The Gibbs Distribution as a Theorem: Deriving Statistical Mechanics from the Recognition Composition Law

We prove that the Boltzmann–Gibbs distribution is a theorem, not a postulate, of the framework governed by the cost functional J(x) = 1/2 (x + x⁻¹) − 1—the unique solution to the Recognition Composition Law J(xy) + J(x/y) = 2J(x)J(y) + 2J(x) + 2J(y). Starting from this single functional equation with zero free parameters, we establish a complete derivation chain: (i) the Ancestor Inequality J(x) ≥ (log x)²/2, proving that J is strictly stronger than squared information content and that Shannon entropy is a second-order shadow of J-cost; (ii) a J-cost divergence DJ(q‖p) = Σ pᵢJ(qᵢ/pᵢ) ≥ 0 that dominates the χ²-divergence; (iii) a many-body counting argument showing that Shannon entropy emerges from microstate multiplicities of N-particle ledgers via Stirling's approximation; (iv) a Lagrange stationarity theorem proving that the unique distribution maximising entropy subject to a J-cost constraint is the Gibbs measure pω ∝ exp(−J(Xω)/TR); and (v) the free energy–KL identity FR(q) − FR(Gibbs) = TR DKL(q‖Gibbs), from which the second law of thermodynamics follows as a corollary. The Recognition Temperature TR arises as the Lagrange multiplier of the optimisation—it is derived, not assumed. The entire construction uses zero adjustable parameters.