Topological Discriminant via Spectral Geometry for Black Hole Echo Signatures

This work resolves the longstanding tension between equidistant echo observational candidates and traditional asymptotic QNM spectral predictions by supplying a closed compact topological explanation built solely on rigorous spectral geometry of connected, boundaryless three-dimensional smooth manifolds. We first solve the scalar Helmholtz eigenproblem on closed \(S^3\) via hyperspherical separation of variables, deriving the closed-form discrete spectrum \(λ_n = n(n+2)\) and proving a controllable monotonic upper bound for relative spectral deviation in any finite low-order mode window. We define the standard concept of apparent equispaced spectrum \(E_obs\) to describe approximate uniform spacing measurable with instrumental precision, and prove that \(S^3\) geometries always satisfy this observational criterion. Three mutually independent rigorous mathematical arguments — scattering pole analysis, WKB semiclassical estimation, and continuous-spectrum operator theory — are deployed to demonstrate that all asymptotically flat open manifolds cannot produce apparent equispaced resonance sequences under any finite observational window. Combining the two core spectral results, we formulate a bidirectional topological discriminant theorem: low-order apparent equispaced echoes uniquely correspond to compact \(S^3\) spacetime topology, while non-uniform resonance spacings confirm standard open Schwarzschild geometry. The theorem is fully model-independent, requiring no modified gravity or horizon microstructure assumptions, and provides quantitative falsification thresholds matched to LIGO and LISA detector resolution benchmarks. All derivations are self-contained with complete auxiliary calculations supplied in appendices; this framework resolves the lingering tension between gravitational wave echo observations and conventional QNM spectral predictions, delivering a purely geometric observable tool for testing global spacetime topology.