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This overview explains how pair-aware compilation represents
one live row variable and one live column variable with four bits.

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Equal token outputs can arise through different
construction routes with different costs and claims.

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Take the two-input and function in true-first order.

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Its four entries are one, zero, zero, zero.

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These four bits describe all two-input behavior only because both the variable identities

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and the input order have been declared.

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For this illustrative encoding, put the first entry in bit zero.

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The resulting integer is one.

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A format that puts the first entry in the high bit would encode the

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same ordered entries differently.

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An integer alone is therefore not enough to interpret the operator; this example does not

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prescribe an internal API format.

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A compact pair representation can be reached through different construction routes.

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Directly recognizing a small supported operator is different from evaluating or reducing a

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larger expression first.

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Equal final bits do not imply that the routes took equal work.

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The current pair API represents one live row variable
and one live column variable with four output bits.

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Structural composition combines recognized small tables using supported rules.

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Retabulation evaluates the four pair assignments when that route is allowed.

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If no eligible pair representation can be built, fallback uses the ordinary evaluator.

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For and, true-first A B order gives four outputs: one, zero, zero, zero.

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Those are the pair result.

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They do not by themselves say how much work the construction route performed.

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Suppose the requested display also includes C, although the rule ignores C.

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For A B one one, both values of C give one.

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For each other A B pair, both values of C give zero.

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Expanding the pair into that display repeats entries; it does not create new dependence.

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The current pair API represents one live row variable
and one live column variable with four output bits.

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It distinguishes pure structural composition,

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a hybrid that may retabulate a subtree, explicit
whole-root retabulation, and ordinary fallback.

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Take A AND NOT B with A on rows and B on columns.

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In true-first order, the pattern is 0100.

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Structural token compilation accounts for B's sign in the common A-B frame.

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Under the pure-structural strategy it must construct
a supported token without a truth-table fallback.

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Under retabulation it explicitly evaluates all 4 cases.

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Those routes can agree on bits while doing different work.

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For the full four-input rule, fix C to zero and D to one.

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The result is NOT of A AND B, pattern 0111 in true-first order.

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The token-only artifact and a lifted dense output remain separate.

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If the pair compiler returns no token, the array API uses its ordinary fallback.

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Work a second complete case with A 1, B 1, C 1, D 0.

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Keep this assignment fixed throughout the calculation.

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Each symbol retains the meaning already introduced; a new value does not

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create a new logical rule.

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The A and B branch gives one.

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The fixed C or D branch also gives one, so the final exclusive-or is zero.

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Under this restriction the pair function is not and, but its complement.

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Its true-true entry must therefore be zero in the small representation.

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This case establishes a specific contrast.

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Fixed values change the pair function.

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The construction context must be preserved.

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Keep the input, intermediate values, and conclusion together when comparing

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it with the previous case.

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The calculated answer belongs to this complete assignment under the stated rule.

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Start from the four-input rule used earlier.

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Fix C and D to zero.

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This is an explicit restriction to a two-input slice, not a claim that the original rule

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always ignores C and D.

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The fixed or branch becomes zero.

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Exclusive-or with zero preserves the other branch, leaving A and B.

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The resulting two-input function can be described by the four and entries

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in the declared order.

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For C one and D zero, the or branch becomes one instead.

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Exclusive-or with one complements A and B.

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The two restrictions produce different pair functions, so the fixed values

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belong to the construction context.

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Unsupported structure must use an eligible fallback rather than being forced into an

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unjustified small-token interpretation.

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Report structural composition, retabulation, and fallback separately.

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Keep the token artifact distinct from a lifted dense output.