  Dimensionality plays a fundamental role in the classification of novel phases
and their responses. In generic lattices of 2D and beyond, however, we found
that non-Hermitian couplings do not merely distort the Brillouin zone (BZ), but
can in fact alter its effective dimensionality. This is due to the fundamental
non-commutativity of multi-dimensional non-Hermitian pumping, which obstructs
the usual formation of a generalized complex BZ. As such, basis states are
forced to assume "entangled" profiles that are orthogonal in a lower
dimensional effective BZ, completely divorced from any vestige of lattice Bloch
states unlike conventional skin states. Characterizing this reduced
dimensionality is an emergent winding number intimately related to the homotopy
of non-contractible spectral paths. We illustrate this dimensional
transmutation through a 2D model whose topological zero modes are protected by
a 1D, not 2D, topological invariant. Our findings can be readily demonstrated
via the bulk properties of non-reciprocally coupled platforms such as circuit
arrays, and provokes us to rethink about the fundamental role of geometric
obstruction in the dimensional classification of topological states.
