A Finite-State Gibbs Construction from a Recognition Cost

TL;DR

This paper introduces a ratio-cost construction for finite-state Gibbs distributions using the Recognition Composition Law, providing new theoretical insights and technical bounds, with comparisons to alternative models.

Contribution

It develops a finite-state ratio-cost framework based on the Recognition Composition Law, including new bounds and comparisons to other models.

Findings

Derived a continuous positive branch function J(x)=cosh(log x)-1

Established the identity F_R(q)-F_R(p)=T_R D_KL(q||p) for the framework

Compared Gibbs law to alternative models with sample-size power calculations

Abstract

On a finite outcome space, the canonical Gibbs distribution is usually obtained by maximizing Shannon entropy at fixed mean of an externally supplied energy functional. This paper studies the finite-state consequences of a ratio-cost construction instead: after adopting the normalized d'Alembert degree-two closure called the Recognition Composition Law (RCL), with unit log-curvature calibration at the reference ratio, the continuous nontrivial positive branch is $J(x)=\tfrac12(x+x^{-1})-1=\cosh(\log x)-1$. Given the induced cost vector $X_\omega=J(r_\omega)$, multinomial counting and convex duality recover the finite-state Gibbs weights and the identity $F_{\mathrm{R}}(q)-F_{\mathrm{R}}(p)=T_{\mathrm{R}}\,D_{\mathrm{KL}}(q\Vert p)$; the entropy-maximization steps are classical once the cost is fixed. New technical content includes a non-asymptotic Stirling bound and soft-shell constrained-type theorems for real-valued costs. A three-state example compares the Gibbs law to squared-log, affinity-as-energy, and Tsallis alternatives at the same cost vector and mean-cost constraint, with sample-size power calculations at fixed RCL ground truth. The framework is conditional on axioms (A1)--(A3) and restricted to finite outcome spaces with strictly positive weights; it does not derive the composition law from a more primitive principle.