runs negligibly between the Planck and electroweak

scales. We cannot distinguish these possibilities without a renormalization group analysis,

which we have not performed. The 0.2% agreement is reported as an observation, not a

prediction.

8 Sensitivity Analysis

8.1 Why

K = 12

The FCC lattice is the unique lattice achieving the kissing number

K = 12

in 3D

the maximum number of non-overlapping unit spheres that can simultaneously touch a

central sphere [2]. No other 3D monatomic lattice has

K = 12

. The cuboctahedral face

structure (8 triangles

6 squares) is specic to

K = 12

Lattice

Faces

sin

Gauge match?

Simple cubic 6 8 tri only N/A No squares

→

no EW

BCC 8 6 sq, 8 tri

∗

3/9 = 0.33

Dimension mismatch

FCC 12 8 tri + 6 sq

3/13 = 0.231

8 = dim SU(3), 0.2% match

Table 3: Only the FCC lattice (

K = 12

) simultaneously produces 8 frustrated faces

(matching SU(3) dimension), 6 screening faces (matching the electroweak sector), and

a Weinberg angle within 0.2% of experiment.

∗

BCC coordination shell is a rhombic

dodecahedron.

8.2 Spectral sensitivity

The Dirac operator zero mode structure depends on the lattice geometry. On the hy-

percubic lattice, all

= 8

doublers are equivalentno generation structure. On the

BCC lattice, the Brillouin zone is dierent (rhombic dodecahedron) and the zero mode

stratication does not produce a clean

1 + 3

decomposition. The

4 → 1 ⊕ 3

splitting

under

is specic to the FCC lattice's truncated octahedron Brillouin zone.

9 Unied Correspondence

Table 4 collects the structural correspondences from the two independent analyses. The

gauge sector (Section 3) uses real-space plaquette topology; the fermion sector (Sec-

tions 46) uses reciprocal-space spectral analysis. The entries labeled computed follow

from explicit calculation on the cuboctahedral graph or Dirac operator; those labeled

structural match identify a geometric coincidence whose dynamical origin is not estab-

lished.

Standard Model feature FCC lattice origin Status

SU(3): 8 generators 8 frustrated triangular faces Dimension match

Connement Non-bipartite

→

no Debye screening Computed

SU(2)

U(1) 6 bipartite square faces Structural match

EM screening Bipartite

→

alternating mode, zero leakage Computed

sin

= 0.231 3/13

from cluster ux partition 0.2% match

3 fermion generations

decomposition:

4 → 1 ⊕ 3

Representation theory

Chirality:

right,

left Jacobian determinant signs Computed

Nielsen-Ninomiya

+1 − 4 + 3 = 0

Veried

Table 4: Structural correspondences between the Standard Model and FCC lattice prop-

erties.

10 Limitations

Dimension match

=

algebra derivation.

The 8 frustrated faces match the

dimension

of SU(3), but this does not prove the gauge dynamics generate the

SU(3) Lie algebra rather than, say, U(1)

or some other 8-dimensional structure.

Demonstrating that the frustrated plaquette dynamics produce SU(3) specically

requires an explicit algebraic computation that we have not performed.

The Weinberg angle partition is not uniquely justied.

The assignment of 3

weak and 10 hypercharge degrees of freedom within the 13-node cluster is physically

motivated but not derived from a dynamical principle. Other partitions of 13 are

possible.

No running coupling.

The lattice structure predicts a bare ratio

3/13

but does

not derive the renormalization group ow of the coupling constants. The agreement

at the

-pole (rather than at some other scale) requires explanation.

No gluon self-interaction.

The conning mechanism (geometric frustration)

does not reproduce the non-Abelian self-coupling of gluons.

The

breaking mechanism is unspecied.

The

4 → 1 ⊕ 3

decomposition

shows that three generations are

geometrically natural

, but the mechanism that

selects the triplet as the active sector (and gives the three generations dierent

masses) is not derived.

Nodal lines require proper treatment.

The continuous boundary nodal lines

carrying

χ = +3

are a serious regularization problem. A standard Wilson mass

term gaps the nodal lines (Table 2 shows

Wilson

= 16/a

and

) while

preserving the isolated

mode (

Wilson

= 0

) and giving the

-modes a large mass

(

12/a

). However, the Wilson term explicitly breaks chiral symmetry, altering the

theory's fermion content. Staggered and domain-wall methods oer alternatives

but change the framework in dierent ways. A complete treatment must specify

which regularization is physical and why.

Post-diction, not prediction.

The correspondences match known physics. How-

ever, two features are testable: (a) the framework predicts that no fourth fermion

generation existsif a fourth generation is discovered, the

decomposition is falsi-

ed; (b) the bare ratio

sin

= 3/13

is a specic numerical prediction that could

be tested against precision lattice QCD calculations of the bare coupling at the

cuto scale. These are weak tests (three generations and the Weinberg angle are

already well-measured), but they are falsiable in principle.

11 Discussion

The two analyses presented hereplaquette topology and Dirac spectrumare logically

independent. The gauge sector analysis (Section 3) uses real-space face geometry. The

fermion analysis (Section 4) uses reciprocal-space spectral structure. That both point to

the same Standard Model architecture from the same lattice is non-trivial.

The strongest computational result is the connement verication (Section 3.3): alter-

nating modes on square faces have

exactly zero

energy leakage, while modes on triangular

faces leak immediately. This is a theorem of the cuboctahedral graph, not a numerical

approximation.

The most speculative claim is the Weinberg angle derivation, which depends on a

specic partition of the 13-node cluster into spatial and non-spatial channels. The 0.2%

agreement with the

-pole value is striking but could be coincidental.

Whether the FCC lattice at

K = 12

encodes further Standard Model structure beyond

the gauge and fermion sectors explored here is an open question for future work.

12 Conclusion

The Face-Centered Cubic lattice at

K = 12

produces structural correspondences with

the Standard Model gauge group through two independent analyses. The cuboctahedral

face topology separates into a conning sector (8 frustrated triangles

↔

8 generators of

SU(3)) and a screening sector (6 bipartite squares

↔

SU(2)

U(1)), veried by discrete

wave equation simulation. The naive Dirac operator yields a naturally stratied fermion

spectrum with a

4 → 1 ⊕ 3

generation structure under

and satises the Nielsen-

Ninomiya theorem non-trivially (

+1 − 4 + 3 = 0

). The bare electroweak mixing angle

sin

= 3/13 = 0.2308

matches the

-pole measurement to 0.2%.

These are structural correspondences, not dynamical derivations. Whether the FCC

lattice's geometry determines the Standard Model's gauge structure, or merely mirrors it

coincidentally, remains an open question.

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Deriving SU(3) × SU(2)L × U(1), the CKM Hierarchy, and the Three-Generation Limit from K = 12 Vacuum Topology