Polynomial Rigidity and a Sharp Degree Bound for Free Quadratic Variation

Abstract

We isolate an elementary highest-degree obstruction to polynomial changes of variables preserving unit free-Brownian diffusion. For every unital moment functional and every nonconstant noncommutative polynomial of degree d, its free quadratic-variation polynomial has degree exactly 2d-2. The leading coefficients form a positive Gram matrix of last-letter derivatives. At any algebraically free self-adjoint tuple, this classifies self-adjoint polynomial unit-covariance maps as affine coisometries. For a standard semicircular m-tuple, the distance of the covariance from the scalars is at least 1/sqrt(m) times the squared norm of the polynomial's highest Wick component. The constant is sharp, and for quadratic polynomials this is a sharp distance-to-affine estimate. The stochastic application concerns a prescribed pathwise transformation in the same filtration; it does not resolve the more general terminal-law steering question in the AIM Free Analysis problem list. This is an unrefereed, self-audited preprint. The exact checker supplements the written proof, not formal verification or independent review. The author used generative-AI assistance and remains responsible for the manuscript. The inherited one-variable observation and established tools are credited. No absolute-priority claim is made; the elementary rigidity statement may be folklore. The full source record AIM-PROBABILITY-0108, including adjacent entropy and pressure questions, remains unresolved by this work.