Quantitative Geometric Amplification Under Hourglass/Triangular Probes

This presents a controlled numerical comparison demonstrating that probe geometry plays a decisive role in detecting near–critical-line instability. Under identical analytic parameters—including residual threshold, two-scale radii, and near-line displacement—the hourglass/triangular probe produces a residual footprint approximately 13.2× larger than that obtained using isotropic circular domains. This amplification is shown to arise purely from geometric configuration rather than numerical resolution or parameter tuning. The results confirm that directional confinement, narrowing-channel effects, and vertical alignment with the intrinsic anisotropy of the critical strip transform localized scale inconsistencies into persistent macroscopic regions. These findings provide quantitative support for the Footprint Lemma (Conjecture A) and significantly strengthen the global rigidity argument by demonstrating that off-critical-line singularities cannot hide through cancellation or localization when evaluated under flow-aligned geometry.