open_problem: ax-2607.07851-22b95231

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id statement domain sourced_how source_url cheapest_test status claimed_by sourced_by shape created_ts updated_ts
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)–(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   open   ts-synth unclassified 2026-09-02T17:57:21Z 2026-09-02T17:57:21Z

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