= Solution
Choose $\phi(x)=\int d^4p\,(2\pi)^{-4}e^{ip\cdot x}\widetilde\phi(p)$ in Euclidean signature. The inverse quadratic kernel is the <smooth-cutoff scalar propagator>
$$
\boxed{\langle\widetilde\phi(p)\widetilde\phi(q)\rangle_0
=(2\pi)^4\delta^{(4)}(p+q)C_\Lambda(p),\qquad
C_\Lambda(p)=\frac1{f_\Lambda((p^2+m^2)/\Lambda^2)}.}
$$
The subscript records the explicit cutoff dependence implied by the condition $f_\Lambda(z)=\Lambda^2z$ at small $z$. For $p^2+m^2\le\Lambda^2$, this reduces to $1/(p^2+m^2)$, the usual Euclidean <scalar propagator>. For $p^2+m^2\gg\Lambda^2$, the kernel $f_\Lambda$ grows rapidly and its inverse tends to zero.
\b[High-momentum modes are strongly suppressed]. For a finite smooth regulator they are not literally identically zero; that statement would require a sharp cutoff. The field variance carried by those <Fourier transform> modes is correspondingly negligible.
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