Gravitational Wave Echoes from Hilbert Space Refactorization: Spin-Dependent Signatures

We extend the Hilbert space refactorization framework to rotating (Kerr) black holesand derive spin-dependent signatures in gravitational wave echoes. In our model, the echo delay time scales as Δtₑcho ≈ (4GM/c³) ln (2M/mP), introducing acharacteristic logarithmic correction that distinguishes it from linear scalingpredicted by firewall scenarios and from power-law scaling (Δt ∝ M^1+β) typicalof fuzzball and generic exotic compact object (ECO) models. We compute the leading-order spin correction f (a) ∝ (a/rₛ) ² to the delay time, showing that prograde and retrograde modes acquire measurably different echointervals — a potential smoking gun for near-horizon quantum structure. Theamplitude of successive echoes decays as Aₙ ∼ exp (−n α ln M), providing amass-dependent damping law testable against LIGO/Virgo/KAGRA ringdown data. Cosmological implications are explored through the scalar spectral index predictionnₛ − 1 ≈ −0. 68/Nₑ, linking Planck-scale horizon microstructure to CMBobservables. We release reproducible Python scripts (echoₛcaling. py) generatingscaling plots and LIGO-band detail visualizations, alongside LaTeX source andcompiled PDF of the manuscript. This work constitutes the third installment in the refactorization series, complementing prior analyses of Schwarzschild echoes and entanglement structure, and provides concrete observational targets for next-generation gravitationalwave detectors (LISA, Cosmic Explorer, Einstein Telescope).