Phase-Synchronized Trajectory Optimization on Curved Manifolds with High-Order Non-Separable Symplectic Thermostats, Adaptive Manifold Restraints, and 3D Braid-Group Spacetime Knot

Abstract We formulate and prove Version 4.0 of the unified geometric, thermodynamic, and topological framework governing relativistic N-body systems evolving on pseudo-Riemannian spacetimes. This work rigorously resolves fundamental challenges across relativistic astrophysics, geometric integration, and mathematical mechanics: 1. Curved-Manifold Langevin Thermostat and Frustrated Kuramoto Dynamics:We formulate the symmetric BAOAB Langevin thermostat on a general Riemannian Cauchy slice (Σ, g) via Levi-Civita covariant parallel transport along geodesics, incorporating the non-conservative 2.5PN Burke-Thorne quadrupole radiation reaction. We prove that the covariant Kuramoto order parameter R_g(t) ∈ [0, 1] exhibits exponential convergence toward phase synchronization with rate λ_g = K[cos(δ_max^g) − 2γ_0 sin(δ_max^g)] − K_c > 0, rigorously accounting for non-linear geometric frustration cross-terms induced by Riemann connection holonomy. 2. Conservative Non-Separable Symplectic Integrator:We establish an extended phase-space symplectic splitting (following Tao [1]) for the conservative Hamiltonian sector, incorporating velocity-dependent relativistic interactions through conservative 2.5 post-Newtonian (PN) next-to-leading order (NLO) spin-orbit coupling and 2PN spin-spin quadrupole-monopole deformations (C_Q). We explicitly derive the Poisson bracket commutators, the exact drift-kick flows, and the spin-precession equations. 3. Strictly Canonical Dynamic Restraint ω(r_min) and Transverse Bounds:To prevent constraint degradation during relativistic periastron passages while preserving exact canonical symplecticity, we decompose the coordinate-dependent restraint flow into a harmonic rotation and an exact symplectic shear kick generated by ∇_q ω(r_min) ||Z_-||². We prove Theorem 3.4, establishing that the steady-state transverse envelope from the physical constraint manifold C = {q = x, p = y} remains uniformly bounded as O(Δt / ω_0) across all pericenter passages with zero secular energy drift over 10⁷ orbits. 4. 3D Spacetime Cylinder Braid Group B_N, Knots, and DFA:We map multi-body timelike worldlines in the spacetime cylinder M = ℝ × Σ into physical braids in the Artin braid group B_N. We calculate the topological winding numbers W_ij = (1 / 2π) ∮ d arg(q_i − q_j), the Jones polynomial V̂_b(t) under Markov closures, and construct a 5-state Deterministic Finite Automaton (B_N-DFA) that strictly classifies chaotic scattering, binary-single exchanges, hyperbolic ejections, and gravitational-wave coalescences. Extensive numerical experiments in double precision (FP64) verify shadow Hamiltonian conservation (ΔH / H_0 < 10⁻⁸) and establish an order-of-magnitude reduction in constraint violation during close black hole encounters. Keywords: Non-Separable Symplectic Integrators, Post-Newtonian Spin Dynamics, Adaptive Constraint Manifolds, Kuramoto Phase Synchronization, Artin Braid Group, Spacetime Cylinder Knots, Topological Scattering DFA