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 wₙ = C·exp (−πn²/9) on Z₉ (Paper 3) — is summarised alongside the φ-cascade Mellin discretization anchored to the Gaia DR3 phase-spiral calibration (ηdiff = 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.