Finite-multipole damping order and endpoint selection in Schwarzschild quasinormal-mode pairs

At leading eikonal order, Schwarzschild quasinormal-mode damping is independent of multipole, but the first finite-L correction fixes how the finite-ℓ spectrum approaches that limit. From the Dolan–Ottewill expansion we derive a sector- and overtone-dependent sign threshold and show that, over any window with the predicted strict order, a pairwise maximum selects a definite multipole endpoint. A hash-recorded calculation generated 1,050 scalar, electromagnetic, and gravitational modes through ℓ=50. In 21 pre-specified confirmation windows, all 879 adjacent damping differences had the predicted sign, and fitted leading coefficients had median relative error 1.25×10⁻⁵. The sign switched between electromagnetic n=0/1 and gravitational n=2/3. A retrospective standalone Leaver calculation now reproduces all 900 roots and all 879 adjacent signs in the 21 confirmation windows. Its largest complex-frequency difference from the original package calculation is 4.87×10⁻¹¹. The two implementations solve the same spectral condition; this is an implementation cross-check, not independent-formulation validation. The 18,837 pair endpoints are a transitive closure of monotonicity, not independent confirmation. The result gives a finite-domain audit rule for nonlinear statistics constructed from paired QNMs. The extracted-package verifier checked stored results for 900 roots and reproduced the summaries of 879 of 879 adjacent signs, 18,837 dependent pairs, and 21 coefficient comparisons. The complete standalone 900-root recomputation had been performed earlier on 26 October 2026; this extraction check did not solve the spectrum again. Median relative coefficient error 1.246772581129783e-5; maximum 1.459216114439132e-4. This is an author preprint and has not been peer reviewed. The supplement preserves nulls, adverse checks and the stated scope.