The Standard Model Gauge Group from PSL(2,7) - SU(3)×SU(2)×U(1) as Representation Theory of the Three-Stratum Decomposition of GL(3,𝔽₂)

Paper 13 in the Merkabit series (Stenberg, 2026). Derives the Standard Model gauge group SU(3)×SU(2)×U(1) from the representation theory of PSL(2,7) ≅ GL(3,𝔽₂), order 168, without imposing the gauge group as an input. Completes the gauge-group identification begun in Papers 11 and 12; exact results left approximate here are closed by Paper 14.

Structure. PSL(2,7) admits a canonical three-stratum decomposition: Binary B₃₁ (orders 1, 2, 4; matter sector), Z₃-boundary Z₆₂ (orders 3, 7; weak force), and Purely Ternary T₇₅ (orders 4, 7; strong force), with |B|+|Z|+|T| = 31+62+75 = 168.

Three exact theorems.

1. S₃ Universality. The symmetric group S₃ ⊂ PSL(2,7) preserves all three strata under conjugation with mutually distinct permutation characters: 8 orbits in B₃₁, 11 in Z₆₂, 14 in T₇₅. S₃ is the universal meta-symmetry above the force level — it sees each force sector differently.
2. Z₃-Boundary Sufficiency. ⟨Z₆₂⟩ = PSL(2,7). The 62 weak-sector elements generate the full group, verified by four independent methods (BFS right-multiplication, BFS both-multiply, direct set closure, closure with inverses).
3. Confinement by Schur Indicator. The colour representations 3 and 3̅ are the unique complex-Schur irreps of PSL(2,7) (indicator 0). All other irreps (1, 6, 7, 8) are real (indicator +1). A state in an isolated complex representation cannot form a gauge-invariant observable. Isolated colour charge is therefore structurally unobservable — confinement as a selection rule, not a dynamical assumption.

SU(3) identification. The tensor product rules of PSL(2,7) satisfy the SU(3) Clebsch-Gordan decompositions identically: 3⊗3̅ = 1+8 (meson channel), 3⊗3 = 3̅+6 (diquark), ∇3(3) = 1 (baryon channel). All multiplicities are exact integers. The 8⊗8 decomposition 1+3+3̅+2(6)+3(7)+3(8) is computed exactly; it differs from SU(3) because PSL(2,7) has no dim-10 or dim-27 irreps. The dim-8 restricted to S₃ decomposes as 1×triv + 1×sign + 3×standard — three copies of the standard 2-dim S₃ representation, corresponding to three colour charges.

Gauge boson locations. W± in the Z₆₂ anti-incidence (dim-7) structure, as the unique 13/13 Z₂-bundle split of 26 order-3 elements. Z⁰ as element 92 = 18², the unique Z₄ midpoint simultaneously in Fix(C) and B₃₁. Photon as the Z₃ grading automorphism of ℤ[ω] (established in Paper 12), fixing Fano point 010 — the unique Fano point whose full order-4 stabiliser lies entirely in T₇₅. Gluons in the dim-8 adjoint acting on B₃₁, with T₇₅ carrying ~73% of the colour content (3 and 3̅ dominant).

Geometric summary. The Standard Model is incidence geometry on the Fano plane PG(2,2): 21 flags (point on line) = matter, 28 anti-flags (point off line) = weak force, oriented lines = colour. The product SU(3)×SU(2)×U(1) is irreducible: PSL(2,7) is simple, so the three gauge factors arise from structurally incompatible mathematical objects — character table identities, stratum bundle geometry, and Eisenstein arithmetic substrate automorphisms — not from a group-theoretic direct product.

Bridge to Paper 14. The commutators of B₃₁'s 21 involutions (transvections of the Fano plane, three per line) land in T₇₅ hitting exactly 8 of 14 S₃-orbits and in Z₆₂ hitting exactly 3, giving the weak:strong ratio 3:8 = N_c:dim(SU(3)) exactly. This exact result, proved in Paper 14, supersedes the approximate rational-multiplicity identification of SU(3) in this paper.

Quantitative predictions (parameter-free; inputs N_c = 3, h = 12, v = 246.22 GeV): sin²θ_W = 3/13 = 0.23077 (measured 0.23122, 0.19%); m_W = 47v/144 = 80.36 GeV (measured 80.377 GeV, 0.017%); m_Z = 91.63 GeV (measured 91.188 GeV, 0.48%); α⁻¹ = 137.035999084 (CODATA 2018: 137.035999084(21), within experimental uncertainty).

Part of the merkabit paper series. Companion to Zenodo10.5281/zenodo.18925475.