Hourglass Geometry, Mean Operators, and the Proof Engine

This component establishes the analytic core of the TDD-RH framework by formalizing the hourglass (triangular) probe geometry and its associated mean operators. It introduces the two-scale hourglass residual, defines the coercive rigidity density, and constructs the global hourglass budget that drives the proof mechanism. The section provides the full potential-theoretic engine underlying the program, replacing numerical intuition with rigorous Green-function identities, harmonic-measure averaging, and explicit geometric lower bounds. Through a sequence of proof “bricks,” it demonstrates how off-critical-line zeros necessarily generate persistent directional mean-value defects that cannot be removed by smoothing, cancellation, or boundary effects. Key contributions of this component include: • A precise definition of hourglass (double-cone) probe domains aligned with the anisotropic geometry of the critical strip. • A rigorous Mean–Green identity linking logarithmic singularities to two-scale averaging defects. • Explicit construction of macroscopic footprint corridors whose area grows linearly with strip height. • Uniform Green-function lower bounds establishing geometric amplification (“no zero can hide”). • A referee-safe transfer from model singularities to the full Riemann \xi-function using distributional identities. • Complete cross-term control showing that cancellation between zeros is analytically impossible under hourglass probing. Together, these results form the central proof engine of the 001 Rigidity Program. They reduce the Riemann Hypothesis to a global stability contradiction: off-critical-line zeros force linear rigidity debt, while the analytic structure of the function permits only logarithmic-linear growth. This component provides the geometric, analytic, and measure-theoretic machinery required for that contradiction to be made fully rigorous.