Algebraic Phase Theory II: The Frobenius Heisenberg Phase and Boundary Rigidity
Joe Gildea
TL;DR
This work develops a representation theory for Algebraic Phase Theory (APT) in strongly admissible regimes where defect and canonical filtration are intrinsic. It introduces filtered representations, defect-induced boundary strata, and APT-indecomposability as replacements for irreducibility in nonsemisimple settings, and uses Frobenius duality to extract the Frobenius Heisenberg algebraic phase (FHAP). The key result is a purely algebraic Stone von Neumann-type rigidity: for finite Frobenius rings, fixing a nontrivial central character yields a unique centrally faithful Frobenius indecomposable representation, realized by the Frobenius Schrödinger model on $S_R = \mathrm{Fun}(R^n,\mathbb{C})$, with all other centrally faithful representations arising as extensions. The rigidity is a boundary phenomenon dictated by defect and canonical filtration, and fails outside the Frobenius class, precisely identifying the structural threshold where Heisenberg rigidity persists.
Abstract
We develop the representation theory intrinsic to Algebraic Phase Theory (APT) in regimes where defect and canonical filtration admit faithful algebraic realisation. This extends the framework introduced in earlier work by incorporating a representation-theoretic layer that is compatible with defect and filtration. In this setting, algebraic phases act naturally on filtered module categories rather than on isolated objects, and classical irreducibility must be replaced by a filtration-compatible notion of indecomposability forced by defect. As a central application, we analyse the Frobenius Heisenberg algebraic phase, which occupies a rigid boundary regime within the broader APT landscape, and show that it satisfies the axioms of APT in a strongly admissible form. We study the representations realising this phase and show that their non-semisimplicity and rigidity properties are consequences of the underlying algebraic structure rather than analytic or semisimple hypotheses. In particular, we establish a Stone von Neumann type rigidity theorem for Heisenberg groups associated with finite Frobenius rings. For each such ring $R$, we construct a canonical Schrödinger representation of the Frobenius Heisenberg group $H_R$, and show that, for a fixed nontrivial central character, every centrally faithful representation of $H_R$ is equivalent to this model. The proof is entirely algebraic and uses no topology, unitarity, Fourier analysis, or semisimplicity. Instead, rigidity emerges as a boundary phenomenon governed by defect and canonical filtration. The Frobenius hypothesis is shown to be sharp: it precisely delineates the structural boundary within APT at which Heisenberg rigidity persists, and outside the Frobenius class this rigidity necessarily fails.