This paper studies symmetric prime-pair search around two structured families of arithmeticcenters: dyadic centers of the form 2n−1 and factorial centers of the form m!. In both settings,candidate pairs are generated by the symmetric template(C − D, C + D),where D is an integer displacement.The contribution is intentionally limited and explicit. First, local admissibility conditionson the displacement variable are isolated in elementary deterministic form. On the nontrivialdyadic branch, if both endpoints exceed 3, then necessarilyD ≡ 3 (mod 6).On the factorial branch, if both m! − D and m! + D are prime and exceed m, then necessarilygcd(D,m!) = 1.Thus the dyadic family yields a universal local congruence filter, while the factorial family yieldsan exact local coprimality lter.Second, finite computations are organized in a unified displacement-centered framework.These experiments show that, after local admissibility filtering, very large symmetric prime pairscan often be generated with remarkably small search effort. On the dyadic side in particular,first successful nontrivial displacements are frequently tiny relative to the center, so that primepairs of enormous size may arise after testing only a small number of admissible candidates.No proof of Goldbach, no asymptotic density statement, and no general existence theorem isclaimed. The paper instead proposes a displacement-centered architecture for symmetric primesearch around structured centers, establishes explicit local obstruction laws, and records nitecomputational evidence indicating that the arithmetic behavior of the displacement variabledeserves independent study.
Ricardo Adonis Caraccioli Abrego (Thu,) studied this question.