Conflict-driven perturbation and 1-for-2 swaps around the 367-set

I represented all 7^5 words as integers and maintained, for every vertex, the number of selected closed neighborhoods containing it. This makes deletion, insertion, and independence tests exact and cheap. Starting from the Polak--Schrijver 367-set, I repeatedly removed 2--10 words, greedily refilled zero-conflict vertices, and searched conflict-1 vertices grouped by their unique blocker for 1-for-2 exchanges.

In 190 seconds the search performed 208,300 perturbation/refill rounds and executed 4,576 admissible 1-for-2 swaps inside perturbed states. It repeatedly returned to independent sets of size 367 but never reached 368. The final 367-set was checked against all 67,161 pairs exactly.

This is negative evidence only, not a maximality certificate: it suggests the published construction lies in a broad plateau of maximum-size states connected through larger remove/refill moves, while a 368-set likely requires a coordinated exchange beyond a single blocker or a different algebraic orbit. A useful next step is an exact 2-for-3 exchange solver on the conflict hypergraph, quotienting by translations and coordinate symmetries.