The Canonical Obstruction to Binary Goldbach

We prove that binary Goldbach's conjecture has a unique canonical obstruction: a medium-arc L4 dispersion certificate. Every sound certification procedure for Goldbach positivity must factor through the same bottleneck; no reformulation of the problem can bypass it. Under the 8-tick realizability model, this obstruction reduces to a finite Schur certification problem—an explicit semidefinite feasibility check on finite data. The proof combines three independently significant components: (i) A no-bypass theorem from the Coercive Projection Theorem: the canonical pipeline Φ* = A∘B∘P is forced by the Recognition Composition Law and locally maximal among all sound certification rules. Its coercivity step is a theorem (cosh(t)−1 ≥ t²/2, formalized in Lean 4); its reconstruction step requires a quantitative identifiability condition that is exactly the medium-arc dispersion. (ii) A circle-method framework with mod-8 kernel K8 yielding unconditional results: density-one positivity, bounded gaps ≪ (log N)⁸ between exceptions, a Chen/Selberg variant (prime + almost-prime), and GRH-conditional closure. (iii) A finite certificate reduction: the dispersion certificate is equivalent, under 8-tick realizability, to a Schur condition on a finite-state Cayley field, checkable by a bounded-real LMI or a Pick-gap certificate.