Universal Generalization of Norm Forms over Real Quadratic Fields: Algebraic Factorization, The Neighbor Theorem, and Cyclotomic Entanglement

For each squarefree d > 1 we attach to Q(√d) the norm-form sequence D_k^(d) = y_k^2 - d = N(y_k + √d), where ε_d^k = x_k + y_k√d for the fundamental unit ε_d; the Pell sequence is d = 2. We develop the theory of this family on three levels. Algebraically, we prove an exact factorization identity D_k^(d) = y_{k-1}y_{k+1} - (d - y_1^2 N(ε_d)^{k-1}), valid for all d and k, which together with the strong-divisibility of the coordinate sequence yields a recursive factorization tower, an intrinsic splitting and a Lucas compositeness test, and an N ± 1 primality-certification scheme generalizing the Pell case. Structurally, we classify when the sequence fails to produce primes: a trichotomy separates unobstructed fields, fields with a single fixed prime divisor (where the Bateman-Horn constant vanishes), and fields covered by a system of small primes (where the heuristic positively fails). We prove that coverings use only split or ramified primes, that the local hit density is bounded by 4/π_p, that the covering residue sets are antipodal, and that the parity family {2, q} is infinite and completely characterized; computation over d < 20000 shows every covering is exact, uses at most three primes, and has modulus at most 12. Analytically, we survey the prime landscape over 2 ≤ d < 200: the regulator R_d sets a budget (correlation 0.998 with 1/R_d) and the local constant C_d a success rate; the case d ≡ 1 (mod 4) requires the maximal order; and a small, stable residual deviation, correlated with the ramification of 2 and with the genus 2-rank, remains and is recorded as an open problem in the 2-adic theory of the unit.