Canonical Minkowski-Reduced Basis, Certified Regulator, and Arithmetic Spectroscopy of the Alpöge–Howell–Claude Elliptic Curve of Rank >= 31

Official canonical lattice dataset and Minkowski-reduced basis for the world-record elliptic curve of rank >= 31 (ICARM Curve #302, announced August 2026 by Alpöge, Howell, and Claude). Deploying the Arithmetic Spectroscopy framework developed at Genesis Spectral Labs, this work provides the first certified canonical geometric resolution of the 31-dimensional Mordell–Weil lattice over Q, resolving the extreme ill-conditioning and catastrophic floating-point cancellation of the Néron–Tate height pairing in dimension 31. Key Certified Results:• Certified 31-dim Regulator: R_31 = 5.520367374821893536678475926502746956624 * 10^39 (certified at 250-bit precision).• Spectral Bounds of Gramian: lambda_min = 0.209324, lambda_max = 290.834711.• Minkowski Trace Compression: Total Gramian trace compressed from 1931.2428 down to 1910.0519 (a net reduction of 21.19 height units).• Refined Minimal Vector: Discovered canonical generator Q_8 = P_15 - P_1 with canonical height h(Q_8) = 52.777548, strictly shorter than the authors' minimal generator (h(P_1) = 56.303293).• Integrality Census: Exactly 19 of the 31 canonical Minkowski generators are proven to be strict integral points on the minimal Weierstrass model y^2 + xy + y = x^3 + x^2 + a_4*x + a_6.• Unimodular Transformation: Full integer transition matrix U in GL_31(Z) with det(U) = -1. Files Included: Canonical_Minkowski_Basis_Rank31_Timakov.pdf: Full academic paper with theoretical derivations, spectral analysis, and complete generator tables. minkowski_basis_rank31.json: Complete machine-readable JSON specification containing coordinates, linear representations, exact heights, and integrality status for all 31 canonical vectors. Author: Andrew Timakov (Independent Researcher, Genesis Spectral Labs)Contact: [email protected]