open_problem: ax-2607.07851-b83ce5f3

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ax-2607.07851-b83ce5f3 Problem 5.9 (Kime action–angle atlas on $\mathcal{O}_{m,s}$ ): Construct an atlas of Darboux charts on $\mathcal{O}_{m,s}$ adapted to the kime fibration, i.e., charts of the form $(x^{i},p_{i};\varphi,S_{z}^{\prime})$ in which the directional factor is the kime cylinder of Lemma 5.2 for the little-group sphere, and quantify the obstruction to a single global chart. The fiber $\mathbb{S}^{2}_{s}$ has $\int_{\mathbb{S}^{2}_{s}}\omega_{s}=4\pi s\neq 0$ , so no global Darboux chart exists, and the minimal atlas is governed by the class $[\omega_{s}]/2\pi\hbar$ , which is integral iff $2s/\hbar\in\mathbb{Z}$ (Weil integrality; ). Make precise, within Assumption 1.1 , the resulting statement that a consistent single-valued kime phase law on the directional fiber exists iff the spin is (half-)integer in units of $\hbar$ , the sharpest available classical bridge to spin- $\tfrac{1}{2}$ , and the direct analogue for Problem (III) of the $2\pi$ -holonomy correction anticipated in Problem 3.12 . 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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