ANP-03: Source-Parent Recapture, Kernel-Inflated Footprints, Weighted Parent-State Extraction, and the C3 Causal Upgrade

Source-Parent Recapture, Kernel-Inflated Footprints, Weighted Parent-State Extraction, and the C3 Causal Upgrade

ANP-02 proves footprint/state representation recursion (REC-F) holds, reducing geometric parent localization to a "local parent" or "leaked residual" dichotomy. The open question: can the nonlinear source packet itself produce an earlier parent state — belonging to the same causal branch, recursively reusable, and quantitatively nontrivial? This round proves a weighted version of this proposition — the key idea is not forcing the parent state into exactly the same hard ball. It absorbs Littlewood–Paley spatial nonlocality into an explicit kernel-inflated causal footprint (mass-conserving, center-of-mass-preserving, aperture-controlled), proves a projected source-localization inequality, bounding sub-shell source contributions by the sub-weighted state times a parent product state measured on the inflated footprint. For a dyadic bilinear source atom, strong source contribution plus bounded sub-residence and bounded partner-amplitude action yields a quantitatively nontrivial earlier weighted high-parent state — this class of source atom is thus upgraded from C2+REC-F to a genuine weighted C3 causal parent edge. If the partner-amplitude action is large, the edge instead routes through the existing driver/dissipation-action branch, rather than ancestry failure.