{
  "schema_version": 1,
  "problem_number": "AIM-PROBABILITY-0103",
  "title": "Hidden Constants after Free Semicircular Regularization",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "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.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.OA",
    "math.PR"
  ],
  "keywords": [
    "free Poincare inequality",
    "conjugate variables",
    "free difference quotient",
    "semicircular regularization",
    "operator algebras",
    "AIM-PROBABILITY-0103"
  ],
  "manuscript_version_date": "2026-10-18",
  "publication_date": "2026-10-17",
  "publication_date_kind": "first public online release",
  "version": "2.0",
  "date_modified": "2026-10-18",
  "presentation_revision_only": false,
  "doi_archived_file_version": "2.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-probability-0103/",
  "pdf_url": "https://eulersolve.org/papers/aim-probability-0103/paper.pdf?v=44f5bc50fafe",
  "doi": "10.5281/zenodo.23442154",
  "zenodo_record_url": "https://zenodo.org/records/23442154",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Complete counterexamples to conjugate-variable sufficiency, including an irreducible inclusion of II1 factors, with exact enlarged derivative kernels and exact failing ratios. The original AIM classification of positive cases remains open. Ito and Lee-Nelson prior work is credited. AI-assisted, self-audited and unrefereed; no independent review, formal certification or absolute-priority claim.",
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  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text.",
  "concept_doi": "10.5281/zenodo.23441917",
  "concept_url": "https://doi.org/10.5281/zenodo.23441917",
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  "revision_note": "Version 2.0 adds the irreducible II1-factor construction; the commuting theorem is retained. This is one paper, not another source closure."
}