= Solution
For $k\gg aH$, the dominant term is
$$
w_k\simeq\frac{e^{ik/(aH)}}{L^{3/2}a\sqrt{2k}}=\frac{e^{-ik\eta}}{\sqrt{2L^3a^3\omega_{\rm phys}}},\qquad\eta=-\frac1{aH},\quad\omega_{\rm phys}=\frac ka.
$$
This is the positive-frequency harmonic-oscillator normalization with the physical box volume $a^3L^3$. Over a period much shorter than $H^{-1}$, the <scale factor> and physical frequency are effectively constant. Equivalently, the rescaled field $aL^{3/2}w_k$ approaches $e^{-ik\eta}/\sqrt{2k}$ exactly in the short-wavelength limit.
For $k\ll aH$, $w_k\to iL^{-3/2}H/\sqrt{2k^3}$ and $\dot w_k=O(x^2)$, so the growing field mode freezes. A single <Fourier coefficient> has <variance> proportional to $k^{-3}$, which by itself is not scale-independent. The number of modes in a logarithmic shell is proportional to $L^3k^3$. Their product gives
$$
\boxed{\frac{d\langle\delta\phi^2\rangle}{d\log k}=\frac{L^3k^3}{2\pi^2}|w_k|^2\longrightarrow\frac{H^2}{4\pi^2}=\left(\frac H{2\pi}\right)^2}.
$$
Thus equal logarithmic intervals have equal <variance> in exact de Sitter. Slow evolution of $H$ or nonnegligible effective <mass> produces a tilt. Integrating an exactly flat spectrum over arbitrarily many logarithmic intervals still requires infrared and ultraviolet cutoffs; the statement concerns power per interval, not a finite all-scale total <variance>.
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