Lemma 4. For every product of finite input alphabets, the rule W has finite range, and each output indicator is a finite linear combination of functions of at most m input coordinates. For independent centered variance-one inputs, whose …

Identity gives the required decomposition for the A report. An attained B report fixes the m leaf values and imposes no condition on u . An attained P^c report fixes u , its selected index i , and the m-1 unselected leaves. At least one of these reported leaves fails its test; hence the reported values already force P^c , whatever the omitted value z_i may be. Its indicator therefore depends on only the m reported coordinates. Unused output labels have zero indicators. This proves the first assertion on every input, including interval endpoints. To estimate the retained variance, predict L by b on A , by ∑_jz_j on B , and by u+∑_ j≠ι(u) z_j on P^c . This predictor is a function of the reported information. On B its error is u ; independence gives 𝔼[1_Bu^2]=ℙ(B) . For the last type of report, fix an index i and let E_i=P^c∩ι(u)=i . This event and its reported data depend only on u and the leaves other than z_i . Consequently 𝔼[1_ E_i z_i]=0, 𝔼[1_ E_i z_i^2]=ℙ(E_i). Summing the squared errors over the three types of report gives ℙ(A^c)+𝔼[1_A(L-b)^2]=1-Δ_b. Conditional expectation has no larger mean squared error . Since L is centered and Var(L)=m+1 , the conditional-variance identity gives .