The (2,3)-Block Framework: From 3 = 1+1+1 = 2+1 to a Universal Identity in Q(i, ω)

We develop an arithmetic framework for ternary prime representations p+q+r = 3n based on the two non-trivial partitions of 3. The symmetric partition 3 = 1+1+1 yields the Eisenstein norm E = a²+ab+b² and an additive master identity governed by ζ_{Q(ω)}. The hybrid partition 3 = 2+1 yields a multiplicative identity H·E_G = (m⁴−d⁴)·r bridging Z[i] and Z[ω] through the conductor-2 bridge form 3c²+d². The S₃ symmetry of each triple produces three hybrid decompositions whose bridge forms are S₃-invariant.

For arbitrary k-prime representations, each decomposition k = 2a+3b yields binary and ternary blocks with a universal identity H_total · E = Q · R and Euler product ζ^{a+b} · L(χ_{-4})^a · L(χ_{-3})^b. The product character χ₁₂ = χ_{-4} · χ_{-3} reveals a sieve complementarity across the four prime classes mod 12, with natural home the biquadratic field K = Q(i, ω).

Applications include: (1) a dark density reduction from 1/2 to 1/4 for blocking primes in mixed decompositions; (2) structurally disjoint blocking sets via a shared-gap construction; (3) unconditional logarithmic-average vanishing of the joint Liouville sum (via Tao–Teräväinen); (4) a deterministic Collatz convergence criterion from Z[i]-splitting with threshold α_c ≈ 0.369, below the stationary split fraction 1/2; (5) an S₃-Chen theorem for ternary representations with a (log N)^{0.288} improvement over single-coset methods; (6) a χ₁₂ coset activity trichotomy for the quartic filter on S₃ ellipse orbits.

This paper contains only rigorously proved results. Heuristic directions and open problems are deferred to separate working notes.