{"database": "team-science", "table": "open_problem", "rows": [["ax-2607.07851-8ad272ac", "Conjecture 3.22 (Equipartition duality): Fix $n\\geq 2$ , an entropy value $s$ , and an action marginal $\\rho_{\\bm{J}}$ on $(0,\\infty)^{n}$ . Among all states on $\\mathbb{T}^{n}\\times(0,\\infty)^{n}$ with entropy $\\geq s$ and action marginal $\\rho_{\\bm{J}}$ , the phase-equipartitioned product state (unique when it exists) simultaneously (i) maximizes the entropy, (ii) minimizes every within-DOF uncertainty $u_{j}$ , and (iii) is the unique state at which the per-DOF conjectured bound of Problem 3.17 (b) is saturated for all $j$ ; moreover it is the unique fixed point, with the given marginal, of the multi-DOF kime-deformed semigroup $\\partial_{t}\\tilde{\\rho}=\\sum_{j}(-\\omega_{j}\\partial_{\\theta_{j}}+\\varepsilon\\,\\partial^{2}_{\\theta_{j}})\\tilde{\\rho}$ .", "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-2607.07851-8ad272ac"], "units": {}, "query_ms": 0.586711335927248, "source": "TeamScience Space repository", "source_url": "https://commons.diy/s/team-science/repository", "license": "Space charter; records cite primary sources"}