= Solution
Using <functional derivatives> of the interaction functional,
$$
\frac{\delta^2e^{-S_1}}{\delta\widetilde\phi(p)\delta\widetilde\phi(-p)}
=\left[\frac{\delta S_1}{\delta\widetilde\phi(p)}\frac{\delta S_1}{\delta\widetilde\phi(-p)}
-\frac{\delta^2S_1}{\delta\widetilde\phi(p)\delta\widetilde\phi(-p)}\right]e^{-S_1}.
$$
Dividing the flow by $e^{-S_1}$ therefore gives
$$
\boxed{\dot S_1=\frac12\int d^4p\,(2\pi)^4\dot C_\Lambda(p)
\left[\frac{\delta S_1}{\delta\widetilde\phi(p)}\frac{\delta S_1}{\delta\widetilde\phi(-p)}
-\frac{\delta^2S_1}{\delta\widetilde\phi(p)\delta\widetilde\phi(-p)}\right].}
$$
In the first term, remove one leg from each of two interaction vertices and join them with a line weighted by $\dot C_\Lambda(p)$. This produces the tree joining of two vertices. In the second, remove two legs from one vertex and contract them with that line, raising the <loop order> by one, with <tadpole diagrams> as the simplest example. The factor one half accounts for interchanging the contracted ends; the displayed signs are the signs in the interaction-action flow.
\b[The varying cutoff replaces an internal propagator by its cutoff derivative.] Repeated tree joins and loop closures express how eliminated high-momentum fluctuations generate the vertices of the <Wilsonian effective action>. The action is not restricted to <one-particle-irreducible Feynman diagrams>: connected tree joins also occur. Field-independent vacuum contributions can again be absorbed into normalization.
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