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Source. Fornal–Sun, Section 2, equations (3)–(8), p. 5 of arXiv v1; the CRT observation is on p. 2.

Setup. Let $V$ be a finite set of $k\ge2$ vertices, each carrying a positive integer modulus $M_v$ and an integer residue $a_v$. Different vertices may have the same modulus. Assume the residue classes are pairwise disjoint, and put

$$d=\max_{v\ne w}\gcd(M_v,M_w).$$

All graphs below are simple: an edge has two different endpoints. Its color is $\gcd(M_v,M_w)$. For $1\le n\le d$, define

$$L_n=\{v:n\mid M_v\},\qquad K_n=L_n\setminus\bigcup_{2\le s\le d/n}L_{sn},$$ $$h(v)=|\{n\le d:v\in K_n\}|,\qquad w(v)=\frac1{h(v)},\qquad w(v,n)=1_{v\in K_n}w(v).$$

Complete proof of the normalization. The set of divisors of $M_v$ in $[1,d]$ is nonempty because it contains 1. Its largest member $n$ has no proper multiple in that set, so $v\in K_n$. Thus $h(v)\ge1$ and $0<w(v)\le1$. Counting each vertex's memberships gives

$$\sum_{n=1}^{d}\sum_{v\in V}w(v,n) =\sum_{v\in V}h(v)w(v)=k.$$

For any partition $[1,d]\cap\mathbb Z=\bigsqcup_i C_i$, define

$$k_i=\sum_{n\in C_i}\sum_{v\in V}w(v,n).$$

Then $k=\sum_i k_i$. No $K_n$ need be disjoint from the others.

Exact arithmetic input. Two residue classes intersect if and only if $\gcd(M_v,M_w)\mid a_v-a_w$. A complete Bézout proof is already in the canonical CRT compatibility result. In particular $d\ge2$: two classes whose moduli are coprime intersect. This lower bound makes the subsequent $\log d$ estimates meaningful.

Dependencies and use. The construction itself uses finite divisor ordering and double counting. It supplies the weights for the structural lemma and the weighted estimate.

Bears on. Problem 202, through the extremal-family consequence.