  We produce a flat $\Lambda$-module of $\Lambda$-adic critical slope
overconvergent modular forms, producing a Hida-type theory that interpolates
such forms over $p$-adically varying integer weights. This provides a
Hida-theoretic explanation for an observation of Coleman that the rank of such
forms is locally constant in the weight. The key to the interpolation is to use
Coleman's presentation of de Rham cohomology in terms of overconvergent forms
to link critical slope overconvergent modular forms with the ordinary part of
1st coherent cohomology of modular curves interpolated by Boxer--Pilloni's
higher Hida theory. We also set up a Galois deformation theory designed to
conform to these critical overconvergent forms and prove "$R = T$" for it. As
applications, we (1) produce a "formal" $\Lambda$-adic interpolation of the
ordinary part of de Rham cohomology, (2) produce a bi-ordinary complex whose
Hecke algebra $T$ is a natural candidate for an "$R = T$" theorem where $R$ is
a deformation ring for 2-dimensional $p$-adic representations of ${\rm
Gal}(\bar{\bf Q}/{\bf Q})$ that become reducible and decomposable upon
restriction to a decomposition group at $p$, (3) produce a degree-shifting
Hecke action on the cohomology of the bi-ordinary complex, and (4) specialize
this degree-shifting action to weight 1 and apply the critical "$R = T$"
theorem to find, under a supplemental assumption, an action of a Stark unit on
the part of weight 1 coherent cohomology over ${\bf Z}_p$ that is isotypic for
an ordinary eigenform with complex multiplication.
