The 𝛼-Connection Structure of the Brody Manifold: Orbit-Universality, Poisson Departure, and the Semiclassical Divide

The Brody distribution is used in thousands of nuclear, atomic, and quantum chaos papers as a diagnostic — yet until now nobody had computed its information geometry. This paper shows that the geometric structure is non-trivial and physically meaningful: the unique connection at which the GOE/GUE duality is "invisible" sits at α≈−1. 14 -1. 14 α≈−1. 14, not the mixture connection α=−1 = -1 α=−1 that naive symmetry would suggest. The deviation is forced by the cubic tensor, which measures precisely how far the Brody family is from an exponential family. The one-point/two-point divide theorem explains, for the first time in geometric terms, why fitting β β from spacings alone cannot detect the GUE critical value βc=πc = βc=π. A recurring theme is the *absence* of the Euler-Mascheroni constant γE≈0. 5772E 0. 5772 γE≈0. 5772 from geometric invariants of the Brody manifold. The ambient Weibull metric components each carry γEE γE explicitly — yet their pullback to the Brody curve at β=0=0 β=0 is γEE γE-free, leaving g (0) =π2/6g (0) = ²/6 g (0) =π2/6 as a pure Basel-constant result. The same cancellation appears in the cubic tensor coefficients N0, N1, N2, N3N₀, N₁, N₂, N₃ N0, N1, N2, N3 via a single-channel mechanism: γEE γE enters only through the first cumulant κ1=ψ (1) =−γE₁ = (1) = -E κ1=ψ (1) =−γE, and centering the score removes it entirely. The geometry "knows" π π and ζ (2) (2) ζ (2), but not γEE γE. Reading order for the series: Each paper is self-contained but the full picture requires all four. (1) Variance Excess paper DOI: 10. 5281/zenodo. 18650473 for the spectral statistics background; (2) Information Geometry DOI: 10. 5281/zenodo. 18879754 for the metric foundation; (3) this paper for the connection structure; (4) The Compressibility of Chaos DOI: 10. 5281/zenodo. 18834609 for the dynamical application.