A Homotopical Geometry of the Multinomial and Negative-Multinomial Potentials
Abstract
We study the one-parameter family of Hessian metrics gα=αh++(1−α)h-, α∈[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) gα≻0 for all α∈[0,1]; its Gaussian curvature (n=2) is −1/4 at α=0 and +1/4 at α=1, but non-constant for intermediate α, changing sign at a closed-form critical value α∗(w), with an analytic uniqueness proof. (2) Viewing gα as the spatial metric of a Kasner-type volume equation, positive-definiteness forces one conserved quantity β≤0, while a second, independent quantity C=4det 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 gα supplies initial data of either sign of C without the trajectories remaining on {gα(θ)} itself. (3) Ψα 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.
Related
- Generalized DCCQ: From Binary Quotients to Multinomial Simplex Geometry and Critical-Strip Coordinates
- Cosmological Correlators from Generating Potentials and Projective Geometry
- Homological and homotopical epimorphisms in monoidal ∞-categories
- Quantum geometry at arbitrary genus - I: Anharmonic potentials
- A homotopical enhancement of Neisendorfer's algebraic models
BibTeX
@misc{yoshizawa2026homotopical,
title = {{A Homotopical Geometry of the Multinomial and Negative-Multinomial Potentials}},
author = {Shintaro Yoshizawa},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.26819},
eprint = {2609.26819},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26819},
}
RIS
TY - UNPB
TI - A Homotopical Geometry of the Multinomial and Negative-Multinomial Potentials
AU - Shintaro Yoshizawa
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2609.26819
UR - https://arxiv.org/abs/2609.26819
AB - We study the one-parameter family of Hessian metrics g_α=αh_++(1−α)h_-, α∈[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≻0 for all α∈[0,1]; its Gaussian curvature (n=2) is −1/4 at α=0 and +1/4 at α=1, but non-constant for intermediate α, changing sign at a closed-form critical value α^∗(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≤0, while a second, independent quantity C=4det 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.
ER -