{"database": "team-science", "table": "open_problem", "rows": [["ax-2607.07851-15e7560c", "Problem 3.17 (Symplectic Schur\u2013Horn problem for within-DOF uncertainties): Characterize, for fixed symplectic spectrum $\\nu_{1}\\geq\\cdots\\geq\\nu_{n}>0$ , the attainable set $\\mathcal{U}(\\nu)=\\Bigl\\{\\bigl(u_{1}(\\Sigma),\\dots,u_{n}(\\Sigma)\\bigr):\\Sigma\\in\\text{the }\\operatorname{Sp}(2n,\\mathbb{R})\\text{-orbit with spectrum }\\nu\\Bigr\\}\\subset\\mathbb{R}_{>0}^{n},$ where $u_{j}(\\Sigma)=\\sqrt{\\det\\Sigma_{jj}}$ . In particular, we need to explore (a) is $\\min_{j}u_{j}\\geq\\nu_{n}$ on the whole orbit (so that no DOF can be squeezed below the smallest symplectic eigenvalue)? (b) in the equipartitioned case $\\nu_{j}\\equiv\\nu$ , is $u_{j}\\geq\\nu$ for every $j$ (the per-DOF form of the conjecture in )? And (c) describe $\\mathcal{U}(\\nu)$ by majorization-type inequalities, in analogy with the Schur\u2013Horn theorem, using the symplectic eigenvalue technology of . Some partial cues include (i) $\\prod_{j}u_{j}\\geq\\prod_{j}\\nu_{j}$ (Theorem 3.16 ); (ii) $u_{j}\\geq\\nu_{n}$ would follow from the interlacing-type bound \u201cevery $2\\times 2$ symplectic compression of $\\Sigma$ has symplectic eigenvalue $\\geq\\nu_{n}$ ,\u201d a statement of exactly the kind studied in ; and (iii) for $n=1$ , $\\mathcal{U}(\\nu)=\\{\\nu\\}$ (Theorem 3.16 ).", "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-2607.07851-15e7560c"], "units": {}, "query_ms": 0.517561100423336, "source": "TeamScience Space repository", "source_url": "https://commons.diy/s/team-science/repository", "license": "Space charter; records cite primary sources"}