A Quadratic Dichotomy Law for the Prime Factors of (a^q+1)/(a+1), with an Eisenstein Cubic Refinement

We prove a quadratic dichotomy law for the prime factors of N_a(q) = (a^q+1)/(a+1) = Φ_2q(a), where a and q are odd primes. If a prime p not dividing 2a(a²−1) divides N_a(q), then ord_p(a) = 2q and, writing p = 2qm+1, the Legendre symbol (p/a) equals +1 unconditionally when a ≡ 3 (mod 4), while for a ≡ 1 (mod 4) it equals +1 exactly when m is even, equivalently when p ≡ 1 (mod 4). The proof uses only Euler's criterion and quadratic reciprocity. The case a = 3 shows that every prime factor p > 3 of F(n) = (3^(2n+1)+1)/4 satisfies p ≡ 1 (mod 3); consequently a cubic refinement via reciprocity in Z[ω] applies to all such factors without restriction, sharpening the Birkhoff–Vandiver congruence from modulo 2q to modulo 6q. A single underlying mechanism (an ℓ-th power level lemma) governs both the quadratic and cubic cases, and explains, via the finiteness of the unit group of Z[ζ_ℓ] for ℓ ∈ {2,3} only, why no elementary congruence analogue is available in higher degree. Motivated by the Sophie Germain pairing of the Mersenne number M(2n+1) = 2^(2n+1)−1 with F(n), we justify the heuristic independence of the two primality events by the linear disjointness of the Kummer extensions Q(ζ_q, 2^(1/q)) and Q(ζ_6q, 3^(1/2q)), and derive an expected count below 2×10⁻² of simultaneously prime pairs with n > 3, consistent with the known instances n = 2, 3. All statements are verified computationally: the dichotomy law on 337 factors across thirteen bases, and the cubic refinement on all 65 common prime divisors of M(2n+1) and F(n) for Sophie Germain primes n ≤ 2×10⁶. This work develops and generalizes the cyclotomic results first presented in DOI 10.5281/zenodo.20624434.