Torsion-Modulated Recursive Branching (TMRB) and Piano Mathematics

Version 4 (May 2026) — Changes from v3 Typographic and cross-reference corrections microtype expansion disabled; set to 2em (bibliography line-break fix) Δλ corrected from 1. 729 to 1. 7287 (exact value: 2π/9κCE) Causal dynamical triangulations (CDT) defined on first body use Bondi–Metzner–Sachs (BMS) defined on first introduction-body use Modular-covariance claim sharpened: Gaussian weight satisfies approximate covariance to corrections O (e^−4π) ≈ 3×10⁻⁶ on finite ℤ₉, not exact invariance Golden-ratio form description corrected: "failed modular covariance" → "was not designed to satisfy modular covariance" Bibliography updated: TMRB Unified Framework citation updated to v3 (doi: 10. 5281/zenodo. 20248650) ; VASIMR DOI corrected to 19268015; M-Theory Bulk Gravity Leakage updated to v9e with full subtitle; Celestial Holography updated to v4 (doi: 10. 5281/zenodo. 20249096) Cross-citation to Paper 1 (Einstein Equations from Prismatic Curvature, doi: 10. 5281/zenodo. 20146627) added Rigour corrections Eq. (3) corrected: The previous equation T₀ = 2π/Δλ ≈ 230 Myr was dimensionally incoherent (2π/1. 7287 ≈ 3. 63, not 230 Myr). Δλ = 1. 7287 is now correctly stated as a dimensionless conformal-weight spacing. T₀ ≈ 230 Myr is identified as the Gaia DR3 observational calibration anchor, not a derived output. The mapping from Δλ to a physical oscillation frequency is labelled an open derivation gap. The falsifiable content of the framework is clarified as the Gaussian-weighted harmonic spectrum T₀/n and ηdiff = 0. 82±0. 02. UV-completeness qualified: "UV-complete closure of the IR triangle" corrected to "conjectured UV-complete closure, subject to the 11D CDT conjecture" — consistent with the Methods caveat that 11D CDT is unproven. BMS displacement memory identification qualified: The Sagittarius pericenter is a Newtonian tidal perturbation, not a gravitational-wave burst at null infinity. The identification with a BMS memory mode is now labelled "proposed"; the formal Newtonian-to-BMS mapping is flagged as an open theoretical gap. ℤ₉ parafermion bijection qualified (×2): "map bijectively onto the nine torsion phases" corrected to "proposed to map bijectively (explicit bijection not yet constructed) " in both §4. 2 and Conclusions. 1. 2σ statistic clarified: Restated as separation of central values in null-uncertainty units; explicit statement that the prediction lies 0. 03 below the null 1σ lower bound of 0. 85. The Milky Way disk is still ringing from the Sagittarius dwarf pericenter passage ∼300–500 Myr ago. This paper applies the Torsion-Modulated Recursive Branching (TMRB) framework to galactoseismology and predicts a fundamental disk-wave crest period of 230 Myr, with the next crest at the present epoch — consistent with the large-amplitude Gaia DR3 phase spiral. The 9-fold discrete torsion structure (θn = 2πn/9, n = 0, …, 8) provides the first discrete realization of BMS superrotation charges at null infinity, closing the infrared triangle among Weinberg’s soft graviton theorem, BMS asymptotic symmetries, and gravitational memory. The 230 Myr crest is the macroscopic displacement-memory signature of this discrete symmetry. v3 corrections and upgrades: Branching weight updated to modular-covariant DFT fixed-point: wn = C exp (−πn2/9) ; torsion entropy 9 ln 9 ≈ 19. 775. κCE = 0. 4038 (this paper) explicitly distinguished from κprop = 0. 48 (VASIMR/QGGPf). Formal null model: ηnull = 1. 0 ± 0. 15 (uniform random phase). TMRB prediction ηdiff = 0. 82 ± 0. 02 requires >3σ resolution over ≥3 crest epochs for detection. CDT 11D extension labelled conjectured; non-SUSY declared throughout. G4/G11 = (5. 214 ± 0. 12) × 10−40 provides stress-tensor normalisation for the disk-wave amplitude. ZF Z9 parafermion primaries (c = 16/11, Gepner–Qiu 1987) map onto the nine torsion phases, connecting galactoseismology to flat-space holography. Single parameter ηdiff = 0. 82 ± 0. 02 spans 1023 m from Planck scale to galactic disk. Three upgraded figures; abstract margin and reference linking fixed; all DOIs hyperlinked. Falsification test: Gaia DR4 kinematics.