Schwarzschild Ringdown as a Resonance-Information Process: A Fixed-Background Case Study of Horizon Distinguishability, Modular Relative Entropy, and Unitary Resonance Compression

Abstract

This record presents a fixed-background case study in the linear axial ℓ=2\ell=2ℓ=2 sector of Schwarzschild perturbation theory. Two independent real Regge–Wheeler evolutions are combined after evolution to form a complex horizon waveform. On the finest numerical grid, the fundamental ringdown fit gives Mγ=0.0889877360M\gamma=0.0889877360Mγ=0.0889877360 and Mω=0.3736490546M\omega=0.3736490546Mω=0.3736490546. The central result compares two descriptions of the information remaining to the future of a horizon cut. After the affine identification V=eκHτV=e^{\kappa_H\tau}V=eκHτ and the canonical Regge–Wheeler/Cunningham–Price–Moncrief horizon-flux normalization, the perturbative horizon distinguishability functional DHD_HDH becomes the coherent modular relative entropy SrelS_{\mathrm{rel}}Srel. The flux coefficient is fixed independently by the standard odd-parity horizon-flux formula and the affine tensor Jacobian; it is not fitted to enforce the equality. The reduced time series agree to machine precision as an implementation check of the analytic identity, while an independent affine-coordinate quadrature agrees with DHD_HDH to 6.51×10−66.51\times10^{-6}6.51×10−6, and the associated differential identity has relative residual 2.07×10−62.07\times10^{-6}2.07×10−6. An exact single-pole control reproduces the quadratic 2γ2\gamma2γ information-decay rate to numerical precision. Fixed-length window shifts of the full Regge–Wheeler signal give ratios between 1.998596 and 2.001479. Finally, the fitted QNM contraction is reproduced by an explicit unitary dilation with maximum moment error 3.72×10−163.72\times10^{-16}3.72×10−16, showing how an irreversible projected ringdown can coexist with reversible total dynamics. The accompanying reproducibility archive contains the self-contained Python calculation, numerical time series, grid-convergence results, affine-quadrature checks, single-pole and fit-window controls, normalization-provenance ledger, figures, claim ledger, and validation reports. The construction is deliberately background-dependent. It does not derive the Schwarzschild geometry, the mass scale, the coefficient 1/(4G)1/(4G)1/(4G), the microscopic origin of the horizon algebra, or Jacobson’s all-local-horizon dynamics contact for reproducibility: kaya@cab-film.com