Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
Dated aleph-one Dowker status
The dated mathematical question is whether ZFC proves the existence of a normal Hausdorff space of cardinality that is not countably paracompact, equivalently a Dowker space of that size.
In Multiple gaps and some finitizations of club and CH, arXiv:2504.15398v1, whose header and submission record are dated 21 April 2025, Cruz Chapital reports this question as open at the start of §9, printed p.26 (the 26th PDF page of the 28-page file). That is a claim about the literature at the source's date. The 2025 paper On Δ-spaces by Leiderman and Szeptycki also reports the ZFC size- problem open on printed p.3.
The scaffold's search record of 9 September 2026 found the 2025 report and the 2024 conditional constructions without verifying a resolution. A follow-up search and source check on 10 September 2026 again located these reports and verified the Cruz Chapital version date and §9 passage. The searches did not establish an authoritative resolution of the general ZFC size- question. This is a bounded literature-search report, not certification that every later publication has been examined.
The 2024 Rinot–Shalev–Todorcevic paper gives additional sufficient hypotheses for small Dowker spaces. Such conditional constructions do not by themselves settle the question in ZFC alone. Neither a dated statement that a problem is open nor failure to find a later solution proves nonexistence, independence, or the impossibility of a ZFC construction. This remark is historical orientation and supplies no mathematical prerequisite for the construction proofs.
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Sources
- Jorge Antonio Cruz Chapital, Multiple gaps and some finitizations of club and CH, arXiv:2504.15398v1, 21 April 2025, §9, printed p. 26 (standard reference, not scraped)
- Cruz Chapital, arXiv submission and version history (standard reference, not scraped)
- Leiderman–Szeptycki, On Δ-spaces (2025), Introduction, printed p. 3, DOI 10.1007/s11856-025-2733-2 (standard reference, not scraped)
- Rinot–Shalev–Todorcevic, A new small Dowker space, Periodica Mathematica Hungarica 88 (2024), 102–117; arXiv version, abstract and Introduction (standard reference, not scraped)