Torsion-Modulated Recursive Branching and the PTRH Framework

Torsion-Modulated Recursive Branching and the PTRH Framework: A Unified Synthesis from Prismatic Cohomology to Trans-Medium Propulsion This paper provides a unified synthesis of the eight-paper Prismatic Torsional Recursive Holographic (PTRH) programme (Paper 2 reserved). The series connects Bhatt–Scholze prismatic cohomology to discrete Z₉ torsion, Bondi–Metzner–Sachs (BMS) superrotation charges, galactic kinematics, sub-femto-g gravity anomaly predictions, M-theory bulk leakage, and trans-medium propulsion engineering. The mathematical spine — a SymPy-verified derivation of the Einstein field equations from prismatic curvature (Paper 1) and the Torsion-Modulated Recursive Branching (TMRB) framework with nine-fold Gaussian weight w_n = C·exp(−πn²/9) on Z₉ (Paper 3) — is summarised alongside the φ-cascade Mellin discretization anchored to the Gaia DR3 phase-spiral calibration (η_diff = 0.82 ± 0.02). The conventional engineering baseline (Rayleigh–Plesset cavity dynamics, VASIMR-class thrust 1–10 N, TRL 2–3, bench-rig cost $10–16k) is presented independently of TMRB assumptions. Eight structural derivation gaps are explicitly enumerated and ranked by tractability. No extraordinary performance claim (435 kn water cruise, sub-femto-g gravity anomalies, BMS holographic encoding) is asserted without the gaps that condition it. Four near-term falsifiable tests are ranked by experimental accessibility. The programme is explicitly non-supersymmetric. Appendices provide a master parameter table, a cross-citation map for all key quantities across Papers 1 and 3–9, and a TRL-3/4 test protocol summary. Open-source firmware and a companion Jupyter notebook are archived at DOIs 10.5281/zenodo.20278598 and 10.5281/zenodo.19664879 respectively.