A Homotopical Geometry of the Multinomial and Negative-Multinomial Potentials

Abstract

We study the one-parameter family of Hessian metrics $g_α=αh_++(1-α)h_-$, $α\in[0,1]$, formed by convexly combining the log-partition functions (not the densities) of the multinomial and negative multinomial Fisher information Hessians. Four results are proved in full. (1) $g_α\succ0$ for all $α\in[0,1]$; its Gaussian curvature ($n=2$) is $-\tfrac14$ at $α=0$ and $+\tfrac14$ at $α=1$, but non-constant for intermediate $α$, changing sign at a closed-form critical value $α^\ast(w)$, with an analytic uniqueness proof. (2) Viewing $g_α$ as the spatial metric of a Kasner-type volume equation, positive-definiteness forces one conserved quantity $β\le0$, while a second, independent quantity $C=4\det K$ governs the equation's branch (bounded oscillatory, parabolic, or unbounded growth) and is not fixed by $β$ or positive-definiteness alone; we show this discriminant dictionary is a theorem about the abstract matrix equation, to which $g_α$ supplies initial data of either sign of $C$ without the trajectories remaining on $\{g_α(θ)\}$ itself. (3) $Ψ_α$ is identified as the log-partition function of a compound distribution: a negative-binomial latent count $M$ (parameter $r=1-α$), augmented by a Bernoulli parity bit and split multinomially, with non-negative base measure for every $α$. (4) The associated Bregman divergence obeys a generalised Pythagorean theorem whose projection onto the symmetric locus is the arithmetic mean, independent of $α$; the endpoints' curvature sign matches the Gauss--Bonnet excess of genuine geodesic triangles. Numerical checks are kept separate from analytic proofs throughout.