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How do larger tensors keep their addresses?

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Two independent Boolean variables give four assignments in a two-by-two matrix.

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Four give sixteen in a four-by-four, and six give sixty-four in an eight-by-eight.

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We will track the complete tensor factors,

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row and column groups, and assignment count so that each larger cell keeps its meaning.

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For the conjunction W and X, the logical matrix is ket W times bra X.

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Its row components are W and not W; its columns are X and not X.

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A second factor for Y and Z follows the same pattern.
Both complete factors are shown before we tensor them.

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The tensor row index combines W and Y; the column index combines X and Z.

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Each entry conjoins one expression from each factor. The upper-right cell contains W,

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not X, Y, not Z. Read the labelled components to see why each literal belongs there.

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Add the E F logical matrix to a tensor already containing A B and C D.

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Each dimension doubles. The rows now index A,

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C, E, and the columns index B, D, F.

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The displayed eight-by-eight overview shows the
structure; we will enlarge one addressed entry next.

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Choose row components one, zero, one and column components zero, one, zero.

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Those choices specify A true, C false, E true, and B false,

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D true, F false. Interleave them in the original variable order to get A,

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not B, not C, D, E, not F.

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The tensor shown is a conjunction basis LM.

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A general Boolean rule chooses a combination of basis matrices according to its truth values.

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For example, ANDing the three pairwise XOR rules uses the tensor of their numeric XOR CMs.

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Other outer operators require their own combination rule.

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For two n independent input bits split evenly, there are n row bits and n column bits.

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That gives two to the n rows and two to the n columns.

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If the split is unequal, a rectangular matrix is equally
valid; the complete assignment count is unchanged.

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Continue the alternating grouping with eight independent bits, A through H.

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Which variables index the rows? How many rows and columns are needed for an even split?

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Predict the shape before considering any particular Boolean operator.

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The row bits are A, C, E, G; the column bits are B, D, F, H.

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Four bits on each axis require sixteen positions, giving two hundred fifty-six cells.

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Next we will transform a larger matrix while keeping its assignment mapping correct.