Fixed Convex-Lens Spectral Constants: Möbius Reduction, Sharp Model Theorems, and Angle-Dependent Bounds
Abstract
For the intersection of two disks meeting at angle 2α, let C(α) 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=λI+N, N2=0: Csq0(α)=πsinα/(2α). A 2×2 matrix attains equality and yields an explicit lens lower-bound certificate. At the right angle, we prove the conjectural √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 |w2|≤κ0<√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.
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BibTeX
@misc{zeng2026fixed,
title = {{Fixed Convex-Lens Spectral Constants: Möbius Reduction, Sharp Model Theorems, and Angle-Dependent Bounds}},
author = {Zijian Zeng},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.25027},
eprint = {2609.25027},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25027},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Fixed Convex-Lens Spectral Constants: Möbius Reduction, Sharp Model Theorems, and Angle-Dependent Bounds
AU - Zijian Zeng
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.25027
UR - https://arxiv.org/abs/2609.25027
AB - For the intersection of two disks meeting at angle 2α, let C(α) 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=λI+N, N^2=0: C_sq0(α)=πsinα/(2α). A 2×2 matrix attains equality and yields an explicit lens lower-bound certificate. At the right angle, we prove the conjectural √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|≤κ_0<√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.
ER -