The grouped blocks and their actual symbol

{Defines the grouped distribution in the source. XIV-J.tex; sec:reference, The grouped cross-line distribution; reported lines 1112–1116; the submersion description follows in the same subsection.}

The source uses the macro H_T, whose expansion is $H_T$. The block tensor is

There is no hatted replacement block in this formula. The symbol $H_T$ is retained rather than rewritten as $H^{\prime }$.

The supplied XIII-C text explicitly states

Thus $H_T$ is the time-ordered line in the supplied product convention. It is not, by that equation, an admissible Hadamard replacement with smooth difference. No definition from XII-A is imported to change its meaning.

In independent relative variables, $D_T$ is a difference of two defined tensor products. The grouped pullback is

The source constructs this pullback by its stated submersion operation. That construction does not multiply ${\mathcal{C}}_{\mathrm{cross}}$ by an internal kernel at $N$.

For the finite vertex differential operators, the source applies derivatives in the two independent relative variables before pullback, and denotes the resulting components by ${{\mathcal{C}}_{\mathrm{cross}}}_D$. The patch keeps that order of operations. It does not expand the grouped difference into separately undeclared mixed products of $H$ and ${G_{\mathrm{act}}}^{\mathrm{A}}$.

The smooth-difference branch of XV-F's cross-line lemma remains conditional. More explicitly, any statement requiring

to be smooth requires an additional hypothesis. Such smoothness is not supplied by (eq:HT), and it is not used here. Keeping a conditional lemma is different from asserting that its hypothesis holds for the frozen time-ordered blocks.

This corrects the disputed replacement notation without changing the grouped target. It also prevents the source's time-ordered difference from being silently replaced by a different smooth-difference problem.