Assume valid replacement forms and a separation design are supplied at every point t=1,...,Q, together with a valid degree bound d_H. Compute J=1+floor(d_H/Q). At each point, combine fixed auxiliary substitutions and restricted measurement sums to obtain J local coefficients. Interpolate the unique polynomial of degree at most d_H consistent with these data, then evaluate it at zero. Because the replacements restore the original inputs at zero, this equals the original count. This closes the recovery argument under its assumptions. It does not yet make the procedure fast: the number of subsets, measurement tuples, local assignments, and the bit cost of rational arithmetic all require bounds.