Two Coefficient Regimes: Where Well-Factorable Machinery Attaches in the Legendre and Goldbach Programs

The recent unconditional level-of-distribution results for the primes (Bombieri–Friedlander–Iwaniec, Maynard, Lichtman, Pascadi) act on well-factorable weights attached to composite moduli. In both prime-detection problems considered here — primes in J_n = [4n²-n, 4n²+n], toward Legendre's conjecture, and the Goldbach problem for primorials — the relevant moduli are composite and well-factorable: the family Q_sc of smooth divisors of integers near 4n² on one side, smooth residue moduli on the other. The two problems therefore do not split at the modulus. They split at the coefficient carried by the binding bilinear form. The primorial hypothesis H(k) carries factorable weights, so the machinery attaches to the binding object itself. The J_n obstacle, Hypothesis R, carries the arbitrary dual coefficient γ_q manufactured by the second Cauchy–Schwarz that terminates the dispersion chain — the sign pattern U(q)-bar / |U(q)|, which no factorable weight can reproduce — placing it in the Bettin–Chandee trilinear class. Well-factorable methods thus reach the Type I distribution input of J_n but not its binding bilinear obstacle, which is further bound by an independent conductor loss |q| = (2n)². The factor-ray geometry provides coordinates for the well-factorable modulus structure on both sides; only on the primorial side does that structure persist into the binding form. This is a technical note; it states a boundary and contains no new theorems.