Bypassing Boundary Collapse in Geometric Complexity Theory via Equivariant Duistermaat-Heckman Volumes

This paper proposes a continuous measure-theoretic formulation for the orbit closure separation problem in Geometric Complexity Theory (GCT). Standard occurrence obstruc- tions dissipate asymptotically. Geometrically, this can be interpreted as a loss of separating information at the boundary of the associated moment polytopes. To address this, we intro- duce the continuous functional that we call the Kirwan-Determinant Potential (KDP). We use the term Kirwan-Determinant Potential to denote the continuous Laplace transform of the Duistermaat-Heckman measure associated with an orbit closure. Despite the name, the construction is defined for arbitrary homogeneous forms and is not specific to determinant orbit geometry. We establish the fundamental analytic properties of the KDP, observing that it coincides with the Moment-Generating Function (evaluating as the Laplace trans- form of the measure) of the orbit’s symplectic distribution. Applying this framework to the base case n = m = 3, we formally juxtapose the exact toric boundary collapse of the poly- topes against the algebraic dimension gap between the orbits. Because the ultimate goal of GCT is proving non-inclusion rather than mere inequality, classical strict separation of these measures is insufficient. We demonstrate via a geometric counterexample (CP1 ⊂ CP2) why naive statistical obstructions (e.g., variance tensor dominance) fail inherently for nested subvarieties due to boundary concentration effects. Instead of relying on discrete occurrence evaluations, we propose shifting the paradigm to a continuous obstruction framework. As a potential application, we hypothesize that topological inclusion may impose a detectable re- striction on analytic localization data. We explicitly acknowledge the missing mathematical bridge required to rigorously bind subvariety inclusion to strict pole restrictions on singular varieties, grounding this hypothesis in the known homological behavior of equivariant Euler classes for smooth inclusions. This work does not claim a new obstruction for GCT, but proposes a continuous geometric language in which potential future obstructions could be formulated. Finally, we outline a compact-orbit Markov Chain Monte Carlo framework to empirically explore geometric distribution profiles in low-dimensional toy regimes.