Variance Excess of Bulk Level Spacings over the Wigner Surmise: A Closed-Form Formula with Critical Point βc = π

The Wigner surmise is exact at precisely one value of the Dyson index: β = π. The surmise has approximated random matrix level spacings for seven decades, but exactly how much does it miss? We show the answer is a single closed-form function of β with a previously unremarked sign reversal between GUE and GSE at this irrational critical point. The 2×2 surmise overestimates the GOE bulk variance by 4.5% — a fact buried in Gaudin's 1961 tables but never given a formula. We derive that formula and discover it changes sign at β = π: the surmise is too narrow below and too wide above. ε(β) = (J/π²)G(β) − G(π), G(β) = (γE/β) ln(K*/β), where J = 0.327919… is the sinc² overlap integral from the French–Mehta–Pandey (FMP) relation between spacing variance and Dyson–Mehta number variance, γE is the Euler–Mascheroni constant, and K* = exp(dC/γE − 1/2) ≈ 19.39 with dC = 2 the Coulomb-gas dimension. The factored structure makes the critical point βc = π — where the Wigner surmise becomes exact — algebraically manifest. Validation against Dumitriu–Edelman β-ensemble simulations (N = 4000, 200 realizations) gives χ²/dof = 0.64 across seven primary β values with no free parameters. A 40-point extended scan confirms βc = 3.1416 ± 0.0066, consistent with π to 0.21%. Universality is established by independent tests on Circular ensembles (COE, CUE), which require no spectral unfolding. v3 additions: (i) The FJM loop-equation recursion generating the expansion polynomials of the spacing variance, with all leading coefficients sj = 5/6 − 2Hj determined in closed form. (ii) A structural derivation of the Brody–Dyson calibration exponent ν = 2/3 from tangency at GOE plus subleading Brody variance asymptotics. (iii) Decomposition of the Brody MLE as q*(β) = q*W(β) + δqcorr(β), where q*W is analytically computable by quadrature and A = 0.944 is shown to be a β-averaged compromise, not a fundamental constant. (iv) The γE-independence theorem: the score-equation transfer function is free of the Euler–Mascheroni constant at every Dyson point. (v) The Wigner rationality lemma: all coefficients in the asymptotic expansion of the Wigner surmise variance are rational. Part of the ICP series on the information geometry of chaos: The ICP (The Instability Compression Principle) framework connects the spectral parameter β to universal compression laws in chaotic systems. It was introduced empirically in the first paper below and derived theoretically in “The Compressibility of Chaos.” Subsequent papers develop the information-geometric machinery on the Brody statistical manifold and identify the symmetry group that organises the underlying geometry. Paper Role DOI The Instability Compression Principle ICP empirical foundation: β → compression scaling across 30 chaotic systems 10.5281/zenodo.18099118 The Compressibility of Chaos ICP theoretical derivation: scaling coefficient α from information geometry 10.5281/zenodo.18834609 This paper - Variance Excess ε(β) formula, FJM recursion, Brody–Dyson calibration 10.5281/zenodo.18650473 Information Geometry of the Brody Distribution Fisher metric, effective dimension, spectral duality theorem 10.5281/zenodo.18879754 The α-Connection Structure of the Brody Manifold Amari–Chentsov tensor, orbit-universal connection 10.5281/zenodo.19151206 Dual Symmetries of the Brody Statistical Manifold and Their Common Fixed Point Z2×Z2 symmetry group, GOE=GUE orbit-equivalence 10.5281/zenodo.19239285 The Duality Web of the Brody Statistical Manifold Four dualities unifying metric, connection, and curvature 10.5281/zenodo.19389065 The Spectral Geoid Convergent mode structure of Wigner surmise residuals, Hellinger detector hierarchy 10.5281/zenodo.19518426 Spiked Random Matrix Signatures of the L–H Transition Experimental application: spectral complexity reduction in tokamak Dα emission 10.5281/zenodo.19423076