One Integer, Three Generations, ~32 Standard Model quantities from a₁ = 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, \ (₏₏₍\), the spectral action coefficients, and several structural identities—are determined by a single integer \ (a₁ = 5\), the unique solution of \ (a₁! = 4\, a₁ (a₁+1) \). The framework has no adjustable parameters beyond the electron mass \ (mₑ\) (a dimensional anchor): falsification of any single prediction invalidates the entire structure. From \ (a₁\) we derive \ (= (1+a₁) /2\) (golden ratio), the 600-cell (\ (a₁! = 120\) vertices), and \ (E₈\) (\ (h = a₁ (a₁+1) = 30\) ). We prove \ (N₆₄₍ = 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\) norm and a universal coefficient \ (C = 2²W/N₆₄₍\) 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 \ (A₅\) with the icosahedral permutation decomposition (\ (11 = 3 + 8\) is the only valid partition). The CKM correction coefficients are ratios of \ (E₈\) Dynkin leg dimensions; the CP phase \ (₂₊₌ = (5) \) (0. 77%) is the Galois angle of \ (\) ; the strong CP problem is resolved by Galois invariance (\ (ₐ₂₃ = 0\) ). PMNS mixing angles follow from a Galois mixing matrix on \ (A₅\) irreps with eigenvalues \ (\0, N₆₄₍, a₁\\) and tribimaximal eigenvectors; \ (A₅\) Clebsch–Gordan corrections give all three angles within \ (0. 8\) of experiment. New in v3. 8 Chirality derived: The McKay graph of the binary icosahedral group is a tree, hence bipartite. The bipartite grading \ (F = (-1) ^2j\) (spin parity of 2I irreps) provides a chirality operator with \ (\F, DF\ = 0\) (exact). The WHITE partition has dimension 16 = fermions per SM generation. Weak isospin \ (T₃\) matches 8/9 fermions. Casimir sum rules: \ (C₂ = 26 = a₁²+1\), giving a third independent derivation \ (²W = b₁/ C₂ = 6/26\). Masses as Wilson lines: Physical masses are path-ordered products on the McKay tree, not eigenvalues of the finite Dirac operator \ (DF\). The mass operator and coupling operator do not commute: \ (V, A 0\) (complementarity theorem). Complete fermionic action: \ (SF =, D\) with \ (D = DM1₃₀ + ₅₎ₑ₌ DF\), chirality \ (= (-1) ᵖ (-1) ^2j\), and balanced chiral subspaces \ ( (H_+) = (H_-) = 39, 600\). New in v3. 9 Galois kernel theorem: The kernel \ ( (b₁ A₅₈₁₄ₑ - A) \) on the McKay graph selects exactly \ (₀ ₁ ₈\) (dim = 9 = \ (N₄₈₆\) ), with \ (b₁ = 6\) being the unique integer producing a nontrivial kernel. The dimensions of the kernel irreps sum to \ (1+2+2 = 5 = a₁\). Alpha equation completed: The product identity \ (L (3) L (5) L (3') = N = 120\) selects \ (a₁ = 5\) uniquely. All three coefficients of the \ (\) equation are derived: topological (\ (2\) ), spectral (\ (N⁴/b₁\) ), normalization (1). \ (N₆₄₍ = 3\) strengthened: Three-line proof from \ (Z\) units: \ (|N (1+b) | = 1 b \0, 1, 2\\), equivalent to "the Fibonacci sequence has exactly three terms equal to 1. " Spectral origin of \ (N₆₄₍\) in mass corrections: The Galois anti-symmetric Cayley eigenvalue \ (Gₖ = (₁₀₀ - ₅₀) /2\) vanishes for WHITE/\ (A₅\) irreps and is nonzero for BLACK/spin irreps. At the strange quark node \ (₄\): \ (|G| = 3 = N₆₄₍\), yielding \ (ₛ = C G/² = -N₆₄₍ C/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 \ (ᵤ = 0\) follows from \ (G (₂) = -6 < 0\) (sign selection). Higgs mass from McKay spectrum: \ (Tr (DF²) = 2|edges| \|Y\|F² = 2 8 12 = 8 = rank (E₈) \) is a universal topological invariant (CG-independent). Combined with the Lucas ratio \ (L (₈) /L (₂) = ²\), this gives \ (mH = mW (- rank (E₈) ) = 125. 08\) GeV (0. 13%). Chiral gauge coupling derived: The McKay bipartite grading \ (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) ₁\) (BLACK), it couples only to left-handed fermions. \ (SU (3) 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\) has \ (||=1\), making mass hierarchy structurally impossible. The framework is rigidly selective, not “infinitely accommodating. ” Key Predictions Quantity Formula Error Status \ (^-1\) (fine structure) \ (2 x² - 4a₁⁴ x + 1 = 0\) 0. 0001% Derived \ (ₛ\) (strong coupling) \ (1/ (2³) \) 0. 11% Derived \ (²W\) \ (b₁/ (a₁²+1) = 6/26\) 0. 19% Derived Gauge group \ (SU (3) SU (2) U (1) \) from \ (A₅\) McKay Exact Derived \ (N₆₄₍\) \ (3\) (Nyquist on icosahedron) Exact Derived 9 fermion masses (corrected) Norm-log: \ (C = 2²W/N₆₄₍\) 0. 11% RMS Derived 6 quarks \ (q = 2T₃ C |N (z) | ^T₃-1/2\) 0. 13% RMS Derived 2 leptons (\ (, \) ) \ (_ = c_ sign (z') |z'|^3/4\) 0. 013% RMS Derived \ (mH/mW\) \ (- rank (E₈) \) 0. 13% Derived \ (m₃\) (heaviest neutrino) \ (2mₑ/^35 = 49. 5\) meV — Prediction \ (nₒ₄₄ₒ₀ₖ = 35\) \ (|A| = | dₖ² \, sgn (ₖ) |\) (spectral asymmetry) Exact Derived CKM angles (all 3) \ ( (^-n (1+c) ) \) < 0. 2% Derived PMNS \ (²₂₃\) \ (4/7\) (\ (A₅\) Clebsch–Gordan) 0. 11% Derived PMNS \ (²₁₂\) \ (2/ (+a₁) \) (TBM + Galois) 2. 3% (\ (0. 8\) ) Derived PMNS \ (²₁₃\) \ (1/ (a₁ N₄₈₆) = 1/45\) 0. 13% (\ (0. 05\) ) Derived \ (₂₊₌\) \ ( (5) \) 0. 77% Derived \ (₏₌₍ₒ\) \ (3 (5) \) 0. 36% Derived Majorana phases \ (₁, ₂\) \ (4/a₁ = 144^\) — Partially derived \ (mₑ/mP\) \ (^4²\) 0. 24% Derived \ (mZ/mₑ\) \ (^25 (mZ) / (0) \) 0. 09% Derived (+running) Jarlskog \ (J\) \ (3. 12 10^-5\) 1. 1% Derived \ (ₐ₂₃\) \ (0\) (Galois invariance) Exact Derived \ (\|Y\|F\) (Yukawa norms) \ (1/2\) universal (McKay CG) Exact Derived \ (Tr (DF²) = 8 = rank (E₈) \) Topological invariant Exact Derived Chirality \ (F = (-1) ^2j\) Bipartite grading of McKay tree Exact (9/9) Derived \ (C₂ = a₁²+1 = 26\) SU (2) Casimir sum over 2I irreps Exact Derived \ (|G (₄) | = N₆₄₍\) (spectral) Galois anti-symmetric Cayley eigenvalue Exact Derived Galois kernel dim = \ (N₄₈₆ = 9\) \ ( (b₁ A₅₈₁₄ₑ - A) \) Exact Derived Beta coefficients \ (bᵢ\) From \ (N₆₄₍=3\), \ (NH=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_\) (new physics) \ (0\) (no new charged particles) — Derived (falsifiable) Dark sector Galois conjugation \ (-1/\) — Derived Seeley–DeWitt \ (cₖ\) \ (cₖ = 2N Aₖ\) ; \ (c₁/ (2c₀) = (h (E₈) +1) / (a₁+b₁) \) Exact Derived Galois norm products \ (Lₖ L₊' Z\): \ (b₁², \; 16a₁, \; 27a₁\) Exact Derived \ (P\) (cosm. const. ) \ (^ (N-N₆₄₍) /2 - \) 0. 33% Pattern The Galois conjugation \ (' = -1/\) 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 \ (P = ^ (N - N₆₄₍) /2 - \) to 0. 13\ (\) (0. 33%). Falsifiable Predictions (Testable by 2030) \ (²₁₃ = 1/45 = 0. 02222\) — JUNO, < 0. 5% precision expected \ (m_ = 0. 058\) eV — EUCLID/DESI, sensitivity ~0. 02 eV \ (₏₌₍ₒ = 197. 7^\) — Hyper-K/DUNE, ±10° \ (mH/mW = - rank (E₈) = 1. 559\) — HL-LHC, ±0. 1% \ (MR mₑ ^35/2 5. 3\) TeV (right-handed neutrino) — FCC-hh direct search \ (a_ (new physics) = 0\) — if anomaly confirmed as BSM, framework is refuted Any single miss falsifies the entire framework. Honest Limitations Ele