= Solution
Separate a smooth large-scale <velocity> $U_L$ from a remaining smaller-scale component $w_L$. If the large-scale variation length is much greater than $r$, their longitudinal increment has the form
$$
\Delta v=rA+\Delta w+O(r^2),\qquad A=\partial_LU_L.
$$
For a positive integer $p$, the leading expansion of its structure function is
$$
\langle(\Delta v)^p\rangle
=\sum_{j=0}^p\binom pj r^j\langle A^j(\Delta w)^{p-j}\rangle
+\text{higher Taylor terms}.
$$
Thus even a <velocity> component at scales much larger than $r$ enters through its <gradients> and through mixed moments with the finer field. It is not generally legitimate to discard those mixed moments: large-scale strain can organize and modulate the finer motions. For example, at fourth order the expansion contains
$$
6r^2\langle A^2(\Delta w)^2\rangle+r^4\langle A^4\rangle,
$$
in addition to the small-scale fourth moment and the generally present odd mixed terms. Even if the components were independent and symmetric, the displayed mixed second-moment term would survive. This explicitly exhibits <large-scale contributions to structure functions> beyond second order.
The implication for <K62 refined similarity> is that an observed structure function need not measure a purely scale-local <velocity> fluctuation. Dividing it by $(\epsilon_r r)^{1/3}$ may still leave dependence on outer-scale <gradients>, correlations or finite-range cutoffs. The universal conditional coefficients $\beta_p$ and the identification of a measured exponent with the <lognormal intermittency model> therefore require additional locality and scale-separation tests. \b[Finite-range mixed contributions can imitate or obscure anomalous scaling.] This is an objection to treating the 1962 dimensional argument as an unconditional law for measured increments; it is not, by itself, a proof that an asymptotic refined similarity regime cannot exist.
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