Computational data for the paper "Boundary balance and the scaling law for the infimum of the truncated Weil form"

Version 1.0 of the computational data accompanying the manuscript of the paper “Boundary balance and the scaling law for the infimum of the truncated Weil form” in its version dated 6 October 2026. The paper (in Russian) studies the infimum λ(c) of the truncated Weil form Qc on the interval [−a, a], a = (ln c)/2, in the even sector and its dependence on c through the boundary coefficient β of the minimizer. The data of this version contain a numerical verification of the following formulas and statements of the paper. The Hadamard formula dλ/dc = −β²/c. The derivative with respect to dilation ln c · β² was checked against the difference of the eigenvalues themselves and against the Hellmann–Feynman formula. The decomposition of ln c · β² = A + R + ℰ into the Archimedean part A, the pole remainder R and the remainder ℰ = −2(S − M) in the prime number theorem, as well as the elementary estimate ln c · β² < 4.01. The boundary balance rc = β²/(2κΘ λ) in units of the scale Θc: rc ∈ [0.936, 1.060] at the points c = 6, 10, 12, 14.5 and rc ∈ [0.81, 1.12] on grids of 81 and 42 nodes. Rigorous mean boundary balances over intervals: ⟨r⟩ on [5, 8] lies in [0.9858, 1.0163], ⟨r⟩ on [5, 31] does not exceed 1.0023 (by the rigorous bounds on λ(c) from the work cited by the paper). Jump identity of the Galerkin approximation at prime powers: βN²(n + 0) − βN²(n − 0) = 2Λ(n) n−1/2 uN(a)², with relative accuracy better than 2·10−8. The exact β² has no jump. The boundary coefficient of the eigenfunctions of the logarithmic Laplacian on (−1, 1): β² = 1/(2a), numerically 0.49997…0.50008. The Osgood constant Kc = 2β²/(λ ln(1/λ)), the values of the Rayleigh–Ritz quotients and the same quantities for the Epstein zeta function of the form m² + 5k². All values were obtained numerically, and the formulas and statements, except for the jump identity, are fully proved in the paper. The numbering refers to the version of 6 October 2026, with the names in the current version given in parentheses: Hadamard's formula: Theorem 15 (Hadamard formula for the infimum), Proposition 5 (finite-difference Hadamard identity), Proposition 6 (Lipschitz continuity of the infimum), Theorem 7 (boundary flux of the infimum); boundary coefficient and its expansion: Theorem 11 (boundary expansion), Corollary 12 (boundary coefficient of the eigenfunctions of the form), Proposition 27 (boundary balance and the function Ψ(x) − x), from which the estimate 4.01 (elementary estimates of the terms) is derived in Section 7; boundary balance: Proposition 18 (boundary balance and scale law), Corollary 19 (mean of the boundary balance), Theorem 20 (reduction to the boundary balance), Lemma 17 (exact expression and asymptotics of the scale); Osgood's constant: Proposition 22 (Osgood's criterion); dilation: Proposition 25 (dilation identity), Theorem 24 (integrability of the derivative of the eigenfunctions); logarithmic Laplacian: the equality β² = 1/(2a) is derived in the caption to Figure 1 from Theorem 15 and dilation (boundary coefficient of the first eigenfunction of the logarithmic Laplacian). The jump identity refers to the Galerkin approximation and has been verified numerically. The exact β² has no jump by Theorem 15(i). The Galerkin and Rayleigh–Ritz numbers are numerical upper estimates of λ(c) obtained without rigorous control of the error, and they are not rigorous bounds. The rigorous lower and upper bounds for λ(c) are taken from the work cited by the paper. The archive of this version was uploaded to the record on 6 October 2026. Checksums of the archive file rh-rc-data-v1.0.zip: MD5 658f8fc0617be6eacac744e4dfd36fe1, SHA-256 b3409ebdc3b4bb1a484e424c61fd8f1f5fb19fc94b8499f91ee9583965110423. The version of the manuscript to which these data belong (PDF dated 6 October 2026) has the SHA-256 checksum 614fc6c425e41481ea3e1327a29a6c7f1ed990356997a8c7ae4c48438d38c15c. The contents of the archive, the formats of all tables and the verification procedure are described in the file README.md inside the archive, and the SHA-256 checksums of all files are in the file SHA256SUMS. Data corresponding to subsequent versions of the paper are published as new versions of this record.