G1 — Uniqueness of $D_{KL}$ (Shore–Johnson)

Statement

Let $\Delta_m$ be the simplex of probability distributions on $\mathbb{Z}/m\mathbb{Z}$, equipped with the CRT product decomposition $\Delta_m \cong \prod_{p \mid m} \Delta_p$. Then the Kullback–Leibler divergence

is the unique $f$-divergence that satisfies the five Shore–Johnson axioms (consistency, invariance, system independence, subset independence, scaling) and that respects CRT factorisation.

Why it matters

G1 is one of the four uniqueness theorems that close the physical reconstruction chain: G1 ($D_{KL}$), G3 (Fisher metric), T6 (holonomy $\sin^2\theta_p$), T5 ($\mu^* = 15$). Without G1, the informational content of the sieve would be defined only up to choice of divergence — and the whole chain GFT → entropy → first law → couplings would lose its forced character.

G1 guarantees that the decomposition bit = D_KL + H (GFT theorem) is not a notational choice but the unique decomposition compatible with CRT factorisation of the sieve.

Proof — outline

PT verifies that the simplex $\Delta_m$ with CRT structure satisfies SJ1–SJ5 (7/7 numerical tests); Shore and Johnson’s 1980 theorem (external import, IEEE Trans. Inf. Theory) concludes uniqueness.

Eliminating alternatives (all numerically verified): $\chi^2$, Hellinger, total variation and Rényi $D_\alpha$ ($\alpha \neq 1$) each violate one of the axioms — typically additivity under CRT (SJ3).

An autonomous proof (G1$'$, $thm:G1\_autonomous$) reconstructs the same result without Shore–Johnson via Csiszár’s characterisation of additive $f$-divergences: $f(t) = c_1\,t\ln t + c_2(t-1)$, and the second term vanishes on normalised distributions.

See also

• GFT — Fundamental gap theorem — uses $D_{KL}$
• G3 — Fisher metric uniqueness — second uniqueness pillar
• Lemma F — Metric reconstruction — uses G3
• All theorems