We introduce a synchronization-resistance geometry constructed from twisted graph Laplacians equipped with non-Abelian parallel transport data. Given a graph G= (V, E), with connection variables U₈₉ U (2), we define a gauge-covariant synchronization distance through the Moore-Penrose pseudoinverse of the twisted Laplacian. The objective of this program is to determine whether the resulting metric geometry lies outside the classical resistance-metric cone generated by positive conductance networks. A sequence of numerical and analytical audits was performed. Current evidence supports exact static metric reducibility for tested N=3 and N=4 systems. Several candidate obstruction mechanisms were investigated and subsequently falsified. The principal unresolved question concerns the behavior of continuous synchronization trajectories relative to positivity constraints in conductance space. This document records the current mathematical state of the framework
John Strother (Mon,) studied this question.