Year: 2026
Type: preprint
Abstract: We develop a computer-assisted approach to Krzyż's conjecture based on validated elimination of the masses in an extremal Herglotz representation. The entropy theorem of Lei and Zhang, together with the Agler--McCarthy reduction, leaves only extremals with fewer than $n$ atoms to exclude. In the critical stratum with $n-1$ atoms, the first-order conditions are linear in the masses. Validated local inversion reduces the regular search to $n$ outer parameters, with a rank-free fallback at singular configurations. This gives a proof of the sixth-coefficient conjecture, including its equality statement; the critical-stratum computation takes 8.2 minutes on an M4 Pro. We also obtain an independent proof for the seventh coefficient, recently established by Xie. The computational exclusions use explicit rounding-error bounds, and the equality cases follow analytically.
Source: arXiv (Cornell University)
Author: Wikström, Frank
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7221793372