Hidden Constants after Free Semicircular Regularization

Abstract

A bounded conjugate variable does not imply a coarse relative free Poincare inequality. Let B be a diffuse tracial von Neumann algebra, let Y be a bounded self-adjoint element commuting with B, and put A=W*(B,Y). If S is a variance-one semicircular element free from A, then X=Y+epsilon S, epsilon>0, generates the entire free product together with B. We prove that X is algebraically free from B, has bounded conjugate variable S/epsilon, and its closed relative free difference quotient has kernel exactly L2(A). If Y is outside B, this kernel is larger than L2(B) and no finite coarse Poincare constant relative to B exists. In a tensor Bernoulli example, equal-trace pinching polynomials have exact squared Poincare ratios n+epsilon^2. This answers Question 7.4 in Ito's arXiv:2409.16498v3 negatively and gives a scoped obstruction for AIM-PROBABILITY-0103, not a classification of all positive cases in the original AIM question. Prior commuting counterexamples and kernel absorption results are credited. The source package contains the proof and supplementary exact rational-arithmetic checks. Unrefereed preprint; AI assistance is disclosed; no independent review, formal verification or absolute priority is claimed.