Poole Defect Calculus: Collected Mathematics of Planar Defects and Outward Frontiers

Poole Defect Calculus studies missing sites in a stable plane of the three-dimensional,synchronous, 26-neighbor B5–7/S5–9 cellular automaton. Its basic observables are open andclosed planar defect footprints. These give exact first-step selectors, a complete classificationof fixed defective monolayers, geometric response laws, and a finite interaction expansion. Weconsolidate these results with proofs, explicit counterexamples to stronger conclusions, and areduction of the fastest outward frontier to a two-dimensional threshold map. The resultingframework connects local geometry, Boolean interaction order, channel-resolved cardinalityreadout, and finite-state convergence computations. We retain the scope of each result: symbolicderivation, recorded exact enumeration, and fresh Wolfram witness replay are distinct evidence.This version restores the historical foundation and shape laws, reconciles conflicting resultidentifiers, and distinguishes exact-real prime sharpening from its rounded channel-filter variant.Original Wolfram programs and independent exact implementations reproduce their finite checks.The accompanying catalogues preserve proof dependencies for further integration of later phases