Paper IV — UN Theory Unconditional Banach-Space Global Action Convexity and Spectral Positivity of S[I₀]

This paper proves Theorem 4 of UN Theory: the full non-local effective substrate action S[I], evaluated on the admissible correlation manifold B_adm \subset C^\infty(M x M), is unconditionally and strictly convex, possessing a unique global minimum at the ground-state kernel I(x,x') = I_0(x,x'). Equipping B_adm with a Sobolev-type Banach norm ||.||B, the paper proves that the functional Hessian operator K_0 \equiv (\delta^2 S / \delta I^2)|{I_0} is densely defined, self-adjoint, and possesses a strictly positive spectrum Spec(K_0) \subset [\mu_0^2, \infty) with \mu_0^2 > 0. This rigorous spectral positivity guarantees absolute vacuum stability, the complete absence of tachyons or Ostrogradsky ghost modes, total suppression of false-vacuum decay, and global unconditional well-posedness for the non-local action functional S[I].