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Can the shape change while answers stay the same?

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Take the exclusive-or of two checks: A AND B, and C OR D.

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This four-input rule has sixteen assignments. We will move
variables between the repository row and column groups,

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track an assignment through square and rectangular
layouts, and check that its Boolean answer is preserved.

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Each row variable doubles the number of row combinations.

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Two row variables give four rows. Two column variables give four columns.

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That is our familiar four-by-four layout, containing sixteen output entries altogether.

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Now put only A in the row group, and put B, C, D in the column group, in that order.

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One row bit gives two rows.

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Three column bits give eight columns. The new two-by-eight rectangle still has sixteen entries.

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Follow one-zero-one-one again. A alone gives row index one.

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The remaining bits, B C D, are zero-one-one.

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Their weights are four, two, and one, so the column index is three. The selected entry remains one.

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For an eight-by-two layout, use A, B, C as row variables and D as the column variable.

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Three row bits give eight rows. One column bit gives two columns.

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Again the product is sixteen, covering the same assignments.

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Our assignment now has row bits one-zero-one, worth four plus zero plus one: index five.

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Column bit one gives index one. The cell still contains one.

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Its address changed because the variable groups changed; its Boolean answer did not.

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Use assignment one-one-one-zero and the two-by-eight layout. What is its new row index?

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What is its new column index? You already know the rule returns zero for this assignment.

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Find where that zero belongs now.

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A gives row one. B C D is one-one-zero, worth four plus two plus zero: column six.

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The entry is zero. The old four-by-four address was row three, column two.

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Both addresses refer to the same complete input assignment.

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To check a repartition, compare answers for matching assignments, not
matching physical positions. Every assignment must retain its output.

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With four variables we can check all sixteen directly. A new
shape does not reduce the number of explicit answers.

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These rectangles belong to the repository row-column layout convention.

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The paper tensor construction we studied was square. The
examples here demonstrate addressing and representation.

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They do not, by themselves, show a speed advantage or a smaller total output.

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You can now read a Boolean rule, understand its matrix entries,

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construct larger paper examples, and track repository assignments through different layouts.

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When a grid looks confusing, begin with its variable groups
and ordering. They are the key to making every cell meaningful.