A Wilson–Fermat Resonance Equation for Primes

We introduce a Wilson–Fermat resonance equation built from the product
X(n) = ((n − 1)! + 1)(2^(n−1) − 1), n ≥ 3.
Instead of studying only divisibility, we decompose X(n) relative to the lattice n2Z as
X(n) = Λ(n) n^2 + ε(n),
where Λ(n) ∈ Z and ε(n) is the centered minimal additive correction. This yields an error
function
A(n) = |ε(n)|.
For primes p, one has A(p) = 0. Computational evidence suggests that small values of A(n)
encode a nontrivial resonance landscape for composite numbers. In particular, powers of two
exhibit an exact quasi-resonance:
A(4) = 1, A(2^m) = 1 for all m ≥ 4.
This motivates the viewpoint that primality corresponds to exact Wilson–Fermat resonance,
while certain composite families occupy the first nonzero resonance level.