Three Research Notes: Parity Base-Space and the Second Division Operator, the Root-Bounded Division Domain, Symmetric Finite Relation Closure

Note A observes that an even N ≥ 4 already carries the divisor 2, and that N/2 is both the result of the compulsory division and the additive centre of any Goldbach pair; Note B rewrites Goldbach in symmetric-offset form (p = m − d, q = m + d): failure ⇔ every offset d is hit on at least one side by a root-bounded proper-divisor witness; Note C gives the positive dual — 2m has a Goldbach pair iff the residual set of the finite relation system (Ω_m, W_m, R_m) is nonempty — and names the candidate representation SRBRD (Symmetric Root-Bounded Relation Domain) with a 10-step development order.