A Finite-State Gibbs Construction from a Recognition Cost

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 $X_ω=J(r_ω)$, 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.