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, $[\bullet] \leftrightarrow P^2(\mathbb{F}_2)$ (Document 000). The primordial symmetry $G = S_4 \times \mathbb{Z}_2^3$ 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 ($125.8 \pm 0.5$ GeV, achieving 0.42%/0.50% accuracy) and the Vacuum Expectation Value ($v = M_Z \times e$, 0.67% accuracy), effectively resolving the Hierarchy Problem without supersymmetry. The $W/Z$ mass ratio is geometrically fixed at $\sqrt{10/13}$ (0.52% error). Complete Fermion Mass Spectrum: Three generations of leptons are derived with 0.002%–0.006% accuracy using PROVEN triadic generation factors $K_n = [720, 170, 204]$. 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 $PG(2,\mathbb{F}_2)$. A new theorem rigorously proves $N_{gen} = 3$ 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 $\heartsuit(6)$, experimental data, and topological constraints. The TTKM theorem ($\det(T(2,n)) = n$) provides topological mass quantization, unifying knot theory with coherence. Information-Mass-Energy Conversion: The framework derives the equivalence 1 bit ≡ 1 coset step on $Q_6$ entirely from SOURCE, without importing Landauer's principle. The coherence cost $\delta_{rel} = 1/|G/H| = 1/64$ and $\hbar = |G|^{-1} = 1/192$ 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 $T_{cc}^+$ 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 $\eta_B = 6.05 \times 10^{-10}$ via $S_4$-geometric leptogenesis (1.2% error, within $2\sigma$ of Planck 2018). SEMF asymmetry ($a_A = \varepsilon_0 \times 19/12$, 0.05%) and pairing ($a_P = \varepsilon_0 \times 23/28$, 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 $210 \pm 8$ GeV and Koheron-896 at $572 \pm 23$ GeV. The framework successfully establishes 12 headline results as PROVEN ($\le 1\%$ 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.