Resolution of the Strong CP Problem: Substrate Cauchy Stress Symmetry, Elimination of Vacuum θ-Angles, and the Vanishing Neutron Electric Dipole Moment in an Elastically Pre-Stressed Vacuum

In non-Abelian quantum chromodynamics (QCD), the Lagrangian allows a P- and CP-violating topological term parameterized by θ_bar ≡ θ_QCD + arg det(M_q). An uncancelled phase θ_bar ≠ 0 induces a permanent electric dipole moment in the neutron (d_n ~ 10^-16 θ_bar e·cm). However, precision ultracold neutron experiments at ILL and PSI establish the strict empirical bound |d_n| < 1.8 × 10^-26 e·cm, requiring an unnatural fine-tuning |θ_bar| < 10^-10 between unrelated strong topological and electroweak Yukawa sectors. The standard remedy—postulating an anomalous global Peccei-Quinn symmetry and an axion—has failed to find empirical confirmation after nearly five decades (ADMX, CAST). Building upon our foundational series on the physical vacuum as an elastically pre-stressed continuum, we show that the Strong CP Problem is an artifact of treating the vacuum as a multi-sheeted gauge-field manifold. In continuum elastodynamics, hadrons are stable topological strain-vortex solitons. Conservation of angular momentum enforces the rigorous symmetry of the Cauchy stress tensor (σ_ij = σ_ji). A permanent electric dipole moment requires a static spatial offset between the center of volumetric dilatation (charge) and the center of vorticity (spin). In an isolated neutron soliton, such an offset creates an asymmetric shear strain that spontaneously relaxes to zero within the substrate lattice relaxation timescale (τ_relax ~ ℓ_0 / c ~ 10^-43 s). We prove that θ_bar ≡ 0 is an exact mechanical equilibrium theorem rather than a dynamical coincidence, resolving the Strong CP Problem under Occam's razor without axions.