Fractional projective-gap inequalities for harmonic Gaussian chaoses

Let F belong to a fixed Gaussian Wiener chaos, normalize E[F²] = 1, and set Z = |DF|²/M, where M is the chaos order. We study the fractional projective defect K_r(F) = E[(F²-Z)Z^r], r > 0. We prove the full shifted inequality for every nonzero element of the second chaos, prove it in every degree for products of mutually orthogonal Gaussian linear forms, and establish the first fully mixed harmonic cubic case in three variables at r = 3/20. The cubic theorem is closed by an exact rational Bernstein certificate whose verifier is included. These results do not assert the inequality for arbitrary chaos order or solve the full nonlinear Kondrat'ev program. This is the author manuscript submitted to the European Journal of Mathematics on 4 September 2026. It has not yet been peer reviewed or accepted.