2.2 Return bridge
If standardized total variance decomposes into the certified systematic term and an aggregate residual contribution bounded by \(\delta \geq 0\), then
\[ \operatorname {Var}(R_q)\leq 1-\mathcal C(q)+\delta . \]
Multiplication by a nonnegative raw scale gives the corresponding raw-variance inequality. The theorem assumes \(\delta \); it does not estimate it or derive residual orthogonality.
Proof
Add the assumed residual budget to the systematic certificate and preserve the inequality under multiplication by the square of the raw scale.