Get started
Work

Paper A-1: Finite-Order Corrections to the GUE Spacing Ratio Mode: Surmise Hierarchy, Tail Asymptotics, and Implications for Many-Body Localisation Diagnostics

Year: 2026
Type: preprint
Abstract: Paper A of the MNZI programme proved algebraically that the mode of the GUE consecutive level-spacing ratio distribution (ABGR surmise) is exactly phi^(-1) = (sqrt(5)-1)/2 (the golden ratio conjugate, approximately 0.6180), via the Buchanan-Atas perfect-square identity (1+r+r^2)^2 - 4r(1+r) = (r^2+r-1)^2. This companion paper analyses two sources of correction to this result: (i) finite Hilbert-space corrections to the large-N_H GUE distribution via the Nishigaki cancellation, and (ii) the systematic error of the 3x3 ABGR surmise relative to the exact distribution. We prove the following four results. (1) The exact curvature of the ABGR density at its mode is p_2''(phi^(-1)) = -5/(16 Z_2) = -405*sqrt(3)/(64*pi) approximately -3.489, derived algebraically from the Buchanan-Atas identity. The peak is extremely flat: the density falls by only approximately 1.6% over a neighbourhood of width 0.08 around the mode, a consequence of the double zero of the perfect-square factor at phi^(-1). (2) Large-N_H corrections are O(N_H^(-4)), which is less than 10^(-17) for all accessible MBL systems (system size L >= 14 spins, Hilbert space dimension N_H = 2^L). The O(N_H^(-2)) term vanishes identically due to an SL(2,R) gauge cancellation in the Nishigaki Bessel kernel correction. For all system sizes accessible to current exact-diagonalisation studies (14 <= L <= 24), phi^(-1) is the effective exact GUE mode to better than 10^(-17). (3) The N x N GUE surmise has tail asymptotics p^(N)(r) ~ C_N * r^(-(N^2-1)/2) as r tends to infinity, proved analytically from the Vandermonde scaling structure of the defining integral. For N=3 this gives r^(-4) (the ABGR tail); for N=5 this gives r^(-12). The tail exponent grows as N^2, explaining the rapid convergence of the surmise hierarchy toward the exponential tail of the exact GUE distribution. (4) Fine-grid Gauss-Laguerre quadrature (200 points, grid spacing 0.002) locates the 5x5 GUE mode at r^(5)_* approximately 0.6123, shifted below phi^(-1) by approximately 0.0057, consistent in sign and magnitude with the perturbation formula of result (1). The dramatically lighter tail of the 5x5 surmise (r^(-12) versus r^(-4)) provides a structural argument that the mode is a more robust GUE diagnostic than the mean for finite MBL systems, where right-tail sampling bias creates systematic upward pressure on mean-based estimators. We propose a two-marker diagnostic combining mode and mean simultaneously: GUE ergodicity requires both the sample mean to converge to approximately 0.530 and the mode estimator to converge to phi^(-1) = 0.6180… (algebraically exact); systematic disagreement between the two markers signals either finite-size tail bias or departure from GUE universality. The exact algebraic value phi^(-1) provides a bias-free reference point for the mode marker that the mean cannot offer. All algebraic results are machine-verified in Lean 4 as sorry-free theorems (file: MNZI/GUE_Spacing_Ratio_Mode.lean, Lean 4 v4.28.0, Mathlib v4.28.0, standard axioms only, verified by lake build). Verified results include: the Buchanan-Atas identity, the mode identity, the tail factorisation p_2(r) = r^(-4) * h(r) with h(r) tending to 1, the mode-shift prefactor, and the finite-size bound 2^(4L) >= 2^56 for L >= 14. KEYWORDS: many-body localisation, level-spacing ratio, GUE universality, golden ratio, ABGR surmise, random matrix theory, surmise hierarchy, tail asymptotics, MBL diagnostic, Lean 4 verification, finite-size effects, Nishigaki cancellation MSC 2020: 15B52, 60B20, 81Q50, 82B44, 11B39 RELATED PAPERS (MNZI PROGRAMME): Paper A: doi:10.5281/zenodo.19160183 Paper P: doi:10.5281/zenodo.19486361
DOI: 10.5281/zenodo.20323744
Language: English
Open Access status: green
OpenAlex ID: W7162013954
Cites:
Cited by:
Related to: