The four-loop sunset endpoint constants: a Mahler measure and a Domb series

Abstract. Maennel and Vanhove reduced the endpoint values of the equal-mass sunset integral in two dimensions to finitely many real constants. With five propagators (four loops) two constants remain, R∞ and S∞, which they leave unevaluated. We prove that R∞ = −2 m(1 + x1 + x2 + x3 + x4), minus twice the logarithmic Mahler measure of the five-term linear polynomial, and that S∞ = (1/8) Σk≥0 Dk/(64k(k + 1)2), where Dk are the Domb numbers. For any number n of propagators the first constant equals −2 m(1 + x1 + ⋯ + xn−1). The proof rotates the oscillatory Bessel integrals that define the constants onto the imaginary axis, where they become regularized moments of I0 and K0, and evaluates the regularized moment of I02K03/t by expanding one factor I0. Combined with Zhou’s evaluations and the quadratic relations of Fresán, Sabbah and Yu, this shows that the two 2 × 2 determinants pairing the vacuum moments ∫0∞ tK05 dt and ∫0∞ t3K05 dt with the two rows of the Broadhurst–Mellit matrix equal −π3R∞/(15√15) and 2π3S∞/(15√15), and that ∫0∞ tK0(t)5 dt = −(5πC/2)(πR∞ + √15 S∞), where C is an explicit product of Γ-values at fifteenths. All the work was done by AI. Generative-AI disclosure and author responsibility. OpenAI ChatGPT-6 Pro was used for derivations, computations and drafting in the research rounds behind this note; Claude Fable 5.2 and Claude Opus 5.5 were used for independent checking and audit of the results and for drafting this note. The author directed and audited the work throughout and takes full responsibility for the mathematics and the final text. This record holds the note (8 pages) and its verification package Matuzas_Sunset_Endpoint_Constants_verification.zip: the supplement second_proof_details.pdf (with its LaTeX source), which gives the second proof of Proposition 5 (the Lagrange concomitant) in full, with every domain, exchange of limits and boundary term; exact SymPy scripts in rational arithmetic; mpmath scripts that recompute the moments, the finite parts and the endpoint constants independently from their definitions; ball-arithmetic values (python-flint) of the moments and constants; the output log of every script; and a SHA-256 manifest. README.md in the package is the entry point and maps every statement of the note to the script that checks it. The package was checked with Python 3.13.18, mpmath 1.3.0 and SymPy 1.14.0. 2020 Mathematics Subject Classification: Primary 33C10; Secondary 11R06, 81T18, 81Q30. Preprint. External peer review is not claimed.