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Turning a lifted tensor contraction into a polynomial in one parameter

Let Y be independent bits on row ports and X independent bits on column ports. Let A(Y) and B(X) multiply the chosen lifted tensor entries on each side. Set l_0(z)=1-z and l_1(z)=z, and define F(T)=sum_{X,Y} A(Y) B(X) product_x l_(Y_x)((W(T)X)_x). Matrix multiplication here means (WX)_x=sum_y W_xy X_y. At T=0 each affine factor is one exactly when its row bit equals the matching column bit, and zero otherwise. Thus F(0) is the lifted contraction. There are M affine factors, giving degree at most M(2L-1), over K[V]/(V^ell).

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Updated 2026-11-08

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