Fixed Convex-Lens Spectral Constants: Möbius Reduction, Sharp Model Theorems, and Angle-Dependent Bounds

Abstract

For the intersection of two disks meeting at angle $2\alpha$, let $C(\alpha )$ be the least constant in the associated spectral-set inequality. We give a self-contained M"obius reduction to the corresponding numerical-range problem on a sector and determine the sharp constant for affine square-zero operators $B=\lambda I+N$, $N^2=0$: $C_{sq0}(\alpha )=\pi \sin \alpha /(2\alpha )$. A $2\times 2$ matrix attains equality and yields an explicit lens lower-bound certificate. At the right angle, we prove the conjectural $\sqrt{2}$ bound in arbitrary dimension for the full palindromic quadratic family, and give exact rational certificates for several larger parameter families, including a complex post-automorphism disk, the complete imaginary diameter, boundary-reaching phase arcs, and symmetric and asymmetric three-node admissible-kernel problems. We also obtain a strict central bound $|w^2|\le {\kappa }_0<\sqrt{2}$, a two-small-zero extension, an exact two-moment criterion for the remaining boundary layer, and a verified angle-dependent envelope. Every computer-assisted assertion has an exact rational verifier. These results are dimension-free but do not determine the unrestricted fixed-lens constant.

44 pages. Contains exact rational/computer-verifiable certificates for all computer-assisted assertions. The unrestricted fixed-lens constant remains open