A proof of Laporta's formula for the second Lüscher–Weisz lattice constant

We prove Laporta's formula for the second Lüscher–Weisz constant of the four-dimensional massless lattice propagator. A differential boundary functional reduces the constant to two six-Bessel moments, with uniform remainder estimates at the singular endpoint. We then identify both lattice constants, after explicit Tate terms, in a rational period matrix of the conifold fibre of the associated lattice family. The determinant-character components of its five singular fibres have rank two and correspond to weight-four modular forms of primitive levels 6, 12 and 8. Their affine Tate factors are determined by the tangent-cone discriminants, and an explicit comparison with three known double octics relates this description to earlier modularity results. We also determine the conifold connection constants, obtain quadratic-twist critical-value identities, and compute the exact vanishing-cycle vector using the positive-real-cycle Gamma theorem. A rational involution splits the central affine extension. Local data at three are derived geometrically; those at two follow from published local compatibility, while a direct characteristic-two derivation remains open. Generative-AI disclosure and author responsibility. OpenAI ChatGPT-6 Pro was used for mathematical derivations, symbolic and numerical computations, literature analysis, and drafting. Claude (Anthropic) was used for independent re-derivation and computational checks. The author directed and reviewed the work and takes full responsibility for the mathematics and the text. 2020 Mathematics Subject Classification: Primary 33C10; Secondary 11F67, 14J32, 81T25. Files: the compiled manuscript Luscher_Weisz_Conifold_Periods.pdf (36 pages) and the verification package Luscher_Weisz_Verification.zip. The package contains manuscript/ with the LaTeX source main.tex and the same PDF; the offline entry point verify.py; checks/ with the formula-preservation, page-reproduction and independent finite-algebra and quadrature checks; reference/calculations.zip with the byte-preserved supporting calculations, data, proofs and earlier verification programs (the earlier drafts it contains are superseded by the current manuscript); review/ with Publication_review.pdf (3 pages), describing the scope, findings and limitations of the publication review, together with the executed review records; and a README, REPLAY.md, requirements.txt and manifest.sha256 with SHA-256 hashes of every other file. Running python verify.py --out DIR, where DIR is a new directory outside the unpacked archive (Python 3.11 or later; dependencies in requirements.txt; pdfLaTeX for the document check), authenticates the package, checks the mathematical source against the authenticated earlier manuscript, recompiles the article and compares all 36 page images, runs the exact finite-algebra checks and a 75-digit Bessel-moment calculation with a 60-decimal-place gate, and replays the supporting scientific programs at reduced precision; python verify.py --document-only --out DIR runs a shorter document, integrity and finite-algebra check. The stored scientific replay returned PASS_WITH_SKIPS with 13 named omissions (private source checks, historical PDF recompilation, and composite wrappers whose component scripts were run instead). The quadratures are not rigorous interval enclosures, and a passing run is not a formal proof or an external peer review.