One Axiom : Masses

Title: ONE AXIOM MASSES: From Higgs to Black Holes – A Complete Theory of Mass (Version 5. 0) Abstract: This document (3A, Version 5. 0) presents a geometric theory of particle masses derived entirely from the SOURCE axiom—the mutual determination of Void and Fano plane, P² (F₂) (Document 000). The primordial symmetry G = S₄ Z₂³ emerges as a lemma. Using the rigorous Dual-Track Methodology (OCR/OER), the framework treats mass as a manifest measure of coherence, eliminating the need for arbitrary Yukawa couplings and free parameters in the Standard Model. Key breakthroughs and results presented in this work include: Higgs Sector & Electroweak Precision: Dual-track derivation of the Higgs mass (125. 8 0. 5 GeV, achieving 0. 42%/0. 50% accuracy) and the Vacuum Expectation Value (v = MZ e, 0. 67% accuracy), effectively resolving the Hierarchy Problem without supersymmetry. The W/Z mass ratio is geometrically fixed at 10/13 (0. 52% error). Complete Fermion Mass Spectrum: Three generations of leptons are derived with 0. 002%–0. 006% accuracy using PROVEN triadic generation factors Kₙ = 720, 170, 204. All six quark masses are calculated with sub-percent accuracy (0. 01%–0. 93%) via charge-sector QCD screening. Fano Geometry & The Three-Generation Bound: The mass triplet structures 0, 1, 6 and 0, 0, 7 are derived directly from the Fano plane PG (2, F₂). A new theorem rigorously proves N₆₄₍ = 3 exactly, structurally excluding a fourth generation. Koide Triple Convergence & TTKM Quantization: The Koide parameter Q = 2/3 is established through three independent paths: dimensional coherence (6), experimental data, and topological constraints. The TTKM theorem ( (T (2, n) ) = n) provides topological mass quantization, unifying knot theory with coherence. Information-Mass-Energy Conversion: The framework derives the equivalence 1 bit ≡ 1 coset step on Q₆ entirely from SOURCE, without importing Landauer's principle. The coherence cost ₑ₄₋ = 1/|G/H| = 1/64 and = |G|^-1 = 1/192 emerge as geometric identities, establishing information as the fundamental currency of mass. Mass Tension Bridge & Exotic Hadrons: The framework resolves exotic hadron formation regimes, confirming the molecular nature of T₂₂^+ and proving the mass and width of the compact tetraquark X (6900) via the newly introduced Fano Penalty Theorem (0. 49% accuracy). Baryogenesis & Nuclear Physics (SEMF): Parameter-free prediction of the baryon asymmetry B = 6. 05 10^-10 via S₄-geometric leptogenesis (1. 2% error, within 2 of Planck 2018). SEMF asymmetry (aA = ₀ 19/12, 0. 05%) and pairing (aP = ₀ 23/28, 0. 37%) coefficients are upgraded to PROVEN status via Fano-line isospin and Fano-group deficiency formulas. Black Hole Thermodynamics & Falsifiable Predictions: Extension of the framework reveals a discrete Hawking emission spectrum comprising 51 channels (derived from the 51: 13 coset partition). The model provides clear falsification criteria through the prediction of two new resonance states: Koheron-832 at 210 8 GeV and Koheron-896 at 572 23 GeV. The framework successfully establishes 12 headline results as PROVEN (1\% error with dual-track convergence), demonstrating that the observed mass spectrum of the universe is the unique geometric realization of coherence constraints on the Fano plane.