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Let $\alpha =\log a$, and define $A=3H+V_{,\varphi }/\Pi$ on a patch with $\Pi \ne 0$. In the shear-negligible long-wavelength approximation in cosmology, the relations just derived givewith ${\partial }_{i}H=-4\pi G\Pi {\partial }_i\varphi$ and ${\partial }_i\Pi =-A{\partial }_i\varphi$. For the nonlinear curvature covector,commuting coordinate derivatives givesThe two terms proportional to ${\partial }_{i}N$ cancel exactly. The remaining expression isSubstituting the spatial relations cancels the gravitational terms and then the $A$ terms separately. ThereforeThis is nonlinear in perturbation amplitude; only the spatial-gradient truncation and the decaying-shear approximation have been used. Conservation does not require a spatially homogeneous lapse. On a uniform-field slice, ${\zeta }_i=-{\partial }_i\log a$, making its interpretation as a conserved local spatial-curvature perturbation transparent, with the sign convention used here.

For ordinary attractor single-field inflation, conservation means that nonlinear curvature correlations already generated near horizon exit are transported to later superhorizon times without additional local growth. It does not prove that the perturbation is Gaussian: horizon-scale interactions, the nonlinear relation to the field perturbation, and quantum initial conditions can generate a nonzero bispectrum. In canonical slow-roll inflation that primordial non-Gaussianity is usually slow-roll suppressed; the single-field inflation squeezed-limit consistency relation describes the effect of a long adiabatic mode on shorter modes under its stated attractor and initial-state assumptions. Strong local superhorizon primordial non-Gaussianity is therefore not generated merely by evolving this conserved variable. Entropy perturbations in multifield models, or non-attractor single-field phases for which the preceding single-clock and shear-neglect assumptions cannot be applied in this form, evade that inference. The result is a conservation theorem under explicit dynamical assumptions, not a blanket assertion that every single-field model has zero or small primordial non-Gaussianity.

Now impose the stipulated negligible shear and work where $\Pi \ne 0$. The Hamiltonian constraint, momentum constraint and local expansion evolution giveTime differentiation of the first equation, followed by $\dot{\varphi }=N\Pi$, gives $\dot{\Pi }=-N(3H\Pi +V_{,\varphi })$. Spatial differentiation of the same equation and use of the momentum constraint givesThese are the temporal and spatial identities needed for the nonlinear curvature covector. Differentiate its definition, keeping the spatially varying lapse function:The terms proportional to ${\partial }_{i}N$ cancel. The terms $-N{\partial }_{i}H$ and $-4\pi GN\Pi {\partial }_i\varphi$ cancel by the momentum constraint, and the remaining two terms cancel by the spatial momentum identity. Thereforeto the retained gradient order. No homogeneous-lapse assumption was needed.

For linear adiabatic cosmological perturbations, expansion of the local scale and scalar field gives ${\zeta }_i={\partial }_i[\Psi +\bar{H}\delta \varphi /\bar{\Pi }]$. For nonzero spatial modes, linearizing the Hamiltonian constraint and the momentum constraint givesThusThis uses the paper's overall sign for the uniform-density curvature perturbation; the opposite overall sign is also common. For an adiabatic single-field attractor, superhorizon conservation of single-field comoving curvature allows the primordial amplitude to be transported through later epochs without following every short-scale process. This proof specifically discarded the shear contribution to the momentum constraint and assumed a usable field clock. It is not a claim that every single-field background conserves every curvature mode: growing curvature perturbation in ultra-slow-roll inflation is a non-attractor counterexample, and $\Pi =0$ makes this particular covector undefined.

For zero shift vector, negative-expansion extrinsic curvature $K=-3H$ and $S^i{}_j={\tilde{K}}^i{}_j$, retain the leading spatial-gradient equations of a minimally coupled scalar. They give $\dot{\varphi }=N\Pi$, $\dot{\Pi }=-N(3H\Pi +V_{,\varphi })$, $\dot{H}=-4\pi GN{\Pi }^2-N{S}^i{}_j{S}^j{}_i/2$, and ${\dot{S}}^i{}_j=-3NH{S}^i{}_j$. The first-gradient momentum constraint is ${\partial }_{i}H=-4\pi G\Pi {\partial }_i\varphi -{\nabla }_j{S}^j{}_i/2$. Shear decays as $a^{-3}$ during expansion. Its neglect after inflationary damping yields the equations underlying conservation of the nonlinear curvature covector.