Set W=(1/2)sum_s(I-Pi_s). On a product-basis vector with j negative choices, W multiplies by j and the full swap V by (-1)^j. Choose an interval I_set of R consecutive integers (R>=1) in {0,...,Q}. Interpolate f(j)=(-1)^j there, with degree at most R-1, and set M=V f(W). Matrix multiplication means composition of the linear maps. Thus f(W) substitutes repeated compositions for powers, with the identity map as W^0. M fixes every basis vector whose j is in the interval. Make C's rows the transposes of the remaining basis vectors. Orthogonality makes the kernel of C exactly the span of the unmeasured directions, which M fixes. Thus only the exceptional directions need measurements; their number is analyzed separately.