= Solution
The renormalizable action for <chiral superfields> $\Phi_i$ and <vector superfields> $V$ has the <superspace integration> form
$$
S=\int d^4x\,d^4\theta\,K(\Phi^\dagger e^{2V},\Phi)
+\left[\int d^4x\,d^2\theta\left(W(\Phi)+\frac14 f_{ab}\mathcal W^{a\alpha}\mathcal W^b_\alpha\right)+\mathrm{h.c.}\right],
$$
where $K$ is a real <Kähler potential>, $\mathcal W_\alpha$ is the <chiral field-strength superfield>, and $W$ is a gauge-invariant <holomorphic function>. For the renormalizable theory choose canonical positive kinetic terms, a polynomial $W$ of degree at most three, and field-independent gauge kinetic coefficients $f_{ab}$. We prove the <non-renormalization theorem> for the local <Wilsonian effective action>, with a nonzero infrared cutoff and a regulator preserving <supersymmetry>. The elementary <chiral superfields> are retained; this is not a claim that eliminating whole massive fields leaves the same polynomial.
Write $W=\sum_A\lambda_A\mathcal O_A(\Phi)$, treating masses and interaction coefficients alike as background chiral <spurions>. A local <F-term> is chiral even when these backgrounds vary, so its coefficient depends holomorphically on $\Phi,\lambda_A$ and the holomorphic gauge couplings. Dependence on $\bar\lambda_A$ would not be chiral. Such dependence can enter <D-terms> or terms with supercovariant derivatives instead. With the infrared cutoff kept fixed, the perturbative local coefficients are regular power series around vanishing masses and interactions. This removes the inverse-mass and infrared singular terms that would spoil the following <spurion selection rule for perturbative non-renormalization>.
Assign every elementary $\Phi_i$ <R-charge> zero, every $\lambda_A$ <R-charge> two, and $\theta$ <R-charge> one. The <superpotential> must have <R-charge> two because $d^2\theta$ has <R-charge> minus two. Holomorphy excludes the conjugate <spurions>, so regularity and this formal <R-symmetry> require exactly one factor of a superpotential <spurion> in any perturbative candidate. Thus $W_{\mathrm{eff}}$ is linear in the $\lambda_A$, with the ordinary flavor <spurion> charges fixing the permitted field monomials. An anomalous formal <R-symmetry> is implemented by also transforming the holomorphic gauge coupling; its anomaly must not simply be ignored.
Gauge interactions cannot provide a perturbative coefficient modifying this conclusion. With $\tau=\vartheta/(2\pi)+4\pi i/g^2$, perturbative diagrams are independent of the topological angle $\vartheta$. A holomorphic coefficient invariant under continuous real shifts of $\tau$ is constant, by the <Cauchy-Riemann equations>. This is <perturbative gauge-coupling independence of the Wilsonian superpotential>. Contributions involving $e^{2\pi i\tau}$ are nonperturbative and are outside this argument. For an Abelian factor the same angle-independent holomorphy argument applies. Hence a putative perturbative coefficient linear in $\lambda_A$ is independent of the gauge coupling and can be evaluated as that coupling tends to zero.
In this free-gauge limit, an interacting chiral loop needs additional superpotential vertices: the free chiral propagator connects $\Phi$ to $\Phi^\dagger$, so a single holomorphic vertex alone cannot produce a loop correcting its local <F-term>. Extra vertices introduce extra <spurions> or their conjugates, both excluded by the preceding charge and holomorphy argument. The surviving coefficient is its tree value. Therefore
$$
\boxed{W_{\mathrm{Wilsonian}}^{\mathrm{pert}}(\Phi)=W_{\mathrm{tree}}(\Phi)}.
$$
For an explicit charge check, take $W=m\Phi^2/2+g\Phi^3/3$ in the <Wess–Zumino model>. Give $(\Phi,m,g)$ ordinary charges $(1,-2,-3)$ and <R-charges> $(1,0,-1)$. A holomorphic monomial $m^a g^b\Phi^n$ of the required charges satisfies $n-2a-3b=0$ and $n-b=2$, giving $a=1-b$, $n=b+2$. Perturbative regularity requires $a,b\ge0$, hence only $(a,b,n)=(1,0,2)$ and $(0,1,3)$ survive. The free limit fixes their coefficients to $1/2$ and $1/3$; there is no room for a coupling-dependent loop correction. This illustrates the <holomorphy argument for superpotential non-renormalization> explicitly.
The <Kähler potential> can acquire <wave-function renormalization>. Restoring <canonical field normalization> changes physical masses and <Yukawa couplings>, so their running does not contradict the boxed statement about holomorphic coordinates. Gauge kinetic <F-terms> have their own renormalization and are not the <superpotential>. Infrared effects in the one-particle-irreducible <quantum effective action>, tree-level elimination of retained fields, and genuine nonperturbative effects must likewise be distinguished from this perturbative <Wilsonian effective action> theorem.
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