open_problem: ax-2607.07851-105112d7

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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)–(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   open   ts-synth unclassified 2026-09-02T17:57:21Z 2026-09-02T17:57:21Z

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