Strict bias for binary congruence conditions

Establish that, when the Euler totient satisfies \(\phi(M)\geq 3\) and every modulus \(l_i\leq 2\), the sign of \(|A_a(x)|-|A_{a'}(x)|\) stabilizes as \(x\to\infty\) for every pair of distinct admissible vectors \(a\) and \(a'\), thereby proving a strict bias among the sets \(A_a(x)\).

Background

For a fixed integer $M\geq 3$, the paper partitions squarefree integers coprime to $M$ according to congruence conditions $n_i\equiv a_i\pmod{l_i}$, where $n_i$ counts prime divisors in each reduced residue class modulo $M$. Theorem 1 proves that all resulting sets $A_a(x)$ have the same leading asymptotic density.

The unresolved issue is whether the lower-order terms nevertheless impose a persistent ordering between the counting functions. Numerical and analytic evidence in the case $l_i\leq 2$ suggests that the sign of the difference between any two distinct counting functions eventually becomes constant. The paper proves extremal comparisons involving the all-zero and all-one vectors, but does not establish the claimed strict bias for every pair.