Proposition: Conditional scalar-rate integration

If the inequality

is available for every interior field on a fixed cube, then (eq:external-soft) follows on that cube for every soft summary.

Proof. For an interior field, put $r_t={\psi }^{-1}(\mathcal{I}(P_{t}g))$. Where $r_t>0$, the chain rule and (eq:F) give

Thus $r_t\le e^{-t}{r}_0$. If information becomes zero, the field is constant by Lemma lem:scalar and remains so.

For boundary-valued inputs apply the result to $(1-\eta )u$, then let $\eta \downarrow 0$ using continuity of finite sums. Correlation zero follows by taking the constant mean. Negative correlations follow from

and invariance of uniform averages under $y\mapsto -y$.