bibkey: balister2019erdos authors: “Paul Balister; Béla Bollobás; Robert Morris; Julian Sahasrabudhe; Marius Tiba” year: 2019 title: “The Erdős-Selfridge problem with square-free moduli” doi: 10.48550/arXiv.1901.11465 url: https://arxiv.org/abs/1901.11465 claim: “A covering system with distinct squarefree moduli must contain an even modulus.” strata_touched:

• D5/S3/Arith/Congruence/TwoOddPrimeUncoveredDensity license: citation-only triage: anchor

The squarefree boundary

Locator: https://doi.org/10.48550/arXiv.1901.11465; https://arxiv.org/abs/1901.11465, submitted 31 January 2019. The arXiv metadata, checked 16 September 2026, links the published article at https://doi.org/10.2140/ant.2021.15.609.

Theorem1.2 gives the geometric form: a cover of the product of the first n odd-prime coordinate sets by proper axis-parallel hyperplanes contains two hyperplanes with the same fixed-coordinate support. Footnote2 identifies exclusion of the empty support with exclusion of the modulus-one progression. Restricting larger distinct odd-prime coordinate sets to these sizes preserves nonempty intersections and fixed supports, so the same nonparallel-cover exclusion applies to any finite set of distinct odd primes. The inspected text was rechecked on27 September2026 for this use.

The original-label top-shadow deduction freezes all lower prime-power digits before applying this squarefree theorem at actual private sources. It obtains same-source lower-shadow collision and top-shell pair-capacity constraints. The numerical capacities use uniform probability on the original period; their unrestricted inventory bound and the transport to a separately chosen supported head law remain unproved.

Section 5 of the arXiv source, read 16 September 2026, defines f21 = c21(3) / mu21; Corollary 5.2 gives the 138.877 threshold. Its surviving-mass denominator cannot be discarded. Transferring the argument to an arbitrary small-prime head is a proposed derivable extension, not the literal statement of Corollary 5.2. A newly chosen probability on the complete old survivor set starts with mass budget one but still needs a valid head-load bound. The separate-cylinder candidate H73 is false; its exact obstruction and the unproved joint-load candidate Γ73 are in the canonical dossier. The dossier proves the arbitrary-head transfer and supplies an exact rational verification of the 138877/1000 continuation through the analytic stopping threshold. The weighted finite rectangle second-moment estimate underlying that transfer is formalized in PrimeRectangleTransfer; the complete CRT and joint-load embedding, survivor-mass recurrence, and numerical continuation are not Lean formalized. The unrestricted universal head bound remains unproved.