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Source. Section 4, printed pp. 1743--1745, physical PDF pp. 5--7.

Statement recorded by the construction

There exists a finite covering system with distinct moduli greater than $1$ such that every integer satisfies at least three of its congruences.

More specifically, the source constructs three covering systems $C_1,C_2,C_3$. Each has distinct moduli greater than $1$, and no modulus used in one $C_i$ is used in another. Their union is therefore a distinct-modulus $3$-covering.

Proof pointer. Section 4 lists the congruences and chooses its auxiliary primes to prevent repeated moduli; the final paragraph takes $p_1=11,p_2=13,p_3=17,p_4=19$. The full list and coverage check were not reconstructed or independently verified here.

Bears on. This proves the lower bound in Question 1.4. It is motivated by Problem 2 but does not alter that problem's status.