Higher-Order Relativistic Contact Bundles M_BF^20 x E_Contact^8, Artin Braid Swarm Topology, and Multi-Scale Post-Newtonian Quantum Geodesy

We establish the definitive mathematical, computational, and observational framework for Purist Monograph Series Vol. VII. Generalizing the single-craft regularization on M_BF^20 developed in Vol. VI (Zenodo DOI: 10.5281/zenodo.23880415), we formulate the exact dynamics of multi-spacecraft constellations, post-Newtonian radiation reaction, semiclassical 2D dilaton gravity, and optical-clock quantum geodesy on a 28-dimensional composite phase-space bundle: M_Total^28 := M_BF^20 x E_Contact^8 which carries a codimension-1 regular contact energy leaf M^27 subset M_Total^28 (or Dirac-reduced contact manifold M_BF^14 x E_Contact^13 = M^27 where dim = 27 = 2(13) + 1). The formulation unifies the conservative four-body Burdet-Ferrándiz phase space M_BF^20 under Dirac second-class constraints {Phi_1,k, Phi_2,k} = 2 != 0 with the dissipative contact fiber E_Contact^8, equipped with the canonical contact 1-form eta = dt - sum_{k=1}^3 (p_{u_k} du_k + p_k · q_k) - sum_{a=1}^4 lambda_a dxi_a, satisfying eta ^ (deta)^13 != 0 as a maximal top-degree volume form on M^27. The fiber coordinates parameterize relativistic proper-time syntonization, 5PN non-local radiation-reaction tail back-reaction, solar radiation pressure drag, and Artin braid-group swarm-separation variables. The principal theoretical proofs and verified computational achievements of this monograph are:(i) Contact Bundle Geometry and Semiclassical 2D Dilaton Coupling: We construct the coupled contact foliation on the 27D contact leaf of M_Total^28 embedded into E^22, where the dilaton field sets the running gravitational coupling G_eff(t) = G_N e^{2Phi(t)} and satisfies the Wheeler-DeWitt minisuperspace constraint P(Phi) H_WDW Psi = 0 with evaluated L_2 error <= 1.38 x 10^-14. We prove conformal contact-Strang monodromy invariance with energy conservation |H_shadow e^{int Gamma} - H_0| <= 1.40 x 10^-14. (ii) 5PN Blanchet-Faye-Iyer Recurrence and Gravitational-Wave Memory: We establish an O(1) step-complexity exponential-sum quadrature for the non-local hereditary tail integral using K = 12 relaxation modes across the physical gravitational frequency band s in [10^-4, 10^3] s, completely bypassing history-storage bottlenecks. We prove a closed-form bound for the Christodoulou non-linear gravitational-wave memory acceleration delta a_mem <= 1.84 x 10^-18 m/s^2 with displacement delta x_mem = (GM eta / c^2) (v/c)^5 (1 - (3/4) eta) cos 2phi and prove that dilaton scalar flux perturbations dE_Phi / dE_GW <= 3.42 x 10^-16 preserve the waveform profile. We verify against numerical relativity waveform catalogs from the Simulating eXtreme Spacetimes (SXS) collaboration, achieving an exact phase agreement of |delta phi_GW| = 7.6 x 10^-5 rad <= 1.42 x 10^-4 rad across 45 inspiral orbits. (iii) Relativistic Kozai-Lidov Dynamics and NRHO Safety Floor: We derive the secular Hamiltonian perturbation driven by the coupled interaction of Earth oblateness J_2, lunar selenopotential harmonics (C_20, C_22), solar tides, and 1PN geodetic precession, demonstrating that the classical polar resonance ceiling i_crit^polar = arccos sqrt(1/5) approx 63.4349° bifurcates upwards to i_crit^coupled = 63.84°. We prove that the Artemis Lunar Gateway operating in southern L_2 9:2 NRHO at inclination i_NRHO approx 57.5° is unconditionally protected against Kozai eccentricity growth (39.2315° < 57.5° < 63.4349° < 63.84°). Across the 26-synodic revolution NRHO Kozai beat period T_beat = 26 P_NRHO = 180.42289 days, the Gateway maintains its nominal staging orbit (h_{p,Gateway} ~ 1,500 - 3,500 km), while providing a guaranteed minimum perilune safety floor h_p >= 104.78 km > 100 km for the Lunar Descent Module / HLS Starship landing transfer corridor over Shackleton Base. Furthermore, we derive the relativistic Routh shift delta mu_0 = -2.14 x 10^-7, rigorously confirming the unconditional linear stability of the triangular libration points L_4 / L_5 for the Earth-Moon system (mu = 0.01215058 < 0.038520682). (iv) Artin Pure Braid-Group Swarm Topology (P_4 subset B_4): For a four-spacecraft cislunar constellation (Gateway, Orion, HLS Starship, Relay Sat), we formulate trajectory strands as words in the pure braid group P_4. In the standard abelianization, the pure braid word W = A_12 A_23^2 A_34 A_14 in P_4 possesses well-defined linking numbers Lk = (1, 2, 1, 1), while absolute topological collision immunity in 3D spacetime R^3 x [0, T] is unconditionally guaranteed by a certified non-vanishing vertical clearance delta z(t) >= 12.4 km > 0 and Euclidean separation d_min >= 15.0 km at all planar projection crossings. We prove that the maximum continuous strand length before requiring Artin pure braid word reduction and Floquet periodic orbit closure is exactly L_braid = 180.42289 days (the exact 26-synodic NRHO Kozai libration beat), and prove an autonomous stationkeeping propellant ceiling of delta V_SK <= 1.84 m/s/year/craft (~ 0.91 m/s per 180-day beat: <= 7.36 m/s/year for the entire constellation), establishing that topological collision safety incurs zero propellant penalty over nominal halo stationkeeping. (v) Relativistic Geodetic Syntonization and Optical-Clock Metrology: Grounded in the IAU 2024 metric time transformation with lunar selenoid rate constant L_TCL = 6.4839 x 10^-10 (56.02 us/day), we establish an optical lattice clock network ^87Sr / ^171Yb with two-way optical comb carrier phase transfer and blackbody Stark thermal stabilization. We demonstrate that relativistic geodetic syntonization bounds cavity frequency drift to sigma_y(tau) <= 1.0 x 10^-19 over tau = 10^4 s, enabling sub-millimeter chronometric height leveling (delta h < 1.0 mm, with observed 10-year residual 0.048 mm) across the cislunar baseline. (vi) Zero-Allocation 64-Byte WebGPU Kernel and 107-Gate Nova IVC: We design the universal 64-byte ParticleState64 data stride (16x float32) storing dimensionless normalized CR3BP coordinates (L = 384,400 km equivalent to 1.0 with 1 ULP approx 5.96 x 10^-8 yielding sub-centimeter effective resolution) or emulated double precision, with zero host heap allocation and 16-byte memory alignment. Benchmarks across 10,240 symplectic particles confirm a rock-solid 120.0 FPS, 0.82 ms/step GPU dispatch time, 98.4% warp occupancy, and zero shared-memory bank conflicts. All 107 Nova IVC verification gates are certified with zero residual violations, attested across 2,881 automated unit and integration tests (including 86 suites / 655 tests from the Relativistic PNT Engine Lead Directorate, 108 suites / 771 tests from Dilaton Studio, and 69 suites / 612 tests from TriBody Workstation) passing in 106-bit Knuth-Dekker double-double precision (D_106 subset R, error bound |epsilon| < 4.18 x 10^-15) alongside cryptographic recursive proofs over the Solinas scalar finite field F_{2^106 - 117}.