Boundary Protocol and Harmonic Remainder Annihilation

This component establishes the referee-safe analytic framework required to transition the Hourglass Rigidity Program from an idealized infinite potential field to the truncated critical strip used in asymptotic analysis. The section formalizes a decomposition of the logarithmic potential into singular contributions arising from zeros and a real-harmonic remainder absorbing Gamma-factor growth, scaling terms, and truncation effects. A boundary protocol is introduced to ensure that the two-scale hourglass residual detects only genuine singularity-induced failures of harmonic mean-value invariance. By defining a submerged interior region and applying a smooth cutoff, all averaging domains are guaranteed to remain fully contained within regions of harmonicity. This prevents spurious contributions from boundary growth or truncation artifacts. The component further specifies axioms for the hourglass averaging kernel, including positivity, normalization, functional-equation symmetry, and moment control. Under these assumptions, an Hourglass Mean-Value Principle is proved, showing that real-harmonic terms are annihilated exactly by the two-scale operator. As a result, boundary effects, Gamma-factor terms, and global background growth contribute identically zero to the rigidity residual. This protocol completes the analytic isolation of singular effects and ensures that any detected instability or “rigidity debt” arises solely from off-critical-line singularities. It provides the technical foundation needed to make the Budget Lemma and Bankruptcy Argument immune to objections based on truncation, boundary leakage, or background analytic noise.