固定凸透镜谱常数：Möbius 约化、精确模型定理与角度依赖界
Fixed Convex-Lens Spectral Constants: Möbius Reduction, Sharp Model Theorems, and Angle-Dependent Bounds
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中文总结 AI 辅助
本文通过Möbius约化将两圆盘交集的谱集不等式转化为扇形数值域问题，确定平方零算子的精确透镜常数，并证明直角情形下全回文二次族的$\sqrt2$界，给出多个参数族的精确有理证书及角度依赖界。
中文摘要 AI 辅助
对于以角度 $2\alpha$ 相交的两个圆盘的交集，令 $C(\alpha)$ 为相关谱集不等式中的最小常数。我们给出一个自包含的 Möbius 约化，将其转化为扇形上的相应数值域问题，并确定仿射平方零算子 $B=\lambda I+N$（$N^2=0$）的精确常数：$C_{\mathrm{sq0}}(\alpha)=\pi\sin\alpha/(2\alpha)$。一个 $2\times 2$ 矩阵达到等号，并给出一个显式的透镜下界证书。在直角情形下，我们证明了全回文二次族在任意维数下的猜想 $\sqrt2$ 界，并为几个更大的参数族给出精确有理证书，包括复自同构后的圆盘、完整虚直径、达边界相位弧，以及对称和不对称三节点容许核问题。我们还获得一个严格中心界 $|w^2|\leq\kappa_0<\sqrt2$、一个双小零点扩展、一个针对剩余边界层的精确二阶矩准则，以及一个经过验证的角度依赖包络。每个计算机辅助断言都有精确有理验证器。这些结果与维数无关，但并未确定无限制固定透镜常数。
英文摘要
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 $B=λI+N$, $N^2=0$: $C_{\mathrm{sq0}}(α)=π\sinα/(2α)$. 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 $|w^2|\leqκ_0<\sqrt2$, 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.
