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≈−1.14α ≈ -1.14 α≈−1.14, not the mixture connection α=−1α = -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 ββ β from spacings alone cannot detect the GUE critical value βc=πβ_c = π βc=π.A recurring theme is the *absence* of the Euler-Mascheroni constant γE≈0.5772γ_E ≈ 0.5772 γE≈0.5772 from geometric invariants of the Brody manifold. The ambient Weibull metric components each carry γEγ_E γE explicitly — yet their pullback to the Brody curve at β=0β=0 β=0 is γEγ_E γE-free, leaving g(0)=π2/6g(0) = π^2/6 g(0)=π2/6 as a pure Basel-constant result. The same cancellation appears in the cubic tensor coefficients N0,N1,N2,N3N_0, N_1, N_2, N_3 N0,N1,N2,N3 via a single-channel mechanism: γEγ_E γE enters only through the first cumulant κ1=ψ(1)=−γEκ_1 = ψ(1) = -γ_E κ1=ψ(1)=−γE, and centering the score removes it entirely. The geometry "knows" ππ π and ζ(2)ζ(2) ζ(2), but not γEγ_E 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.