One Integer, Three Generations: Deriving Particle Physics from a1 = 5

We present a rigid mathematical framework in which 27 independent Standard Model quantities—the gauge group, three coupling constants, three generations, eight fermion mass ratios, 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, and \(γPPN\)—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 \(φ = (1+√5)/2\) (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. 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 to < 0.4%. 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 \(m_H/m_W\) \(φ - 8α\) 0.09% Derived Gauge group \(SU(3)× SU(2)× U(1)\) from \(A_5\) McKay Exact Derived \(Ngen\) \(3\) (Nyquist on icosahedron) Exact Derived Fermion masses (bare) \(m_e · φ⁵ᵃ⁺⁶ᵇ\) 3–21% Derived Fermion masses (corrected) Holonomy corrections per sector 4.4% RMS Alg. motivated CKM angles (all 3) \((φ⁻ⁿ(1+c))\) < 0.2% Derived PMNS \(sin^2θ₂₃\) \(4/7\) (\(A_5\) Clebsch–Gordan) 0.32% Derived PMNS \(sin^2θ₁₂\) \(2/(φ+a_1)\) (TBM + Galois) 0.26% Derived PMNS \(sin^2θ₁₃\) \(1/(a_1 · Neig) = 1/45\) 0.87% Derived \(δCKM\) \((√5)\) 0.77% Derived \(δPMNS\) \(3(√5)\) 0.36% 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 Dark sector Galois conjugation \(φ → -1/φ\) — 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. 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° \(m_H/m_W = φ - 8α = 1.559\) — HL-LHC, ±0.1% Any single miss falsifies the entire framework. Honest Limitations Holonomy correction sector assignments are algebraically motivated, not derived from a single principle (Proposition 7: no-go for static geometry) Cosmological constant formula is a strong pattern without a spectral action mechanism Dark matter abundance ratio \(ΩDM/Ω_b = 7 - φ\) is speculative Electron mass \(m_e\) remains one free dimensional parameter Yukawa edge weights on the McKay graph (8 parameters) remain open Reproducible code: github.com/razvananghelina/One-Integer-Three-Generations — five self-contained Python scripts verifying coupling constants, 600-cell spectrum, fermion masses, Berry phase, and spectral action.