Defect Necessity via Effective Throat Instability under Arithmetic Incompatibility

We establish a variational mechanism showing that defect formation is not merely admissible but energetically necessary under supercritical arithmetic incompatibility. Introducing an effective throat variable that parametrises the local neck geometry, we define an associated effective energy functional combining elastic, mismatch, and holonomy contributions. We prove that, once the arithmetic incompatibility functional exceeds a critical threshold, the smooth orientation-preserving configuration ceases to be a local minimiser of the effective energy. In particular, no energy-decreasing evolution can remain within the smooth class. Furthermore, we show that orientation-reversing (Möbius-type) configurations provide a strictly lower-energy channel, forcing the system into an open-throat regime corresponding to defect formation. This yields a conditional justification of the defect formation principle underlying the working assumption: defect emergence arises as a consequence of the instability of the smooth energy branch under arithmetic forcing, rather than as an independent hypothesis.