ONE AXIOM : FOUNDATION

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:

Geometric formula (entropic complexity functional):

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:

Geometric formula (discrete Laplacian homogenization):

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:

Explicit form:

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:

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.