= Solution
A light <inflaton> behaves approximately as a scalar field in de Sitter space while a wavelength is well inside the <Hubble radius>. Its <Bunch-Davies vacuum> has <quantum fluctuations>. <Cosmic inflation> stretches each <Fourier mode> until $k=aH$, after which its physical wavelength exceeds the <Hubble radius>. The nearly constant growing field mode has a typical fluctuation per logarithmic wavenumber interval $\delta\phi\sim H/(2\pi)$.
A field fluctuation changes the local position on the rolling background trajectory. Neighboring regions therefore reach the same field value, and the end of inflation, at slightly different times. A clock displacement of magnitude $|\delta\phi/\dot\phi|$ becomes a difference in local expansion of order $H|\delta\phi/\dot\phi|$. Equivalently, in a conventional sign choice the <comoving curvature perturbation> is related to the field fluctuation on spatially flat slices by
$$
\boxed{\mathcal R=\frac{H}{\dot\phi}\,\delta\phi_{\rm flat},\qquad |\mathcal R|\sim\frac{H}{|\dot\phi|}\frac{H}{2\pi}.}
$$
Changing the sign convention for spatial curvature changes the sign of $\mathcal R$, but not its spectrum. This is the <inflaton clock-shift origin of curvature perturbations>. For a single-field slow-roll attractor there is no independent entropy mode. On a <super-Hubble scale>, gradient terms are negligible and the <superhorizon conservation of single-field comoving curvature> preserves the growing <adiabatic mode>, so fluctuations generated near exit persist as primordial curvature perturbations. This conservation requires the attractor and adiabatic assumptions; a freely chosen non-attractor background would not have the same conclusion.
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