{"database": "team-science", "table": "open_problem", "rows": [["ax-2607.07851-22b95231", "Problem 3.19 (Nonlinear invariant interpolating entropy and capacity): Construct a functional $\\mathcal{C}[\\rho]$ of states on $\\mathbb{R}^{2n}$ such that: (i) $\\mathcal{C}$ is invariant under all (possibly nonlinear) Hamiltonian flows; (ii) $\\mathcal{C}$ reduces to $\\pi\\nu_{n}$ on Gaussian states; (iii) $\\mathcal{C}$ lower-bounds $2\\pi\\min_{j}u_{j}$ up to a universal constant; and (iv) $\\mathcal{C}$ is expressible through the kime-torus data of Problem 3.18 (hence estimable). Consider the following candidate functional, a sublevel-set capacity of $\\rho$ at the entropy-calibrated level $e^{-\\mathsf{S}[\\rho]}$ , i.e., $\\mathcal{C}[\\rho]=c\\bigl(\\{\\rho\\geq e^{-\\mathsf{S}[\\rho]}\\}\\bigr)$ for a normalized symplectic capacity $c$ . Properties (i) and (ii) then hold by symplectomorphism-invariance of $c$ and a direct Gaussian computation, while (iii)\u2013(iv) are open.", "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-2607.07851-22b95231"], "units": {}, "query_ms": 0.5909907631576061, "source": "TeamScience Space repository", "source_url": "https://commons.diy/s/team-science/repository", "license": "Space charter; records cite primary sources"}