For the intersection of two disks meeting at angle 2α, let C(α) be the least constant in the associated spectral-set inequality, uniformly over operators for which each disk separately is a spectral set. The exact value of C(α) is unknown except at the disk endpoint α = π/2. We give a self-contained Möbius reduction to the corresponding numerical-range problem on a sector and compute the sharp constant on the infinite-dimensional class of affine square-zero operators B = λI + N, N2 = 0: Csq0(α) = πsin α/2α. A 2 × 2 matrix and a conformal extremal attain equality and yield an explicit lens lower-bound certificate. At the right angle we prove the conjectural 2 estimate, in arbitrary dimension, for the full palindromic quadratic family. Exact rational matrix certificates further cover the complex post-automorphism disk |c| ≤ 19/20, the complete imaginary diameter and a transverse cusp, and boundary-reaching phase arcs whose union misses only 9.17 degrees of the parameter circle near -1. For every symmetric three-node set in the right-angle disk coordinate, we also prove the sharp identity-multiplier estimate on the complete admissible-kernel cone; the proof combines an exact extreme-ray rank bound, scalar Pick interpolation on the rank-one faces, and a two-variable Bernstein certificate on the rank-(2, 2) face. A nested exact certificate extends this result to the genuinely asymmetric two-parameter patch ρ = (−a, 0, b), 1/6 ≤ a, b ≤ 5/6. On a rank-one localized face we also prove a three-real-parameter singular-boundary family: for Φ(c) = eiθc2, sin θ > 1/√2, the sharp target is positive for every quadratic Blaschke product w(w − a)/(1 − a¯w) with a ∈ D. At the central square, a polynomial bidisk extension and Ando’s theorem prove ∥w2∥ ≤ κ0 < √2, where 276889/127200 − 6103√2/10600 = 1.362560 . . .. We also derive an exact two-complex-parameter consequence: the same construction proves a strict √2 bound for every bαbβ with |α|, |β| ≤ 1/150, allowing two independently phased nonzero zeros. We also derive 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.