Native Extractor Classes and Comparative Reversible Extraction

This paper studies three finite-dimensional congruence-based extractor regimes for structured matrices: Pascal extraction, geometric/differencing extraction, and hybrid extractors generated from both. It proves exact diagonal collapse for binomial-convolution kernels Bₙ(w) = Pₙ diag(w₀, …, wₙ₋₁) Pₙᵀ, exact identity collapse for cumulative covariance kernels Σₙ(q) = Gₙ(q) Gₙ(q)ᵀ, and exact hybrid collapse for the mixed family Mₙ^mix = Pₙ Σₙ Pₙᵀ. It also gives an obstruction family Ω₂ₘ = diag(J, …, J), with J = [[0, 1], [−1, 0]], showing that diagonal collapse fails for all congruence-type extractors on alternating blocks. The paper’s main contribution is a comparative classification of when Pascal, differencing, and hybrid extractor classes are structurally native to a matrix family.