This paper develops a framework for studying prime pairs through the quartic factorization n⁴ − d⁴ = (n² − d²)(n² + d²), which realizes the Dedekind zeta function identity ζ_{Q(i)}(s) = ζ(s) · L(s, χ₋₄) at the level of individual Goldbach candidates. Starting from the symmetry 2 = 1+1, we unify the Goldbach conjecture and the twin prime conjecture as two projections of a single problem in Z[i] arithmetic. We introduce the bridge number m(p,q) = (q−p)/2 · (q+p)/2 and the norm association graph G whose edges arise from shared Gaussian integer norms n²+d². We prove G is connected on {7, ..., N} for all N ≥ 7, establish a quartic root dichotomy classifying prime obstructions via χ₋₄, and identify seven explicit correspondences between the combinatorial sieve structure and the Euler product of ζ_{Q(i)}. Main results (76 theorems/propositions/lemmas/corollaries across 67 pages): Theorem A (Euler Product Realization): The quartic identity n⁴−d⁴ = H·E, the Ω-additivity, blocking bounds, and Legendre symbol refinements are the arithmetic realization of ζ_{Q(i)} = ζ · L(χ₋₄), through seven correspondences. Theorem B (Chen-Type Theorem, unconditional): For all sufficiently large k, the primorial P_k = p₁···p_k has a coprime factorization P_k = xy with y−x prime and y+x a product of at most two primes. Proved using a bilinear sum bound exploiting the multiplicative structure of sieve remainders specific to divisor sequences, with full verification of the Rosser-Iwaniec sieve axioms. Theorem C (Computational Prime-Pair Theorem, unconditional): For every k with 2 ≤ k ≤ 15 (primorials up to P₁₅ ≈ 6.15 × 10¹⁷), exhaustive computation verifies that P_k has at least one coprime factorization producing a prime pair, with exponential growth N₂(k) ≈ 0.10 · 2^{k−1}. Theorem D (Conditional Prime-Pair Theorem, under GRH): For all k ≥ 2, N₂(k) > 0. Proved via Sathe-Selberg theorem for Ω-distribution among rough numbers, unconditional CRT decorrelation, and Paley-Zygmund inequality. Parity Separation Theorem (new in v5, unconditional): The Liouville function λ(H) = (−1)^{Ω(H)} perfectly separates Goldbach pairs (λ = +1) from Chen-not-Goldbach pairs (λ = −1). This is an exact algebraic fact following from Ω(v) + Ω(w) = 1+1 = 2 (even) versus 1+2 = 3 (odd). Chen Identity (new in v5, unconditional): N₂ = (N_Chen + L_Chen)/2, where L_Chen = Σλ(H) over Chen candidates. This transforms the remaining gap from a 30× sieve constant improvement (the subtraction N₂ = N_Chen − N⁻_Chen) into a sign problem for a restricted Liouville sum: does L_Chen > −N_Chen? Computationally, L_Chen/N_Chen ∈ [+0.20, +1.00] for k = 5,...,12 (always positive). Under GRH, |L_Chen| = o(N_Chen), closing the gap. Disjoint Blocking for Inert Primes (new in v5, unconditional): For primes ℓ ≡ 3 (mod 4), the v-blocking set {x : ℓ | v(x)} and the w-blocking set {x : ℓ | w(x)} are disjoint, and exactly one is non-empty. This holds for approximately 50% of all sieve primes. Proof: simultaneous membership would force ℓ | 2P_k, contradicting gcd(ℓ, P_k) = 1 and ℓ > 2. This means the resources needed for v-primality and w-primality are non-overlapping at inert primes. Equivalent Formulations (new in v5): N₂ > 0 ⟺ L_Chen > −N_Chen ⟺ not all Chen w are semiprime ⟺ Σλ(w) 0. Added restricted Liouville sum as new open question (Question 11.23). Updated G1 description from "improve sieve constants by 30×" to "prove L_Chen > −N_Chen". Updated Introduction §1.2 to reflect the transformed problem. Fixed remaining "approximately half transparent" to "approximately a quarter" in §11.5.