The sunset endpoint constants at every loop order: coordinates of the vacuum Bessel moments

Abstract. Maennel and Vanhove reduced the endpoint values of the equal-mass sunset integral in two dimensions with n propagators to d = ⌊(n − 1)/2⌋ real constants b1, …, bd, of which the first is −2m(1 + x1 + ⋯ + xn−1). We prove that for every n ≥ 3, ∫0∞ tK0(t)n dt = −Σi=1d n!/((n − i)! 2i) bi ∫0∞ tI0(t)iK0(t)n−i dt. When n is not divisible by four, the same coefficients serve the powers t3, …, t2d−1. One contour rotation evaluates every regularized mixed moment by the bi and even zeta values; its imaginary part gives the degree bound of Maennel and Vanhove. The quadratic relations of Fresán, Sabbah and Yu then give the coordinates. For seven propagators the two further constants are expressed by the five-loop constants, the Mahler measure, and two series of rational numbers paired with six-Bessel moments. For eight propagators, Zhou's evaluations express the vacuum moment through the critical values L(f, 4) and L(f, 5) of the weight-six, level-six newform, with endpoint constants as coefficients. A determinant of Broadhurst and Mellit equals 35/128 times a regularized moment. We also state a conjectural evaluation of the three seven-Bessel moments by critical values of a weight-three, level-525 newform, verified to more than one hundred digits. All the work was done by AI. Generative-AI disclosure and author responsibility. The statements of the main theorems, the outline of their proofs and Conjecture 7 were first obtained with Claude Fable 5.2 and Claude Opus 5.5, which were also used for computations and for the audit of the proofs. OpenAI ChatGPT-6 Pro wrote the complete proofs, found the extension to numbers of propagators divisible by four, checked the sources, produced the certified numerical verification and drafted this note in the research rounds behind it. The author directed and audited the work throughout and takes full responsibility for the mathematics and the final text. This record holds the note (15 pages) and its verification package Matuzas_Sunset_Endpoint_Constants_Every_Loop_Order_verification.zip: the supplement proof_details.pdf (30 pages, with its LaTeX source), which proves the rotation with explicit bounds on every arc, the complete period pairing when the number of propagators is not divisible by four, the exceptional bases and the scalar relation for every number of propagators, the ladder from six to seven Bessel functions with rational tail bounds, and the critical values of the weight-three and weight-six forms by bounded Mellin series; exact SymPy scripts for the rotation coefficients, the coordinates, the exceptional projection, the ladder and the eight-propagator algebra; a short mpmath suite that recomputes the moments, the regularized moments and the identities from their definitions; ball-arithmetic producers (python-flint) for certified enclosures of the seven- and eight-propagator integrals and of the modular L-values; a PARI/GP script that regenerates the exact coefficient tables of the level-525 form; JSON receipts recording every value as a ball and every comparison with its status; the output log of every program; 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 3.12.14, mpmath 1.3.0, SymPy 1.14.0 and python-flint 0.9.0; the coefficient tables were regenerated with PARI/GP 2.19.0. Companion notes: J. Matuzas, The four-loop sunset endpoint constants: a Mahler measure and a Domb series, doi:10.5281/zenodo.24122507; J. Matuzas, The five-loop sunset endpoint constants: a Mahler measure and a second Domb series, doi:10.5281/zenodo.24139789. 2020 Mathematics Subject Classification: Primary 33C10; Secondary 11F67, 11R06, 81T18, 81Q30. Preprint. External peer review is not claimed.