The biquadratic norm framework I algebraic foundations and sieve analysis

We study the field norm of the biquadratic cyclotomic field K = Q(ζ₁₂) as a tool for analyzing prime distribution. The norm N_{K/Q}(α) for elements α ∈ O_K = Z[i,ω] is expressed via an explicit master form, a triple factorization through three quadratic subfields, and the Eisenstein invariant system (R, Q, b). We prove the exponent parity theorem (all primes ℓ ≢ 1 mod 12 appear with even exponents in the norm) by three independent methods, compute exact sieve blocking counts, and establish that the effective sieve dimension is κ = 1 — the linear sieve regime. We further compute the unit group, regulator, lattice theta series (a weight-2 modular form for Γ₀(12)), Dedekind zeta factorization with explicit L-values, and the Bombieri–Vinogradov type level of distribution for the ideal counting function. This is Part I of a series; Part II develops the transfer principle toward connecting the κ = 1 norm sieve to Goldbach-type questions.