Trident Reconstruction Operator (TRO) v1- symmetry-closure for missing-variable recovery, coherence gating, and external-blockage localization (TRO×RID)

This work introduces the Trident Reconstruction Operator (TRO v1), a symmetry-closure method for reconstructing missing or unobserved variables in six-branch, boundary-constrained systems exhibiting approximate $C_6$ rotational symmetry. The approach exploits the over-determined nature of such systems by identifying a minimal independent basis—three non-collinear branches separated by $120^\circ$ (the trident basis)—from which the full state can be generated through symmetry-consistent interpolation. A pre-reconstruction coherence test (the Trident Gate) is defined to assess whether boundary conditions and symmetry assumptions remain valid prior to closure. Post-reconstruction consistency is evaluated via a ring-closure error that quantifies deviation from local symmetry constraints. When combined with a ratio inversion diagnostic (RID), the framework additionally enables localization of external perturbations or misplaced constraints that break otherwise expected symmetry relationships. The operator is domain-agnostic and applies to scalar, vector, or feature-based branch variables, with interpolator choice determined by whether multiplicative (ratio-dominant) or additive structure governs the system. TRO v1 is intended as a lightweight diagnostic and reconstruction layer for structured datasets, physical systems, and conceptual or analytical frameworks where symmetry and boundary constraints dominate system behaviour. Keywords Symmetry closure; rotational invariance; missing variable reconstruction; boundary-constrained systems; trident basis; hexagonal symmetry; ratio-first interpolation; diagnostic geometry; external constraint detection; structural coherence