Information Geometry of the Brody Distribution: Exact Fisher Metric, Effective Dimension, and Spectral Duality

The Brody distribution has been the standard one-parameter summary of level-spacing statistics in quantum chaos since 1973. It has been fitted to nuclear spectra, quantum billiards, condensed matter systems, and coupled oscillators across hundreds of papers. Yet in over fifty years, nobody has studied its information geometry — the Riemannian structure that measures how distinguishable nearby spectral statistics actually are. This paper fills that gap. We derive the exact Fisher-Rao metric on the Brody manifold in closed form, revealing that the Basel constant ζ (2) = π²/6 dominates the geometry at every value of the chaos parameter β, contributing at least 62% of the metric from the Poisson limit through GOE to GSE. The Amari α-connection structure proves the Brody family is not an exponential family — the cubic tensor is strictly negative — and the geodesic arc-length coordinate takes a functional form identical to the IGAC compression coefficient from information-geometric approaches to classical chaos. The central results are a pair of theorems. First, we prove that the effective statistical dimension of nearest-neighbor spacings in β-ensembles is dₑff = (β+2) /β, established through three independent routes: the χ² decomposition at N=2 (Dumitriu-Edelman), the Dyson-Mehta number variance at N→∞, and center-of-mass decoupling at all N. Second, we prove a spectral duality theorem: the Weibull dimension dW = β+1 and the effective dimension satisfy (dW − 1) (dₑff − 1) = 2, where 2 is the Coulomb embedding dimension. The self-dual point β* = √2 produces the silver ratio √2 − 1 as the compression exponent. These results have immediate consequences. The variance excess ε (β) — the systematic departure of the Wigner surmise from exact bulk statistics — turns out to be exactly linear in dₑff, providing a direct experimental route to the effective dimension from spectral data. The framework also identifies a semiclassical divide: the effective dimension governs quantum spectral fluctuations, while classical compression coefficients depend on phase-space geometry through dₚhase, not dₑff — a distinction invisible at GOE where both happen to equal 3. This paper provides the mathematical foundation for the The Instability Compression Principle framework developed in the companion papers: "Variance excess of bulk level spacings over the Wigner surmise" (DOI: 10. 5281/zenodo. 18650473) — derives the zero-parameter formula ε (β) with critical point βc = π "The compressibility of chaos" (DOI: 10. 5281/zenodo. 18834609) — establishes the Ordo ab Chao theorem connecting autoencoder compression of chaotic attractors to spectral statistics via the effective dimension Together, these three papers connect information geometry, random matrix theory, rate-distortion theory, and dynamical systems into a unified framework for understanding Why and How chaotic systems compress. //Jon. Wiberg@Live. se