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The sparsity of the product of a Laver table and a Hadamard matrix only follows from the periodicity of the Laver tables since the non-distributive but still periodic fake Laver tables (see this question for a more general construction) still exhibit this phenomenon. I have used computer calculations to test this phenomenon with the fake Laver tables. My computer calculations indicate that the product with a fake Laver table matrix and a Hadamard matrix is just as sparse as the product of a real classical Laver table and a Hadamard matrix.