= Solution
Let $P(k,\tau)=\langle|\delta_{C,\mathbf k}|^2\rangle$ be the <matter power spectrum> in the normalization of the question. The <Harrison-Peebles-Zeldovich spectrum> gives $P\sim\mathcal C\tau^4k$ before entry. Using the order-of-magnitude <cosmological horizon crossing> criterion $k\tau_H\sim1$, rather than tracking a wavelength factor of $2\pi$, gives
$$
P(k,\tau_H)\sim\mathcal C k^{-3}.
$$
The mass-fluctuation <variance> on radius $k^{-1}$ is therefore
$$
\boxed{\left\langle\left(\frac{\delta M}{M}\right)_k^2\right\rangle_H
\sim k^3P(k,\tau_H)\sim\mathcal C,}
$$
independent of <wavenumber>. Equal logarithmic ranges of scales enter with comparable dimensionless fluctuation strength. This is the <scale invariance> encoded by the <HPZ spectrum>; it is not the claim that the dimensional <matter power spectrum> is independent of $k$ at a fixed time.
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