Quantitative Comparison Results and Interpretation

This component presents the numerical validation of the Hourglass (Triangular) Geometry extension within the 001 Rigidity Program. Using identical analytic parameters and varying only the averaging-domain geometry, the study compares traditional isotropic circular probes against anisotropic hourglass and triangular domains. The results demonstrate a substantial geometric amplification effect. For near-critical displacements, hourglass probes produce footprint areas more than an order of magnitude larger than those generated by circular averaging domains. This amplification arises from three structural mechanisms: directional alignment with the critical strip’s vertical flow, confinement enforced by narrowing triangular boundaries, and enhanced detection of asymmetric potential-field propagation. These findings provide quantitative support for the Hourglass Extension by showing that off-critical-line singularities cannot be dissipated through isotropic smoothing. Instead, geometric confinement forces scale inconsistency through a narrowing channel, magnifying its contribution to the rigidity budget. The numerical evidence confirms that hourglass and triangular domains constitute strictly stronger detectors of off-line instability than circular probes. This strengthens the solvability framework of the 001 program by accelerating the accumulation of rigidity debt and reinforcing the interpretation of the Riemann Hypothesis as a global confinement condition governed by geometry rather than zero enumeration.