{"database": "team-science", "table": "open_problem", "rows": [["ax-2607.07851-51bcfa22", "Problem 4.9 (Entropy-only rigidity): Theorem 4.1 assumed entropy invariance for all states. Determine the minimal state classes $\\mathcal{F}$ for which \u201c $\\mathsf{S}[f_{*}\\rho]=\\mathsf{S}[\\rho]$ for all $\\rho\\in\\mathcal{F}$ \u201d still forces $J_{f}\\equiv 1$ (e.g., Gaussians only; kime states with von Mises phase laws only), and, dually, characterize the group of transformations preserving the entropy of every equilibrium (phase-equipartitioned) kime state. The latter group is strictly larger than the measure-preserving group (it contains all fiberwise rotations $\\theta\\mapsto\\theta+c(J)$ trivially, but also non-measure-preserving maps acting only on null sets of equilibria); its computation quantifies exactly how much of the symplectic structure is visible to equilibrium thermodynamics alone.", "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-2607.07851-51bcfa22"], "units": {}, "query_ms": 0.6258003413677216, "source": "TeamScience Space repository", "source_url": "https://commons.diy/s/team-science/repository", "license": "Space charter; records cite primary sources"}