  The bulk-boundary correspondence is a hallmark feature of topological phases
of matter. Nonetheless, our understanding of the correspondence remains
incomplete for phases with intrinsic topological order, and is nearly entirely
lacking for more exotic phases, such as fractons. Intriguingly, for the former,
recent work suggests that bulk topological order manifests in a non-local
structure in the boundary Hilbert space; however, a concrete understanding of
how and where this perspective applies remains limited. Here, we provide an
explicit and general framework for understanding the bulk-boundary
correspondence in Pauli topological stabilizer codes. We show -- for any
boundary termination of any two-dimensional topological stabilizer code -- that
the boundary Hilbert space cannot be realized via local degrees of freedom, in
a manner precisely determined by the anyon data of the bulk topological order.
We provide a simple method to compute this "obstruction" using a well-known
mapping to polynomials over finite fields. Leveraging this mapping, we
generalize our framework to fracton models in three-dimensions, including both
the X-Cube model and Haah's code. An important consequence of our results is
that the boundaries of topological phases can exhibit emergent symmetries that
are impossible to otherwise achieve without an unrealistic degree of fine
tuning. For instance, we show how linear and fractal subsystem symmetries
naturally arise at the boundaries of fracton phases.
