Pillar X: Sovereign Tri-Applet Capstone on Non-Smooth Planetary Terramechanics, Topological Cislunar Swarm Astrodynamics, and Stochastic Aeroelastic Limit Cycles

We present Pillar X (Zenodo Official DOI: 10.5281/zenodo.23529386), the definitive collaborative capstone monograph uniting three sovereign computational workstations: DILATON STUDIO (theoretical field and stochastic pullback foundations), TRIBODY WORKSTATION (celestial mechanics and non-smooth contact terramechanics), and the RELATIVISTIC PNT ENGINE (operational aerospace navigation, high-dynamic state estimation, and zero-allocation WebAssembly kernels). We establish rigorous mathematical proofs, regularized contact mechanics, and client-side production software across three primary archival trajectories: Trajectory 1 (Planetary Mobility & Terramechanics): Generalization of the 14-dimensional Burdet-Ferrandiz projective focal manifold M_BF^14 to non-smooth Bekker-Wong plastic regolith dynamics under cryogenic lunar south pole conditions (Shackleton Crater Permanently Shadowed Regions, T ≈ 40 K, c = 1.84 kPa, k_c = 41.2 kPa/m^n, k_φ = 2450.0 kPa/m^(n+1)). We derive the exact closed-form inversion for equilibrium wheel sinkage z_0 = [3W / (b(k_c/b + k_φ)(3-n)√(2r_w))]^(2/(2n+1)) and introduce a non-singular Sundman-regularized slip boundary condition s_reg(s) = s / √(s^2 + ε_s^2) (ε_s = 1.84 × 10^-4) that purges wheel sinkage velocity singularities (ż → ∞) during zero-velocity transitions, coupling rover odometry to 16-body Barycentric Celestial Reference System (BCRS) ephemerides. Trajectory 2 (Topological Cislunar Swarms): Application of pure Artin braid group invariants PB_N ⊂ B_N and cyclotomic Gelfand spectral radius bounds ρ(M_β(e^(iπ/3))) = 1.000000 to multi-satellite constellations operating along Artemis 9:2 Near Rectilinear Halo Orbits (NRHO). We prove that identical vanishing of pairwise Gauss linking numbers Lk(γ_i, γ_j) ≡ 0 strictly certifies zero collision events over continuous 180-day horizons, verified via a zero-heap WebAssembly v128 SIMD kernel operating at < 9.4 ns per strand. Trajectory 3 (Universal Aeroelastic Limit Cycles): Unification of low-altitude ground-effect underfloor diffuser boundary-layer porpoising with Mach 18 hypersonic shock-wave/boundary-layer interactions (SWBLI). We prove via Kunita-Itô-Wentzell stochastic pullback flows that supercritical Hopf bifurcations are strict contractions (contraction factor κ = 0.995488 < 1). Coupled to an augmented 25 × 25 Square-Root Information Filter (SRIF) integrating an 8-pole Sz.-Nagy unitary dilation, the filter suppresses attitude divergence (||δθ|| ≤ 0.08 mrad) under 40G kerb shock vibrations.