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 Mₙᵐix = 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.
David Betzer (Sat,) studied this question.