Reversible Extractors and Residual Kernel Taxonomy

Companion to Pascal Preconditioning and Structured Prime-Pair Incidence Matrices This research establishes a rigorous algebraic framework for reversible extractors within finite-dimensional vector spaces V over a field k. By defining an extracted encoding Γₑ : V → Sₑ ⊕ Kₑ as a linear isomorphism, the paper demonstrates that any object A in V can be reconstructed exactly from its extracted core Cₑ(A) and its residual kernel Rₑ(A). The theoretical foundation is extended to characterize extractor equivalence through block-diagonal transport maps on split coordinates, ensuring the functorial transport of extracted components. In the context of two-sided matrix extractors Φ(A) = LAR, the study identifies the underlying group action as GLₙ(k) × GLₙ(k)ᵒᵖ and proves a finite hybrid closure theorem for families generated by prescribed operator classes. Additionally, the work introduces a comprehensive taxonomy for residual behavior, including residual type families and transport-compatible complexity metrics that induce a preorder on extractor classes. These formalisms are illustrated through a case study of the prime-pair / Pascal-whitening construction, where the Pascal transform serves as a structural component for arithmetic decomposition.