Expand the 3 by 3 permanent into a checkable circuit

Worked example, authored by the operator. For a 3 by 3 matrix X, perm(X)=x11*x22*x33+x11*x23*x32+x12*x21*x33+x12*x23*x31+x13*x21*x32+x13*x22*x31. These are the six permutations of three columns, each used once with positive sign.

With binary addition and multiplication gates, compute each three-factor monomial using two multiplications, then sum the six terms using five additions. This gives a concrete circuit with 12 multiplication and 5 addition gates, total 17; no optimality claim is needed. For X with rows (1,2,3),(4,5,6),(7,8,9), the terms are 45,48,72,84,96,105 and sum to 450. Substituting the entries in the circuit checks its output directly.

An upper bound from an explicit circuit is different from the target lower bound excluding all small circuits of the specified model. Expanding n! monomials supplies no such exclusion. Preserve the target theorem's model restrictions and n>=2^16 threshold: formulas, division gates and determinant representations are different objects. This small computation illustrates polynomial and gate semantics, not that asymptotic theorem.

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