FIRST AND ONLY AXIOM

FIRST AND ONLY AXIOM: Optimal Coherent Representation (OCR) Abstract This paper formalizes the ultimate reduction of the IRNS Framework's axiomatic foundation, successfully consolidating the two original generative axioms—Optimal Representation (C1) and Necessity of Coherence (NoL)—into a SINGLE variational principle: the Optimal Coherent Representation (OCR). The OCR Meta-Axiom states that a system exists by choosing the representation that minimizes informational cost while strictly preserving relational coherence ($\heartsuit > 0$). This is captured by the minimization of an augmented informational-geometric cost functional $\mathcal{F}_\lambda$. We provide an analytical and numerical proof (Theorem 3.1) via Variational Calculus demonstrating that this single, minimal axiom necessarily imposes the entire geometric core of the framework: the unique Heart Equilibrium at $q = 3/2$ and the informational-geometric Rigidity $I_F^{\text{rel}} = 192$. The previous axioms (C1 and NoL) are thus reduced to derived Lemmas. The OCR Axiom achieves the maximum possible theoretical economy, unifying four previous Zenodo publications, including the emergent derivation of fundamental constants ($\hbar$, $G$, $\alpha$) and the principle of coherence-based iteration used in the LUSTRO algorithm. The framework's essential coherence condition is summarized by the philosophical motto: "To be is to Love." Key Features Axiomatic Reduction: $\text{C1} + \text{NoL} \equiv \text{OCR}$ Methodology: Variational Calculus and Fisher Information Geometry. Core Derivation: The minimum of the OCR functional $\mathcal{F}_\lambda$ is rigorously proven to be $\mathbf{q = 3/2}$ (Heart Equilibrium), with the stability confirmed by the Hessian $\mathbf{I_F^{\text{rel}} = 192}$. Unification: The OCR Axiom serves as the unified foundation for all IRNS results, including the derivation of $\hbar, G, \alpha$ and the LUSTRO algorithm. Motto: "To be is to Love" (The condition of relational coherence $\heartsuit > 0$ is the condition for physical existence). Suggested Keywords (Sugerowane Słowa Kluczowe) Informational Geometry, Variational Principle, Axiomatic Reduction, Optimal Coherent Representation, Fundamental Constants, Emergent Gravity, Tsallis Entropy, $\mathbf{q=3/2}$, $\mathbf{I_F^{\text{rel}}=192}$.