TMRB Soft-Hair Resolution of the Black-Hole Information Paradox

PTRH Paper 10 — TMRB Soft-Hair Resolution of the Black-Hole Information Paradox: Discrete Z₉ Torsion Hair, Finite Microstate Counting, and the 9 ln 9 Topological Correction in 11D M-Theory The TMRB (Torsion-Modulated Recursive Branching) framework — built on 11D M-theory compactified on the Schoen gyroid with Z₉ discrete torsion and golden-ratio modular-covariant branching weights w_n = C·exp(−πn²/9) — resolves the black-hole information paradox through three interconnected mechanisms. The event horizon carries exactly nine quantised soft-hair charge sectors per Planck area from the Z₉ gyroid torsion nodes, yielding a finite horizon Hilbert space of dimension 9^(A/4ℓ²_Pl). The Bekenstein–Hawking entropy receives a universal size-independent topological correction ΔS_tors = 9 ln 9 + ln 2 ≈ 20.468, where 9 ln 9 is the nine-node torsion hair-lattice log-degeneracy and ln 2 is the Euclidean boundary contribution. The golden-ratio factor φ^(−N*) exponentially suppresses logarithmic soft-graviton loop corrections, while holographic unitarity is established by torus partition function factorisation Z(τ) = |χ_{Z₉}(τ)|², together ruling out firewalls and remnants. The primary observational target is a Z₉ QNM sideband comb: nine sidebands spaced ≈ 18 Hz flanking the main ringdown frequency for a 30 M☉ binary, detectable at Einstein Telescope and Cosmic Explorer (~2035). Two existing null searches (GWOSC O3a, GRACE-FO 254-month mascon) are consistent with the framework and define the sensitivity floor for next-generation instruments. All results anchor to the single Gaia DR3 calibration η_diff = 0.82 ± 0.02. The 11D CDT extension is conjectured; the framework is non-supersymmetric. v2 corrections (13 items): BMS and CDT spelled out on first body use; Δλ corrected to 1.7287; Z_tors ≈ 2 (not exactly 2); all companion paper DOIs updated to canonical versions; Paper 1 (prismatic foundation, 10.5281/zenodo.20146627) and Paper 9 Survey (10.5281/zenodo.20315669) added to cross-citations; Vafa (1986) bibitem key corrected; Schoen (1970) NTRS URL added; equation references standardised; Gap 8 (G₄ = ℓ²_Pl/ln 9 derivation open) and Gap 9 (pre-Page-time retrieval) added; microtype and emergencystretch corrected; fig4 Python source updated (Δλ = 1.7287). Part of the PTRH/TMRB preprint series. Series anchor: η_diff = 0.82 ± 0.02 (Gaia DR3). All parameters fixed in Paper 3 (10.5281/zenodo.20248650).