Resolution of the Strong CP Problem: Static Cauchy Stress Balance, Deterministic Vanishing of the Neutron Electric Dipole Moment, and the Elimination of Peccei-Quinn Axions

The Strong CP Problem constitutes one of the most prominent fine-tuning crises in theoretical particle physics. In quantum chromodynamics (QCD), non-trivial topological vacuum configurations permit a P- and CP-violating term θ_bar = θ + arg det(M_q) of natural order unity. However, high-precision laboratory searches for the neutron electric dipole moment (nEDM) establish a strict upper bound |d_n| < 1.8 × 10^-26 e·cm, enforcing |θ_bar| < 10^-10 without an identified dynamical symmetry in the Standard Model. The orthodox resolution—the Peccei-Quinn mechanism—posits an unobserved global U(1)_PQ symmetry and a hypothetical pseudo-Goldstone boson (the axion) that has eluded empirical detection for over four decades while suffering from the severe theoretical "axion quality problem" at the Planck scale. Building upon our foundational series on the physical vacuum as an elastically pre-stressed continuum, we show that θ_bar ≡ 0 is an exact, non-perturbative mechanical identity mandated by the internal stress equilibrium of the baryon soliton. In our ontology, a neutron is a composite toroidal strain knot in an elastic medium endowed with bulk modulus K_v and pre-stress P_0. An electric dipole moment corresponds to an axial spatial separation between the centroids of volumetric dilation and shear strain. We prove that any nonzero θ_bar induces an uncompensated internal Cauchy stress divergence (∇ · σ ≠ 0), generating an immense mechanical restore force that forces the ground state to strictly preserve parity. The Strong CP Problem is thus resolved under Occam's razor as an artifact of point-particle field theory, eliminating the need for hypothetical axion fields.