Triangular Geometry as the Canonical Hourglass Motivation: Why Geometry Matters

This component establishes triangular (hourglass) geometry as the natural analytic domain for the TDD–RH rigidity framework. While classical circular probes test local harmonicity, they are isotropic and fail to reflect the intrinsic directional structure of the critical strip, where symmetry acts horizontally and analytic growth propagates vertically. As a result, circular averaging allows off-critical-line defects to dissipate through lateral cancellation. Triangular and hourglass domains resolve this limitation by aligning the averaging geometry with the strip’s anisotropic flow. Anchored at the critical line apex, these narrowing domains enforce confinement: any horizontal displacement from the neutral manifold generates asymmetric flow that is amplified rather than smoothed. This geometric structure converts local scale inconsistency into a persistent footprint, enabling the 001 rigidity scheme to function as a global exclusion mechanism. Within this geometry, the critical line emerges as the only stable manifold capable of hosting singularities without destabilizing the analytic field.