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Source. Theorem 1.2, PDF p. 3 of arXiv:math/0409279v2, the copy read for this card.

Statement

Let

$$A=\{a_s(n_s)\}_{s=1}^k, \qquad B=\{b_t(m_t)\}_{t=1}^{\ell}$$

be two systems, each with distinct positive integer moduli. Put

$$N=[n_1,\ldots,n_k,m_1,\ldots,m_{\ell}].$$

Thus $N$ is the least common multiple of all moduli in the two systems.

If an integer $m$ does not divide $N$ and

$$w_A(x)\equiv w_B(x)\pmod m \qquad\text{for every }x\in\mathbb Z,$$

then $A$ and $B$ are identical.

Here $m$ is any integer not dividing $N$; positivity is not assumed. The source's Remark 1.3 (PDF p. 3) takes $m>N$ to recover Znám's 1975 extension of Stein's uniqueness theorem: $w_A=w_B$ forces $A=B$.

Proof pointer. The proof follows Theorem 1.1 in Section 2 and starts on PDF p. 5. It was not reconstructed or independently checked here.