  Mixed-parity module emerges for instance when a de Rham Galois representation
is being tensored with a square root of cyclotomic character, which produces
half odd integers as the corresponding Hodge-Tate weights. We build the whole
foundation on the $p$-adic Hodge theory in this setting over small $v$-stacks
after Scholze and we also consider certain moduli $v$-stack which parametrizes
families of mixed-parity Hodge modules. Examples of the small $v$-stacks in our
mind are rigid analytic spaces over $p$-adic fields and moduli $v$-stack of
vector bundles over Fargues-Fontaine curves. The preparation implemented at
this level will be expected to provide further essential foundationalization
for generalized Langlands program after Langlands, Drinfeld, Fargues-Scholze.
One side of the generalized Langlands correspondence in the geometric setting
is the perverse motivic derived $\infty$-category over $\mathrm{Moduli}_G$
related to smooth representations of reductive groups, while the other side of
the generalized Langlands correspondence in the geometric setting is the
corresponding derived $\infty$-category over the stack of mixed-parity
$L$-parametrizations (i.e. from two-fold covering of the Weil group into
$\ell$-adic groups) related to the representations of Weil group in our setting
into Langlands dual groups. Although after Scholze and Fargues-Scholze our
generalized Langlands program can go along $\ell$-adic cohomologicalization to
immediately achieve various solid derived $\infty$-categories
$\mathrm{DerCat}_\text{\'et}(\mathrm{Moduli}_G,\square)$,
$\mathrm{DerCat}_\mathrm{lisse, \blacksquare}(\mathrm{Moduli}_G,\square)$,
$\mathrm{DerCat}_{\blacksquare}(\mathrm{Moduli}_G,\square)$ and so on with
well-established formalism regarding 6-functors, we already provide certain
$p$-adic cohomologicalization of the story over $\mathrm{Moduli}_G$.
