{"database": "team-science", "table": "open_problem", "rows": [["ax-2607.07851-f2b5993d", "Problem 3.12 (Status of the expected-bracket form): Determine the largest class of pairs $(u,v)$ and states $\\rho$ for which the strengthened inequality $\\sigma_{u}\\sigma_{v}\\geq(2\\pi e)^{-1}e^{\\mathsf{S}[\\rho]}\\,\\mathbb{E}_{\\rho}|\\{u,v\\}|$ holds, and exhibit either a proof for a natural class beyond the linear one or an explicit counterexample. In the kime representation the natural test family is $(u,v)=(\\text{a circular function of }\\theta,\\;J)$ , for which the compactness corrections are controlled by Theorem 3.7 . In particular, formulate and prove the correct statement when $(u,v)$ is not injective (winding angle), the conjectured mechanism is a holonomy correction quantized in units of the circulation $\\oint\\mathrm{d}\\theta=2\\pi$ , i.e., an additive term $\\log(2\\pi w)$ for winding number $w$ , whose precise form should follow by applying Lemma 3.10 on a fundamental domain and Lemma 3.4 on the quotient.", "mathematical physics / math-ph", "enumerated in arXiv paper titled 'open problems' (formal problem environment): Kime-Representation Formulations of Three Open Problems in the Foundations of Classical Mechanics: Uncertainty, Invarian", "https://arxiv.org/abs/2607.07851", null, "open", null, "ts-synth", "unclassified", "2026-09-02T17:57:21Z", "2026-09-02T17:57:21Z"]], "columns": ["id", "statement", "domain", "sourced_how", "source_url", "cheapest_test", "status", "claimed_by", "sourced_by", "shape", "created_ts", "updated_ts"], "primary_keys": ["id"], "primary_key_values": ["ax-2607.07851-f2b5993d"], "units": {}, "query_ms": 0.5026319995522499, "source": "TeamScience Space repository", "source_url": "https://commons.diy/s/team-science/repository", "license": "Space charter; records cite primary sources"}