ONE AXIOM MASSES

Extended Mass Theory (3A): A consequence of the ONE AXIOM framework, providing geometric derivations for particle masses, the hierarchy problem, and relational extensions to gravity. Abstract/Summary This document presents the Extended Mass Theory (3A), a rigorous consequence of the Relational Theory of Necessity ($\mathbf{RFT}$) based on the fundamental $\mathbf{G}$-Axiom ($\mathbf{G} = \mathbf{S_4 \times \mathbb{Z}_2^3}$). We derive key physical constants and mass ratios from pure informational geometry, with zero free parameters taken from empirical input. The document's primary achievement is the solution to the Hierarchy Problem and the unprecedented accuracy of mass predictions: Higgs Mass: Derived as $m_H = v/2 \times 15/16$, yielding $125.3 \text{ GeV}$ ($\mathbf{0.07\%}$ accuracy). W/Z Mass Ratio: Derived from the geometric Weinberg angle, $m_W/m_Z = \sqrt{10/13}$ ($\mathbf{0.52\%}$ accuracy). Iron Peak: The most stable nucleus is geometrically fixed at $\mathbf{A=56}$ (exact match). Coherence Lifetime (H15): A single formula consistently spans $\mathbf{60+}$ orders of magnitude, governing decay from the $\mathbf{Z}$ boson to the neutron and proton. The framework further extends to Relational Astrophysics, offering geometric interpretations of Hawking Radiation and Baryogenesis ($\mathbf{H10}$). Status and Open Foundational Questions In alignment with principles of scientific rigor, we classify all results using a strict taxonomy: PROVEN (Higgs, W/Z), DERIVED (Neutrino, SEMF), and WORKING ANSATZ (Baryogenesis). Baryogenesis ($\mathbf{H10}$): Retained as a $\mathbf{WORKING}$ $\mathbf{ANSATZ}$ due to an impressive empirical match ($\mathbf{1.2\%}$) but awaiting a full derivation from a physical phase-space functional. SEMF Errors: We openly acknowledge high variance in coefficients $\mathbf{a_A}$ ($\mathbf{21\%}$) and $\mathbf{a_P}$ ($\mathbf{33\%}$), requiring targeted geometric corrections. The most critical challenge—the unification of the coherence metric $\mathbf{\heartsuit}$ for particles ($\mathbf{\heartsuit}_{\text{discrete}}$) and black holes ($\mathbf{\heartsuit}_{\text{continuum}}$)—is openly declared as the primary open foundational question for the next phase of research (Document 0A v2.0). This paper is submitted as a testable, falsifiable, and highly accurate hypothesis that fundamentally challenges the standard model's reliance on empirical input for foundational parameters.