ONE AXIOM : FOUNDATION

ONE AXIOM Foundation: Complete Derivation of Fundamental Physical Constants via Primordial Symmetry Description: This foundational document establishes the mathematical basis for the ONE AXIOM framework, providing the first complete derivation of fundamental physical constants ($\hbar$, $c$, $G$) from a single primordial symmetry group $G=S_{4}\times\mathbb{Z}_{2}^{3}$ with zero free parameters. 1. Planck’s Constant ($\hbar$) – The Coherence Quantum Primary: independent formula for $\hbar$ In natural units ($\hbar=1$), Planck’s constant emerges from entropic minimization: $$\hbar_{\text{natural}} = \arg \min_{\rho \in \Delta_N} \mathcal{S}_{\text{ent}}(\rho) = 1$$ Geometric formula (entropic complexity functional): $$\mathcal{S}_{\text{ent}} : \Delta_N \to \mathbb{R}, \quad \mathcal{S}_{\text{ent}}(\rho) = \left. \frac{1 - \sum_{i=1}^{N} p_i^{q^*}}{q^* - 1} \right|_{q^*=3/2}$$ Components: $\Delta_N$ (probability simplex with $N=24$ vertices from $IF/2$), $q^*=3/2$ (heart equilibrium), $\rho^*$ (uniform distribution). Physical meaning: $\hbar$ is the unit of relational coherence. 2. Speed of Light ($c$) – The Propagation Invariant Primary: independent formula for $c$ In natural units ($c=1$), the speed of light emerges from Laplacian homogenization: $$c_{\text{natural}} = \beta \sqrt{C_{\text{eff}}} = 1$$ Geometric formula (discrete Laplacian homogenization): $$\mathcal{L}_{R}^{base} \xrightarrow[\ell \to 0]{\Gamma\text{-conv}} C_{\text{eff}}(-\Delta), \quad C_{\text{eff}} = 1 \text{ (isotropic cubic)}$$ Components: $\mathcal{L}_{R}$ (discrete Laplacian on relational network), $\beta$ (intrinsic velocity scale), $C_{\text{eff}}$ (effective conductivity). Physical meaning: $c$ is the maximal coherence propagation rate — the upper bound on information speed. 3. Gravitational Constant ($G$) – The Curvature Invariant Primary: independent formula for $G$ In natural units ($G=1$), the gravitational constant emerges directly from Fisher–Rao geometry: $$G_{\text{natural}} = S^{-N_{\text{exact}}} = S^{-\left(N_{\text{OER}} + \frac{1 + \pi / C_{\text{obs}}}{S - \pi}\right)}$$ Explicit form: $$G_{\text{natural}} = S^{-\left(68 + \frac{1 + \pi/150}{S - \pi}\right)} = (18.68)^{-68.0657} = 2.91 \times 10^{-87}$$ Components: $S$ (geometric capacity from Tsallis manifold), $N_{\text{OER}}=68$ (coherence partition), $C_{\text{obs}}=150$ (observable modes), $\pi$ (Fisher–Rao per-mode curvature). Physical meaning: $G$ is the squared ratio of Planck time to unit time. The framework operates on the Dual-Track Derivation principle, utilizing the Optimal Coherent Representation (OCR) as the system's Generator and the Optimal Epistemic Resolution (OER) as the observer's Reflector. This document serves as the "Mathematical Kernel" and primary ontological map for the entire ONE AXIOM project, bridging information geometry, category theory, and fundamental physics.