G3 — Fisher metric uniqueness (Čencov)

Statement

On the sieve state space $\Delta_2 = \{(p, 1-p) : p \in (0,1)\}$, the Fisher metric

is the unique Riemannian metric monotone under all Markov maps of the sieve (in particular the transfer matrix $T_m$ and its marginals). “Monotone” means that for every stochastic matrix $T$, $g_{T(P)}(T_*v, T_*v) \leq g_P(v,v)$.

Why it matters

G3 is the metric uniqueness pillar of the PT reconstruction chain. Together with G1 (uniqueness of $D_{KL}$), it pins down the information geometry of the sieve: with no arbitrary choice, $\Delta_2$ carries one and only one Riemannian metric.

This uniqueness propagates downstream: Fisher → Lemma F (metric reconstruction) → Lorentzian Riemannian manifold (beyond the threshold $\mu_c$) → Einstein equations. Without G3, the Riemannian geometry of the relativistic branch would be constructed, not forced.

Proof — outline

External import. Čencov (1982, Statistical Decision Rules and Optimal Inference, AMS) proved that the Fisher metric is the unique Riemannian metric monotone on the probability simplex, up to a positive constant.

PT contribution.

1. Verify that $\Delta_2$ is a standard probability simplex (by T1, two non-zero residue classes mod 3);
2. Verify that the transfer matrix $T_m$ is stochastic (rows non-negative summing to 1);
3. Verify monotonicity numerically: all sieve projections give a max ratio $= 1.000000$.

Čencov’s theorem then applies directly.

Elimination. Euclidean metric (396/1200 tests fail monotonicity), $\chi^2$ metric $ds^2 = dp^2/p^2$ (not Markov-contractive), weighted metrics $ds^2 = w(p)\,dp^2$ (only $w = 1/(p(1-p))$ — Fisher — survives).

See also

• G1 — Uniqueness of $D_{KL}$ — first uniqueness pillar
• Lemma F — Metric reconstruction — uses G3 for Fisher → Riemann
• All theorems