Polynomial Root Variance as Arithmetic Invariant: Degree Independence, Exact Formulas, and a Primality Characterization

For integers n ≥ 2 and k ≥ 2, we study the variance Vk(n) = ∑c,d ∈ ℤ/nℤ (Nk(c,d;n) − 1)2, where Nk(c,d;n) counts roots of f(x) = xk + cx + d in ℤ/nℤ. Our main result is that Vk(n) is independent of the degree k (a phenomenon whose proof hinges on a purely combinatorial coincidence count), and that for prime p the universal value is V(p) = p(p − 1). We establish the closed-form formula V(n) = ∏pa || n p2a−1(ap + p − a) − n2 for all n, and derive the primality criterion: n is prime if and only if V(n) = n(n − 1). For semiprimes n = pq, the variance determines the factorization explicitly. Over prime fields, we obtain a flat Fourier spectrum with exact phase formula and compute the moments of the cubic root count distribution through order four by a complete combinatorial proof, showing non-Poisson structure. Moments of order five and six are proved by the same stratification method. All results are verified computationally with zero failures across extensive ranges.