Fixed Convex-Lens Spectral Constants: M\"obius 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 $B=\lambda I+N$, $N^2=0$: $C_{\mathrm{sq0}}(\alpha)=\pi\sin\alpha/(2\alpha)$. A $2\times2$ matrix attains equality and yields an explicit lens lower-bound certificate. At the right angle, we prove the conjectural $\sqrt2$ 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 $|w^2|\leq\kappa_0<\sqrt2$, 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.