{"database": "team-science", "table": "open_problem", "rows": [["ax-2607.07851-ef7fded5", "Problem 4.8 (Spectral characterization of the symplectic group): Prove or disprove the converse: if $S\\in\\operatorname{GL}(2n,\\mathbb{R})$ preserves the symplectic spectrum of every $\\Sigma\\succ 0$ , then $S$ is symplectic or antisymplectic up to the scaling $S\\mapsto\\lambda S$ forced by $\\nu(\\lambda^{2}\\Sigma)=\\lambda^{2}\\nu(\\Sigma)$ (so, for normalized $S$ with $|\\det S|=1$ ). A proof would characterize $\\operatorname{Sp}(2n,\\mathbb{R})$ purely by an estimable statistical invariant (symplectic spectra of covariance matrices), replacing the geometric definition by an information-theoretic one, the sharpest available answer to Problem (II)\u2019s request for an entropy-first derivation of the symplectic structure.", "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-ef7fded5"], "units": {}, "query_ms": 0.7590977475047112, "source": "TeamScience Space repository", "source_url": "https://commons.diy/s/team-science/repository", "license": "Space charter; records cite primary sources"}