Prime Pairs, Quartic Factorizations, and the Arithmetic of Z[i]: From 2=1+1 to a Chen-Type Theorem for Primorials

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 (71 theorems/propositions/lemmas/corollaries across 64 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 for our sifting sequence.

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 by combining the GRH-derived first moment (via the Sathe-Selberg theorem for Ω-distribution among rough numbers) with the unconditional CRT decorrelation bound Var/E² → 0 and the Paley-Zygmund inequality. Under GRH, this simultaneously yields infinitely many Goldbach representations and infinitely many prime pairs with bounded gaps, from a single mechanism.

Additional results include: a quartic parity filter λ(H) = λ(n⁴−d⁴)·λ(E) excluding ~50% of candidates; cross-vertex CRT independence for the second moment method; graph diversity clearing with exponential decay 2^{−L}; an interpretation of the Riemann Hypothesis as Ω-budget balance; Gaussian prime 2× enrichment of Goldbach success rate; twin prime isolation in the graph; Ω-Lipschitz property (mean |ΔΩ| = 0.92 across edges) with Goldbach potential quasi-Lipschitz corollary; primorial optimality among bridge numbers; R⁺ = 8 Goldbach supernodes; expander graph properties (Cheeger constant h ≥ 28.6, spectral gap ratio 42%, independence number α/|V| ≤ 8.5%); topological redundancy (first Betti number β₁ ≈ 13|V|); quartic parity flip rate 48.6% ≈ 1/2 across edges; and a Chen-to-Goldbach propagation theorem via the E-side parity mixing mechanism.

The remaining unconditional gap is identified as a ~30× mismatch in sieve constants between the Chen lower bound and the Selberg semiprime upper bound for rough numbers: a quantitative rather than structural obstacle.

Version 4 changes: Corrected the blocking bound proof (added explicit condition ℓ > 2^{k−1} for injectivity from residue classes to divisors). Made the Chen-type theorem (Theorem B) fully unconditional by adding explicit verification of Rosser-Iwaniec sieve axioms in the proof. Replaced heuristic conditional survival estimate with honest Remark labeling. Fixed multiplicative remainder factorization proof with complete algebra. Strengthened the impossibility theorem proof with three-step argument. Corrected transparent prime fraction from ~47% to ~25% throughout. Upgraded conditional closure proof to cite Sathe-Selberg theorem (Tenenbaum, Ch. II.6) for Ω-distribution among rough numbers, labeled as proof sketch. Added Tenenbaum reference. Fixed all cross-references for downgraded Proposition→Remark.