Instantiation, Aggregation, and Observer Variance Regimes: A Diagnostic Framework for Resolution-Dependent Geometry

This work introduces and empirically validates a Variance Triangle diagnostic framework that distinguishes three fundamentally different regimes of variance organization in complex systems: Instantiation Variance Triangles (IVT) — built from local, transition-based observables, Aggregated Variance Triangles (AVT) — built from cumulative, global observables, Observer-Resultant Variance (ORV) — emerging at the interface between local instantiations and global aggregation. Through systematic stress testing across number theory (prime distributions), constraint satisfaction problems (XOR-SAT), and synthetic controls, we show that IVT and AVT are not two views of the same geometry, but represent distinct mathematical regimes with different resolution behavior. Key findings include: IVT ratios amplify under refinement, indicating scale-dependent, emergent structure (AD = 3 behavior). AVT ratios stabilize under refinement, exhibiting resolution invariance and admitting a duality structure organized by the involutive map D(x) = e/x. The dual correspondence does not appear as fixed-point clustering at \sqrt{e}, but rather as geometric-mean invariance between stable dual pairs. Synthetic controls fail to reproduce these effects, confirming that the observed structures are not artifacts of normalization or construction The results clarify why local correlation observables and global cumulative observables behave differently under scaling, explain prior persistence-gradient anomalies, and provide a unifying diagnostic language applicable across discrete mathematics, computational complexity, and physical data analysis. The framework is explicitly falsifiable and avoids claims of universality beyond tested domains.