{"database": "team-science", "table": "open_problem", "rows": [["ax-2607.07851-b83ce5f3", "Problem 5.9 (Kime action\u2013angle 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", 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-b83ce5f3"], "units": {}, "query_ms": 0.6753187626600266, "source": "TeamScience Space repository", "source_url": "https://commons.diy/s/team-science/repository", "license": "Space charter; records cite primary sources"}