Resolution of the Proton Stability Paradox: Homotopic Knot Invariants, Infinite Lifetime of Free Baryons, and the Elimination of GUT Leptoquarks in an Elastically Pre-Stressed Vacuum

Standard Grand Unified Theories (GUTs) based on SU(5) or SO(10) place quarks and leptons in common multiplets, mediating baryon-number-violating decays (p -> e+ π0) and predicting proton lifetimes τ_p ~ 10^30–10^32 yr. However, multi-decade null results at Super-Kamiokande establish strict empirical lower bounds τ(p -> e+ π0) > 2.4 × 10^34 yr and τ(p -> ν_bar K+) > 5.9 × 10^33 yr, ruling out minimal unification and creating the Proton Stability Paradox. Building upon our foundational series on the physical vacuum as an elastically pre-stressed continuum, we show that the absolute stability of the free proton is an exact, non-perturbative topological theorem. In our ontology, a baryon is a three-dimensional closed topological vortex knot with an integer winding index B ≡ Q_topo = π_3(S^3) ≅ Z = 1, whereas leptons are simple vortex rings (B = 0) and mesons are non-topological strain oscillations. Under Cauchy momentum conservation in an elastic continuum with finite shear modulus G_v and holding pre-stress P_0, the displacement field u(x, t) is a C^2-differentiable manifold whose topological winding index is strictly conserved (dQ_topo/dt ≡ 0). Transitioning from B = 1 to B = 0 requires mechanical fracture of the substrate, presenting an insurmountable energy barrier E_fracture ≥ M_Planck c^2 ≈ 1.22 × 10^19 GeV that suppresses tunneling to zero. The free proton's spontaneous decay lifetime is strictly infinite (τ_p ≡ ∞), resolving the paradox under Occam's razor without requiring ad-hoc R-parity, doublet-triplet tuning, or unobserved GUT monopoles.