%
\label{def:tau-exterior-derivative}
The \textbf{$\tau$-exterior derivative} is the operator
\[
    \boxed{%
    d_\tau
    \;\colon\;
    \Omega^k_\tau(M)
    \;\longrightarrow\;
    \Omega^{k+1}_\tau(M),}
\]
defined as follows.

\smallskip
\noindent\textbf{On $0$-forms.}
For $f \in \Omega^0_\tau(M) = \mathcal{O}_\tau(M)$,
\[
    d_\tau f
    \;:=\;
    \frac{\partial f}{\partial A}\, dA
    \;+\;
    \frac{\partial f}{\partial B}\, dB
    \;+\;
    \frac{\partial f}{\partial C}\, dC
    \;+\;
    \frac{\partial f}{\partial D}\, dD,
\]
where the partial derivatives
are defined \textbf{stagewise}:
at stage~$k$, the function
$f_k \colon \mathbb{Z}/P_k\mathbb{Z} \to H_\tau$
is defined on a finite set,
and the partial derivative
with respect to a coordinate~$X$
is the discrete difference operator
\[
    \frac{\partial f_k}{\partial X}(x)
    \;:=\;
    f_k(x + e_X) - f_k(x),
\]
where $e_X$ is the unit vector
in the $X$-direction
within the CRT decomposition.
These discrete derivatives
are tower-coherent:
$\pi_{k,k+1} \circ (\partial f_{k+1}/\partial X)
= \partial f_k / \partial X$.

\smallskip
\noindent\textbf{On $k$-forms.}
For $\omega = f_{I}\, dX^{i_1} \wedge \cdots \wedge dX^{i_k}$
(a monomial $k$-form with coefficient $f_I \in \mathcal{O}_\tau$),
\[
    d_\tau \omega
    \;:=\;
    (d_\tau f_I)
    \;\wedge\;
    dX^{i_1} \wedge \cdots \wedge dX^{i_k}.
\]
Extend by $H_\tau$-linearity
to all $k$-forms.