The ONE AXIOM Foundation (0A): Geometric Derivation of $h, c$, and $G$ from the Primordial Symmetry Abstract: This is the final version (4.6, 28 December 2025) of Document 0A, the foundational pillar of the ONE AXIOM framework. It presents complete, independent, and non-circular geometric derivations of the three fundamental constants — Planck’s constant ($h$), the speed of light ($c$), and the gravitational constant ($G$) — all emerging from a single primordial symmetry group $\mathbf{G} = S_4 \times \mathbb{Z}_2^3$. No empirical fitting is used. Each constant emerges autonomously from its own mathematical domain via two rigorously defined pathways: OCR (Optimal Coherent Representation) and OER (Optimal Epistemic Resolution). 1. Planck’s Constant ($\hbar$) – The Coherence Quantum Primary: independent formula for $\hbar$ In natural units ($\hbar = c = 1$), Planck’s constant emerges from entropic minimization: $$\hbar_{\text{natural}} = \arg \min_{\rho \in \Delta_N} \mathcal{S}_{\text{ent}}(\rho) = 1 \quad (233)$$ Geometric formula (entropic complexity functional): $$\mathcal{S}_{\text{ent}} : \Delta_N \to \mathbb{R}, \quad \mathcal{S}_{\text{ent}}(\rho) = \left. \frac{1 - \sum_{i=1}^{N} p_i^{q^*}}{q^* - 1} \right|_{q^*=3/2} \quad (234)$$ Components (all from geometry, no $c$ or $G$): $\Delta_N$: probability simplex with $N = 96$ vertices (from IF/2) $q^ = 3/2$*: heart equilibrium (unique minimum of tension functional) $\rho^ = (1/N, \dots, 1/N)$*: uniform distribution (unique $\mathbf{G}$-invariant minimum) Physical meaning: $\hbar_{\text{natural}}$ is the unit of relational coherence. 2. Speed of Light ($c$) – The Propagation Invariant Primary: independent formula for $c$ In natural units ($\hbar = c = 1$), the speed of light emerges from Laplacian homogenization: $$c_{\text{natural}} = \beta \sqrt{C_{\text{eff}}} = 1 \quad (268)$$ Geometric formula (discrete Laplacian homogenization): $$\mathcal{L}_{R}^{base} \xrightarrow[\ell \to 0]{\Gamma\text{-conv}} C_{\text{eff}}(-\Delta), \quad C_{\text{eff}} = 1 \text{ (isotropic cubic)} \quad (269)$$ Components (all from geometry, no $\hbar$ or $G$): $\mathcal{L}_{R}^{base}$: discrete Laplacian on relational network $\beta$: intrinsic velocity scale from relational dynamics $C_{\text{eff}} = 1$: effective conductivity (unique for isotropic networks) $\Gamma$-convergence: rigorous homogenization theorem Physical meaning: $c_{\text{natural}} = 1$ 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 ($\hbar = c = 1$), the gravitational constant emerges directly from Fisher–Rao geometry: $$G_{\text{natural}} = S^{-N_{\text{exact}}} = S^{-\left(N_{\text{OER}} + \frac{1 + \pi / C_{\text{obs}}}{S - \pi}\right)} \quad (143)$$ Explicit form: $$G_{\text{natural}} = S^{-\left(68 + \frac{1 + \pi/150}{S - \pi}\right)} = (18.68)^{-68.0657} = 2.91 \times 10^{-87} \quad (144)$$ Components (all from geometry, no $c$ or $\hbar$): $S = 6\pi - e/16 = 18.68$: geometric capacity from Tsallis manifold $N_{\text{OER}} = 68$: OER coherence partition $C_{\text{obs}} = 150 = 96 + 64 - 10$: observable modes minus metric DOF $\pi/C_{\text{obs}}$: Fisher–Rao per-mode curvature $S - \pi$: quantum geometric excess Physical meaning: $G_{\text{natural}} = \tau_P^2 / t_0^2$ is the squared ratio of Planck time to unit time. Triangle Identity (TI): A Consistency Check, Not a Derivation After independent derivations, the three constants satisfy the Master Identity: $$\frac{G\hbar}{c^5} = \frac{1}{S^N}$$ This serves exclusively as an internal consistency verification. TI does not define or constrain any constant; it confirms the coherence of the dual-track architecture. Key Guarantees Against Fitting: Disjoint Domains: Each constant is derived on a separate mathematical chart (simplex, graph, interval). Singleton Solutions: The possibility space is a singleton for each — no free parameters. PSP Triviality: The Possibility Selection Principle (PSP) acts trivially on fundamentals — selection is forced by geometry. SI Anchors: SI numerical values serve as unit translation anchors, not as inputs to derivation. Keywords: ONE AXIOM, fundamental constants, geometric derivation of G, Planck constant, speed of light, $\mathbf{G} = S_4 \times \mathbb{Z}_2^3$, OCR/OER duality, Fisher–Rao curvature, primordial symmetry, dimensional emergence, Triangle Identity.