Hidden Constants after Free Semicircular Regularization

A bounded conjugate variable does not imply a coarse relative free Poincare inequality, even for an irreducible inclusion of II1 factors. We first give a commuting-extension construction: for a diffuse coefficient algebra B, a bounded self-adjoint Y commuting with B, and A=W*(B,Y), the variable X=Y+epsilon S with S scalar-free semicircular generates A*W*(S) together with B and has bounded conjugate variable S/epsilon, but its closed derivative kernel is L2(A). The tensor Bernoulli example has exact squared Poincare ratios n+epsilon^2. Version 2 adds an irreducible factor example. For the bilateral tensor Bernoulli shift alpha on the hyperfinite II1 factor B and A=B crossed_alpha Z, the variable X=u+u*+epsilon S satisfies B' intersect W*(B,X)=C and has closed derivative kernel exactly L2(A), strictly larger than L2(B). Twisted pinching polynomials have exact squared ratios 2^m+1+epsilon^2. Both examples answer the sufficiency question in Ito's arXiv:2409.16498v3, Question 7.4, negatively. This is not a classification of all positive cases of the original AIM question. Earlier commuting obstructions and kernel absorption results are credited. The source package includes proofs and exact supplementary checks. Unrefereed preprint; AI assistance is disclosed.