The Spectral Geoid: Convergent Mode Structure of Wigner Surmise Residuals

For fifty years, random matrix theorists have decomposed spacing distribution residuals into polynomial modes weighted by the Wigner surmise — and for fifty years, that decomposition has been silently divergent above β = 1. Somewhat unnoticed, because everyone truncated the series and reported power fractions as if they were meaningful. They weren't: at β = 4 (GSE), the polynomial coefficients grow by a factor of 400 over ten modes. The "kurtosis dominance" reported at large β was never a physical transition — it was the highest retained mode absorbing the divergence. This paper identifies the problem and fixes it. The natural decomposition variable is the Hellinger deviation h = √(P_bulk/P_W) − 1, which lies in L²(P_W) at all β by a trivial but apparently unnoticed bound. In the convergent Hellinger basis, the variance mode dominates at every repulsion strength — from 80% at GOE to 51% at GSE — meaning the Wigner surmise residual is always primarily a width change, never a shape distortion. The "dominance reversal" was an artifact of expanding a function that doesn't belong to the space. Three additional results: An exact, closed-form identity connecting the variance mode coefficient to the variance excess ε(β) via gamma-function ratios (Proposition 2), explaining the empirically observed c₂/ε ≈ 0.73 as a near-cancellation. A mode hierarchy inversion in Berry–Robnik mixed spectra at chaotic fraction ρ ≈ 0.6, with a measurement horizon at ρ ≈ 0.97 beyond which the geoid becomes featureless. A negative result on β_c = π: five natural geometric constructions fail to detect it as a curvature feature; the critical value is algebraic (a property of the variance excess formula), not geometric (a property of any statistical manifold). The paper completes the two-point layer of the information-geometric programme for spectral statistics, complementing the one-point Brody manifold geometry developed in the companion papers. Paper Role DOI The Instability Compression Principle ICP empirical foundation 10.5281/zenodo.18099118 The Compressibility of Chaos (Ordo ab Chao) ICP theoretical derivation: α₀ = 5π² 10.5281/zenodo.18834609 Variance Excess ε(β) formula, β_c = π — the quantity this paper decomposes 10.5281/zenodo.18650473 Information Geometry of the Brody Distribution Fisher metric, one-point layer — the layer this paper complements 10.5281/zenodo.18879754 The α-Connection Structure of the Brody Manifold Amari–Chentsov tensor, γ_E cancellation → ζ-stratification origin 10.5281/zenodo.19151206 Dual Symmetries of the Brody Statistical Manifold Z₂×Z₂ symmetry, β*=√2 — the one-point special value vs this paper's β_c=π 10.5281/zenodo.19239285 The Duality Web of the Brody Statistical Manifold Conjugation-singularity theorem, interval arithmetic methods 10.5281/zenodo.19389065 Spiked Random Matrix Signatures of the L-H Transition Experimental Krylov bridge endpoint: σ²(bₙ) diagnostic connected here via §6.3 10.5281/zenodo.19423076 The Spectral Geoid Two-point layer: Hellinger convergence theorem, mode hierarchy, ζ-stratification this paper