Boolean-output filtered γ₂ equals the general adversary value

For A=J−F, the defining factorization equations for γ₂(A|Δ) are exactly Σ_{i:x_i≠y_i}⟨u_{x,i}|v_{y,i}⟩=1 for opposite outputs and 0 for equal outputs. The target has zero diagonal, so [F2] gives a finite attained value.

The dual-vector theorem [F4] gives the minimum of precisely these equations and the same maximum row/column squared-norm cost. Its single-family form has cost max{C_0,C_1}. Assigning u_{x,i}=w_{x,i} and v_{y,i}=w_{y,i} for every input gives a factorization of identical cost, while the dual-vector proof constructs the single family by taking the row family on D_0 and the column family on D_1, then applying the Boolean output-label code. Therefore the two minima coincide and equal Adv±(f).

In the equivalent cross-output-only formulation, tensoring by the two-code label yields equal-output zeros while preserving cross-output constraints. The associated label Gram matrix is PSD, and [F5] confirms its entrywise product with each original Gram matrix is PSD; thus this equivalent realization remains a valid finite Gram factorization with unchanged vector norms. The Boolean output alphabet makes the label inner product exactly 1 on required cross-output mismatches and 0 on same-output mismatches, so there is no output-size loss factor.