= Solution
For a statistically homogeneous <comoving curvature perturbation>, define the connected <primordial bispectrum> by
$$
\boxed{\langle\zeta(\boldsymbol k_1)\zeta(\boldsymbol k_2)\zeta(\boldsymbol k_3)\rangle_c
=(2\pi)^3\delta^{(3)}(\boldsymbol k_1+\boldsymbol k_2+\boldsymbol k_3)B(k_1,k_2,k_3).}
$$
<Statistical isotropy> makes $B$ depend only on the three magnitudes, which must form a triangle. A <Gaussian random field> has zero connected bispectrum. Canonical attractor <single-field slow-roll inflation> with the usual vacuum produces only slow-roll-sized <primordial non-Gaussianity>. A measurable signal can arise from additional light fields and nonlinear conversion of <isocurvature perturbations>, or from enhanced interactions such as a small sound speed, departures from an attractor, or suitable features/excited initial states. The resulting model must still reproduce the observed nearly scale-invariant <power spectrum> and remain under perturbative control, with acceptable backreaction and late-time conversion to the observed perturbations. Large non-Gaussianity is therefore possible but is not automatic in a viable model.
Put $c=3f_{\mathrm{NL}}/5$ and use the consistent <Fourier transform> convention with $\zeta_G(\boldsymbol x)$, rather than the repeated momentum argument printed in the integrand. The quadratic part has transform
$$
\zeta(\boldsymbol k)=\zeta_G(\boldsymbol k)+c\int\frac{d^3p}{(2\pi)^3}
\zeta_G(\boldsymbol p)\zeta_G(\boldsymbol k-\boldsymbol p)
-c\langle\zeta_G^2\rangle(2\pi)^3\delta^{(3)}(\boldsymbol k).
$$
The subtraction sets the mean to zero and removes the internal contraction of a single quadratic factor. At first order in $f_{\mathrm{NL}}$, choose one of the three external factors to be quadratic. For example, the two connected <Wick contractions> at the third leg pair its two fields with the first two legs, giving
$$
2c(2\pi)^3\delta^{(3)}(\boldsymbol k_1+\boldsymbol k_2+\boldsymbol k_3)P(k_1)P(k_2).
$$
Summing the three placements proves
$$
\boxed{B^{\mathrm{loc}}=\frac65f_{\mathrm{NL}}\left[P(k_1)P(k_2)+P(k_2)P(k_3)+P(k_3)P(k_1)\right]+O(f_{\mathrm{NL}}^3).}
$$
This is the leading local <primordial bispectrum>. The factor $6/5$ is the quadratic coefficient $3/5$ multiplied by the two cross-pairings. Terms of order $f_{\mathrm{NL}}^2$ have an odd Gaussian moment and vanish.
The literal quadratic local model also has a connected cubic-in-$f_{\mathrm{NL}}$ contribution, from one quadratic factor at every leg:
$$
B_{\mathrm{loop}}=8c^3\int\frac{d^3p}{(2\pi)^3}
P(p)P(|\boldsymbol k_1-\boldsymbol p|)P(|\boldsymbol k_2+\boldsymbol p|).
$$
This is the <loop correction to the local primordial bispectrum>; regulators may be needed for idealized spectra. Thus the PDF's displayed formula is the tree/leading-order result, not an exact identity for arbitrary $f_{\mathrm{NL}}$. The weak-non-Gaussian expansion assumes these loop terms are small. The local shape is enhanced in the <squeezed bispectrum configuration> when a long-wavelength perturbation modulates small-scale power.
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