  Essential to the description of a quantum system are its local degrees of
freedom, which enable the interpretation of subsystems and dynamics in the
Hilbert space. While a choice of local tensor factorization of the Hilbert
space is often implicit in the writing of a Hamiltonian or Lagrangian, the
identification of local tensor factors is not intrinsic to the Hilbert space
itself. Instead, the only basis-invariant data of a Hamiltonian is its
spectrum, which does not manifestly determine the local structure. This
ambiguity is highlighted by the existence of dualities, in which the same
energy spectrum may describe two systems with very different local degrees of
freedom. We argue that in fact, the energy spectrum alone almost always encodes
a unique description of local degrees of freedom when such a description
exists, allowing one to explicitly identify local subsystems and how they
interact. As a consequence, we can almost always write a Hamiltonian in its
local presentation given only its spectrum. In special cases, multiple dual
local descriptions can be extracted from a given spectrum, but generically the
local description is unique.
