Solution (source code)

= Solution

For an attractor with one adiabatic clock and a regular <Bunch-Davies vacuum>, a long <comoving curvature perturbation> $\zeta_L$ acts at leading order as a spatial dilation. The <single-field inflation squeezed-limit consistency relation> therefore fixes
$$
\lim_{k_L/k_S\to0}\frac{B_\zeta(k_L,k_S,k_S)}{P_\zeta(k_L)P_\zeta(k_S)}=-(n_s-1).
$$
For the local convention $\zeta=\zeta_G+(3/5)f_{\rm NL}(\zeta_G^2-\langle\zeta_G^2\rangle)$, comparison with its squeezed <primordial bispectrum> gives
$$
\boxed{f_{\rm NL}^{\rm squeezed}=\frac5{12}(1-n_s)}.
$$
The matter-era negative gravitational potential is $\varphi=3\zeta/5$, so this is also the conventional $f_{\rm NL}$ of part (i). In <single-field slow-roll inflation>, the <spectral tilt> is small and the effective local coupling is of slow-roll size, typically of order $10^{-2}$ for a tilt of a few percent. This is a statement about the normalized coupling: the squeezed <primordial bispectrum> still contains the large long-mode <power spectrum>.

Formally putting this small coupling into the local template of part (ii) predicts only a tiny correction on accessible scales, rather than an order-unity local signal. For a physical halo abundance there is a stronger qualification. The consistency-relation modulation at fixed coordinate wavelength is a dilation; the coordinate size of a fixed physical halo must be dilated too. Those responses cancel. This is the <absence of primordial scale-dependent halo bias in single-clock inflation>: the leading consistency-relation term produces no physical $k^{-2}$ halo bias. The first physical long-mode response contains two spatial derivatives, so dividing it by $\delta_L\propto k_L^2\varphi_L$ does not produce the local-template enhancement. Thus the leading primordial large-scale galaxy <power spectrum> remains the Gaussian-bias result, apart from ordinary transfer, bias, and projection effects.

The cancellation can be seen with a <top-hat filter>. At fixed local time, a constant long curvature rescales physical lengths by $e^{\zeta_L}$, so the coordinate radius of a fixed physical halo is $R_c=e^{-\zeta_L}R_0$, and the coordinate short <density contrast> is $\delta_s(\mathbf x)=\delta_s^{(0)}(e^{\zeta_L}\mathbf x)$. Changing variables $\mathbf y=e^{\zeta_L}\mathbf x$ gives
$$
\frac{3}{4\pi R_c^3}\int_{|\mathbf x|<R_c}d^3x\,\delta_s^{(0)}(e^{\zeta_L}\mathbf x)
=\frac{3}{4\pi R_0^3}\int_{|\mathbf y|<R_0}d^3y\,\delta_s^{(0)}(\mathbf y).
$$
Thus the smoothed field and its <variance> at fixed physical mass are unchanged by the constant long mode.

The assumptions matter: a non-attractor such as <ultra-slow-roll inflation>, additional fluctuating fields, or an excited initial state need not obey this squeezed-limit conclusion. A sizable physical local modulation would test the stated single-clock attractor assumptions, rather than every possible one-field model.