= Solution
For a statistically homogeneous isotropic dimensionless field, write its two-point <autocorrelation function of a random field> as $C(r)$ and
$$
C(r)=\int\frac{d^3k}{(2\pi)^3}P(k)e^{i\mathbf k\cdot\mathbf r}.
$$
The formal real-space <scale invariance> condition is $C(\lambda\mathbf r)=C(\mathbf r)$ for all $\lambda>0$. Changing variable $\mathbf q=\lambda\mathbf k$ in the first expression and equating the nonzero-mode <Fourier transforms> gives $\lambda^{-3}P(q/\lambda)=P(q)$, equivalently $P(\lambda k)=\lambda^{-3}P(k)$. Taking $\lambda=k/k_0$ yields
$$
\boxed{P(k)=A/k^3,\qquad \Delta^2(k)=k^3P(k)/(2\pi^2)=A/(2\pi^2).}
$$
Thus a <scale-invariant inflationary power spectrum> has equal <variance> per logarithmic <wavenumber> interval, rather than equal power per Fourier volume.
There is an important mathematical qualification to the literal <covariance> condition. A nonzero $k^{-3}$ <power spectrum> over all scales has $C(0)=A\int dk/k/(2\pi^2)$, divergent at both endpoints; it is not the <covariance> of a finite-variance ordinary field. Moreover, exact dilation invariance of a continuous isotropic $C(r)$ makes it constant for $r>0$ and, by continuity, at zero: its spectrum can then consist only of a zero-mode delta measure. The nontrivial cosmological statement consequently concerns nonzero modes, with cutoffs or subtraction of an unobservable constant. For example the finite subtracted <covariance>
$$
C(r)-C(r_0)=\frac A{2\pi^2}\int_0^\infty\frac{dk}{k}\left[\frac{\sin kr}{kr}-\frac{\sin kr_0}{kr_0}\right]= -\frac A{2\pi^2}\log(r/r_0)
$$
is invariant under simultaneous rescaling of $r,r_0$. This is the precise <infrared qualification of a scale-invariant covariance>; a regulated unsubtracted $C(r)$ generally shifts by an additive constant under dilation. The formal derivation above gives the intended nonzero-mode scaling, with this qualification rather than an impossible finite-variance premise.
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