Sharp Oscillation and Component Bounds for Coupled Geometric Phase Projections

Two planar triangles sharing a variable side generate coupled phase projections whose squared product may have several peaks and disconnected sublevel sets. We determine sharp global complexity bounds on a selected positive Gram branch. The four geometric parameters are positive and belong to the relative closure of the family with distinct positive turning points inside the common strictly feasible domain; both phases are arbitrary. On every compact strictly Gram-feasible interval, the product has at most four strict interior local maxima, and every closed sublevel set has at most five connected components, including isolated points. Both bounds are attained by an explicit rational nine-point witness on a shared positive turning-point face and, by parameter perturbation, with distinct positive turning points. The proof combines central phase envelopes, exterior geometric-parameter curves, phase-type compatibility, and uniform approximation on a fixed interval. Exact polynomial coefficient certificates and rational square-root enclosures make the technical sign inequalities and attaining example reproducible. These sharp bounds quantify the complexity of closed sublevel sets for the specified geometric family.