{"database": "team-science", "table": "open_problem", "rows": [["ax-2607.07851-105112d7", "Problem 4.11 (Volume-preserving vs. symplectic for $n\\geq 2$ ): Corollary 4.4 pins down the measure but, for $n\\geq 2$ , not the finer symplectic structure: $\\mathrm{SDiff}(\\mathbb{R}^{2n})\\supsetneq\\mathrm{Symp}(\\mathbb{R}^{2n})$ . Identify the weakest statistical requirement that reduces the invariance group from volume-preserving to symplectic. Candidates, in increasing strength: (i) invariance of all within-DOF uncertainties $u_{j}$ at equilibrium; (ii) invariance of the linear symplectic capacity of covariance ellipsoids (see the capacity remark 3.4 ; by this fails for generic volume-preserving linear maps once $n\\geq 2$ ); (iii) invariance of the whole symplectic spectrum (Problem 4.8 ). Determine which of (i)\u2013(iii) are equivalent, and which are estimable from kime-tomographic data in the sense of Problem 3.18 .", "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-2607.07851-105112d7"], "units": {}, "query_ms": 0.5252617411315441, "source": "TeamScience Space repository", "source_url": "https://commons.diy/s/team-science/repository", "license": "Space charter; records cite primary sources"}