Fisher–Rao Geometry of a Thermal Kaluza–Klein Sector: Exact Information Geometry, Formal Dilaton Analysis, and Phenomenological Decoherence Outlook

Abstract

We present a closed-form analysis of the Fisher–Rao information geometry of the statistical (not thermal-field-theoretic) sector of a Kaluza–Klein compactification, consisting of N_A free bosonic degrees of freedom on the flat torus T⁸ at thermal equilibrium, embedded in a (4 + 8)-dimensional product spacetime. The “thermal” in our title refers to a statistical ensemble of macrostates parametrized by \((\beta ,{{m}^{2}})\), not to a thermal quantum field theory in the usual sense. Using the score-covariance definition of Fisher information for the Gibbs family parametrized by \((\beta ,{{m}^{2}})\), we derive the metric components at the analytically tractable massless locus m² = 0. The family is exponential in \(\beta \) but not a natural two-parameter exponential family in m², because \({{\omega }_{k}} = \sqrt {{{k}^{2}} + {{m}^{2}}} \); hence the metric is not introduced as a naive Hessian of lnZ in both variables. At m² = 0 we prove strict positive definiteness from the inequality \({{I}_{9}}{{I}_{5}} > I_{7}^{2}\) among standard Bose integrals, and compute the Gaussian curvature \(K = 4{{I}_{5}}{\text{/}}({{I}_{9}}{{I}_{5}} - I_{7}^{2}) \cdot 1{\text{/}}(\eta {{T}^{8}}) \approx 1.165{\text{/}}s\), where it is strictly positive. Restricting to the equilibrium submanifold reproduces \({{G}_{{ss}}} = 1{\text{/}}(8s)\) exactly. These constitute the paper’s core, exact results at the massless locus; the full \((\beta ,{{m}^{2}})\) dependence away from m² = 0 remains open. We then pursue two formal extensions that we treat with explicitly different epistemic status. First, formally promoting the Fisher manifold \({{\mathcal{M}}_{2}}\) to a sigma-model target, we show that the positivity of \(K\) requires a nontrivial dilaton. We obtain an exact partial solution: the profile \(\Phi = {{\Phi }_{0}} + ({{K}_{0}}{\text{/}}4)\ln (s{\text{/}}{{s}_{0}})\) solves the trace equation \({{\Delta }_{{{{\mathcal{M}}_{1}}}}}\Phi = - K\) restricted to the one-dimensional equilibrium submanifold \({{\mathcal{M}}_{1}}\). We further prove, as an obstruction theorem at the massless Fisher-geometry slice, that no \(\Phi (\beta )\)-only ansatz satisfies all tensorial beta-function components simultaneously; the finite-m² 2D elliptic dilaton problem, requiring the full \({{G}_{{ab}}}(\beta ,{{m}^{2}})\) and \(K(\beta ,{{m}^{2}})\), remains the primary open problem. A full one-loop-consistent background has therefore not been established. The associated central-charge deficit of 20 (bosonic) or 6 (superstring) is quantified and three possible but unresolved closure routes are outlined. Second, as a phenomenological outlook, we couple a composite scalar \(\chi (x)\) to a four-dimensional two-level system and derive, within a Born–Markov and Ohmic-bath approximation, a curvature-sensitive decoherence-rate shift \(\Delta \Gamma {\text{/}}{{\Gamma }_{0}}\, = \,g\langle \chi \rangle (1\, + \,\ell _{\chi }^{2}K)\). Using the Seeley–DeWitt heat-kernel expansion as a geometric ansatz, we further motivate a non-Markovian \({{t}^{3}}\) correction to the coherence decay with coefficient proportional to K. Both results are parametric benchmarks derived from an approximate bath model, not first-principles predictions; the exact Kaluza–Klein spectral density (governed by Jacobi theta functions) is deferred to future work. A matched power-spectral-density protocol with dynamical-decoupling filter functions is proposed as a concrete experimental discriminator for any future completion of the construction.