Equally Spaced Black Hole Echo Harmonics from S³ Topological Eigenspectrum with Supplementary Derivations and Observational Test Specifications

We identify a binary observational discriminant for black hole gravitational-wave echoes: equal-spacing versus unequal-spacing in the harmonic frequency sequence. For echo mechanisms governed by S3S3 topological resonance, the discrete eigenspectrum λn=n(n+2)λn=n(n+2) yields strictly equal spacing, fn=n⋅f0fn=n⋅f0, in the low-frequency long-wavelength limit. This contrasts sharply with all mainstream echo models (QNM, string-inspired membranes, loop quantum gravity corrections, modified gravity), which universally predict unequal, nn-dependent spacing. The two classes of models produce opposite predictions; detection of multiple resolvable harmonics in a single event can distinguish them. Future LISA and next-generation ground-based detectors can perform this test; explicit falsification criteria are provided.