  For a scheme $X$ defined over the length $2$ $p$-typical Witt vectors
$W_2(k)$ of a characteristic $p$ field, we introduce total $p$-differentials
which interpolate between Frobenius-twisted differentials and Buium's
$p$-differentials. They form a sheaf over the reduction $X_0$, and behave as if
they were the sheaf of differentials of $X$ over a deeper base below $W_2(k)$.
This allows us to construct the analogues of Gauss-Manin connections and
Kodaira-Spencer classes as in the Katz-Oda formalism. We make connections to
Frobenius lifts, Borger-Weiland's biring formalism, and Deligne--Illusie
classes.
