One Integer, Three Generations, ~32 Standard Model quantities from a_1 = 5 with zero free parameters

We present a rigid mathematical framework in which ~35 independent quantities—the gauge group \(SU(3)\!×\!SU(2)\!×\!U(1)\), three coupling constants, three generations, nine fermion mass ratios (with zero-parameter corrections to 0.11% RMS), six mixing angles, two CP phases, the strong CP angle, the Higgs-to-\(W\) mass ratio, the \(Z\)-to-electron mass ratio, the electron-to-Planck mass ratio, one neutrino mass, \(γPPN\), the spectral action coefficients, and several structural identities—are determined by a single integer \(a_1 = 5\), the unique solution of \(a_1! = 4\,a_1(a_1+1)\). The framework has no adjustable parameters beyond the electron mass \(m_e\) (a dimensional anchor): falsification of any single prediction invalidates the entire structure. From \(a_1\) we derive \(φ = (1+√a_1)/2\) (golden ratio), the 600-cell (\(a_1! = 120\) vertices), and \(E_8\) (\(h = a_1(a_1+1) = 30\)). We prove \(Ngen = 3\) via Galois invariance and derive all nine fermion mass exponents through Galois conjugation and Casimir eigenvalues. A norm-log correction formula using the \(Z[φ]\) norm and a universal coefficient \(C = 2sin^2θ_W/Ngen\) predicts all six quark masses to 0.13% RMS; a complementary Galois-conjugate mechanism with exponent \(k = 3/4 = 1 - 1/(spacetime)\) corrects lepton masses to 0.013% RMS—yielding all nine charged fermion masses to 0.11% RMS with zero free parameters. The gauge group is the unique compact Lie group whose adjoint restricts to \(A_5\) with the icosahedral permutation decomposition (\(11 = 3 + 8\) is the only valid partition). The CKM correction coefficients are ratios of \(E_8\) Dynkin leg dimensions; the CP phase \(δCKM = (√5)\) (0.77%) is the Galois angle of \(φ\); the strong CP problem is resolved by Galois invariance (\(θQCD = 0\)). PMNS mixing angles follow from a Galois mixing matrix on \(A_5\) irreps with eigenvalues \(\{0, Ngen, a_1\}\) and tribimaximal eigenvectors; \(A_5\) Clebsch–Gordan corrections give all three angles within \(0.8σ\) of experiment. New in v3.8 Chirality derived: The McKay graph of the binary icosahedral group is a tree, hence bipartite. The bipartite grading \(γ_F = (-1)²ʲ\) (spin parity of 2I irreps) provides a chirality operator with \(\{γ_F, D_F\} = 0\) (exact). The WHITE partition has dimension 16 = fermions per SM generation. Weak isospin \(T_3\) matches 8/9 fermions. Casimir sum rules: \(∑ C_2 = 26 = a_1^2+1\), giving a third independent derivation \(sin^2θ_W = b_1/∑ C_2 = 6/26\). Masses as Wilson lines: Physical masses are path-ordered products on the McKay tree, not eigenvalues of the finite Dirac operator \(D_F\). The mass operator and coupling operator do not commute: \([V, A] ≠ 0\) (complementarity theorem). Complete fermionic action: \(S_F = Ψ, DΨ\) with \(D = D_M⊗1₃₀ + γform⊗ D_F\), chirality \(γ = (-1)^p ⊗ (-1)²ʲ\), and balanced chiral subspaces \((H_+) = (H_-) = 39{,}600\). New in v3.9 Galois kernel theorem: The kernel \((b_1 Âfiber - Â)\) on the McKay graph selects exactly \(ρ_0 ⊕ ρ_1 ⊕ ρ_8\) (dim = 9 = \(Neig\)), with \(b_1 = 6\) being the unique integer producing a nontrivial kernel. The dimensions of the kernel irreps sum to \(1+2+2 = 5 = a_1\). Alpha equation completed: The product identity \(L(3) · L(5) · L(3') = N = 120\) selects \(a_1 = 5\) uniquely. All three coefficients of the \(α\) equation are derived: topological (\(2π\)), spectral (\(Nφ^4/b_1\)), normalization (1). \(Ngen = 3\) strengthened: Three-line proof from \(Z[φ]\) units: \(|N(1+bφ)| = 1 ⇒ b ∈ \{0,1,2\}\), equivalent to "the Fibonacci sequence has exactly three terms equal to 1." Spectral origin of \(Ngen\) in mass corrections: The Galois anti-symmetric Cayley eigenvalue \(G_k = (λ₁₀ₐ - λ₅ₐ)/2\) vanishes for WHITE/\(A_5\) irreps and is nonzero for BLACK/spin irreps. At the strange quark node \(ρ_4\): \(|G| = 3 = Ngen\), yielding \(δ_s = C · G/φ^2 = -Ngen C/φ^2\) exactly. The number of generations appearing in the mass correction is a computed Cayley-graph eigenvalue, not a free parameter. The up quark's \(δ_u = 0\) follows from \(G(ρ_2) = -6φ < 0\) (sign selection). Higgs mass from McKay spectrum: \(Tr(D_F^2) = 2|edges| · \|Y\|_F^2 = 2 · 8 · 1/2 = 8 = rank(E_8)\) is a universal topological invariant (CG-independent). Combined with the Lucas ratio \(L(ρ_8)/L(ρ_2) = φ^2\), this gives \(m_H = m_W(φ - rank(E_8) · α) = 125.08\) GeV (0.13%). Chiral gauge coupling derived: The McKay bipartite grading \(γ_F = (-1)²ʲ\) is a \(Z/2\) ring homomorphism. All dim-2 irreps (BLACK) flip chirality under tensor product; all dim-3 irreps (WHITE) preserve it. Since \(SU(2) ↔ ρ_1\) (BLACK), it couples only to left-handed fermions. \(SU(3) ↔ ρ_2\) (WHITE) is vector-like. This resolves the open question of why \(SU(2)_L\) is chiral. Uniqueness among regular polytopes: All six regular 4D polytopes tested against seven SM criteria (generations, gauge group, Weinberg angle, fine-structure constant, mass hierarchy, mixing angles, anomaly cancellation). Result: 600-cell 7/7; the four independent alternatives (5-cell, 8-cell, 16-cell, 24-cell) score 0/7. The 120-cell (dual, same algebra) scores 6/7, failing only the gauge test (vertex degree 4, no color octet). The 24-cell's ring \(Z[ω]\) has \(|ω|=1\), making mass hierarchy structurally impossible. The framework is rigidly selective, not “infinitely accommodating.” Key Predictions Quantity Formula Error Status \(α⁻¹\) (fine structure) \(2π x^2 - 4a_1φ^4 x + 1 = 0\) 0.0001% Derived \(α_s\) (strong coupling) \(1/(2φ^3)\) 0.11% Derived \(sin^2θ_W\) \(b_1/(a_1^2+1) = 6/26\) 0.19% Derived Gauge group \(SU(3)× SU(2)× U(1)\) from \(A_5\) McKay Exact Derived \(Ngen\) \(3\) (Nyquist on icosahedron) Exact Derived 9 fermion masses (corrected) Norm-log: \(C = 2sin^2θ_W/Ngen\) 0.11% RMS Derived 6 quarks \(δ_q = 2T_3 · C · ln|N(z)| · φT_3-1/2\) 0.13% RMS Derived 2 leptons (\(μ, τ\)) \(δ_ = c_ · sign(z') · |z'|3/4\) 0.013% RMS Derived \(m_H/m_W\) \(φ - rank(E_8)·α\) 0.13% Derived \(m_3\) (heaviest neutrino) \(2m_e/φ³⁵ = 49.5\) meV — Prediction \(nseesaw = 35\) \(|η_A| = |∑ d_k^2 \,sgn(λ_k)|\) (spectral asymmetry) Exact Derived CKM angles (all 3) \((φ⁻ⁿ(1+c))\) < 0.2% Derived PMNS \(sin^2θ₂₃\) \(4/7\) (\(A_5\) Clebsch–Gordan) 0.11% Derived PMNS \(sin^2θ₁₂\) \(2/(φ+a_1)\) (TBM + Galois) 2.3% (\(0.8σ\)) Derived PMNS \(sin^2θ₁₃\) \(1/(a_1 · Neig) = 1/45\) 0.13% (\(0.05σ\)) Derived \(δCKM\) \((√5)\) 0.77% Derived \(δPMNS\) \(3(√5)\) 0.36% Derived Majorana phases \(α1,2\) \(± 4π/a_1 = ± 144^∘\) — Partially derived \(m_e/m_P\) \(α4φ^2\) 0.24% Derived \(m_Z/m_e\) \(φ²⁵ · α(m_Z)/α(0)\) 0.09% Derived (+running) Jarlskog \(J\) \(3.12 × 10⁻⁵\) 1.1% Derived \(θQCD\) \(0\) (Galois invariance) Exact Derived \(\|Y\|_F\) (Yukawa norms) \(1/√2\) universal (McKay CG) Exact Derived \(Tr(D_F^2) = 8 = rank(E_8)\) Topological invariant Exact Derived Chirality \(γ_F = (-1)²ʲ\) Bipartite grading of McKay tree Exact (9/9) Derived \(∑ C_2 = a_1^2+1 = 26\) SU(2) Casimir sum over 2I irreps Exact Derived \(|G(ρ_4)| = Ngen\) (spectral) Galois anti-symmetric Cayley eigenvalue Exact Derived Galois kernel dim = \(Neig = 9\) \((b_1 Âfiber - Â)\) Exact Derived Beta coefficients \(b_i\) From \(Ngen=3\), \(N_H=1\), gauge group Exact Derived SU(2)\(_L\) chirality McKay bipartite \(Z/2\) ring homomorphism Exact Derived Polytope uniqueness 600-cell: 7/7 SM tests; all others: 0/7 Exact Derived \(a_μ\)(new physics) \(0\) (no new charged particles) — Derived (falsifiable) Dark sector Galois conjugation \(φ → -1/φ\) — Derived Seeley–DeWitt \(c_k\) \(c_k = 2N · A_k\); \(c_1/(2c_0) = (h(E_8)+1)/(a_1+b_1)\) Exact Derived Galois norm products \(L_k · Lk' ∈ Z\): \(b_1^2,\; 16a_1,\; 27a_1\) Exact Derived \(Λ_P\) (cosm. const.) \(α^{(N-Ngen)/2 - φ}\) 0.33% Pattern The Galois conjugation \(φ → φ' = -1/φ\) generates a dark sector: all coupling constants become complex or negative, yielding 9 stable, electromagnetically dark particles interacting only gravitationally—a dark matter candidate with zero additional parameters. Gravitational freeze-in is identified as the unique viable production mechanism. The cosmological constant matches \(Λ_P = α^{(N - Ngen)/2 - φ}\) to 0.13\(σ\) (0.33%). Falsifiable Predictions (Testable by 2030) \(sin^2θ₁₃ = 1/45 = 0.02222\) — JUNO, < 0.5% precision expected \(∑ m_ν = 0.058\) eV — EUCLID/DESI, sensitivity ~0.02 eV \(δPMNS = 197.7^∘\) — Hyper-K/DUNE, ±10° \(m_H/m_W = φ - rank(E_8)·α = 1.559\) — HL-LHC, ±0.1% \(M_R ~ m_e φ³⁵/2 ≈ 5.3\) TeV (right-handed neutrino) — FCC-hh direct search \(a_μ(new physics) = 0\) — if anomaly confirmed as BSM, framework is refuted Any single miss falsifies the entire framework.