Bypassing Boundary Collapse in Geometric Complexity Theory via Duistermaat-Heckman Moments

This paper proposes a measure-theoretic framework for orbit closure separation in Geometric Complexity Theory (GCT). Standard occurrence obstructions dissipate asymp- totically due to the boundary collapse of Kirwan moment polytopes. To address this, we define the continuous Kirwan-Determinant Potential (KDP) as an integral transform over the Duistermaat-Heckman measure of the moment polytopes. We establish the fundamental ana- lytic properties of the KDP, proving it functions strictly as the Moment-Generating Function of the orbit’s symplectic volume distribution. Applying this framework to the strict base case n = m = 3, we formally juxtapose the exact toric boundary collapse of the polytopes against the topological dimension gap between the orbits. We emphasize that the Fubini-Study volumes originate from projective varieties of strictly different dimensions (63 vs. 75). Consequently, pushing them forward onto the identical Birkhoff polytope imposes fundamentally different upper bounds on the maximal degrees of their piecewise polynomial densities. While a highly non-generic cancellation could theoretically equate these measures, the underlying topological gap strongly suggests they are distinct. Relying on the analytic uniqueness of the Moment- Generating Function, we conjecture that evaluating the higher-order derivative tensors of the KDP provides a robust, continuous volumetric obstruction capable of separating the orbits. Finally, we outline a Markov Chain Monte Carlo framework to empirically test this hypothesis in low-dimensional regimes.