ANP-02: Recursive Edge Compatibility, Adjoint-Footprint Aperture, Canonical Footprint Nodes, and the Parent-Localization Gap

Recursive Edge Compatibility, Adjoint-Footprint Aperture, Canonical Footprint Nodes, and the Parent-Localization Gap

ANP-01 proves weighted output-local source–core ancestry (SCPB-W), obtaining a genuine C2 forward PDE causal edge, but leaves two open problems: geometric parent/source localization (SCPB-G) and recursive reuse (SCPB-R). This round, under the Type-I L_t^∞ L_x^(3,∞) bound, proves a second-moment aperture estimate for the terminal core adjoint footprint, obtaining quantitative core-scale mass recapture over a parabolic time window; proves an exact adjoint-semigroup ancestry and introduces canonical footprint spectral nodes — whose spatial weight and full dyadic state profile remain stable under backward recursion, closing the footprint/representation component of recursive-edge compatibility. Derives an exact shell-by-shell weighted vorticity identity and a monotone square-tail observable. A functional-analysis NO-GO shows: adjoint-weight localization alone is not enough to force localization of the weighted nonlinear source — so weighted footprint recursion is proven, but geometric source-parent recapture remains an open problem.