  Opinion is divided about the nature of state dependence in the black hole
interior. Some argue that it is a necessary feature, while others argue it is a
bug. In this paper, we consider the extended half-sided modular translation
$U(s_0)$ (with $s_0 > 0$) of Leutheusser and Liu that takes us inside the
horizon. We note that we can use this operator to construct a modular
Hamiltonian $H$ and a conjugation $J$ on the infalling time-evolved wedges. The
original thermofield double translates to a new cyclic and separating vector in
the shifted algebra. We use these objects and the Connes cocycle to repeat
Witten's crossed product construction in this new setting, and to obtain a Type
II$_\infty$ algebra that is independent of the various choices, in particular
that of the cyclic separating vector. Our emergent times are implicitly
boundary-dressed. But if one admits an ``extra'' observer in the interior, we
argue that the (state-independent) algebra can be Type I or Type II$_1$ instead
of Type II$_\infty$, depending on whether the observer's light cone contains an
entire Cauchy slice or not. Along with these general considerations, we present
some specific calculations in the setting of the Poincare BTZ black hole. We
identify a generalization of modular translations in BTZ-Kruskal coordinates
that is pointwise (as opposed to non-local) and is analytically tractable,
exploiting a connection with the covering AdS-space. These evolutions can reach
the singularity.
