A Symplectic Measure Approach to Geometric Complexity Theory: The Continuous Kirwan-Determinant Potential in Finite Dimensions

This paper introduces a formal variational framework for orbit closure separation within Geometric Complexity Theory (GCT) using the symplectic volumes of continuous weight spaces. We define the continuous Kirwan-Determinant Potential (KDP) as an integral transform over the Duistermaat-Heckman (DH) measure of Kirwan polytopes, aiming to bypass the asymptotic dissipation of discrete occurrence obstructions. To eliminate degree mismatch and dimension scaling anomalies, we introduce homogeneous padding structures. We evaluate this framework on a concrete computational model for n = 3 over C. By juxtaposing the fundamental topological dimension gap between the projective orbits (dim = 76 vs. dim = 64) against their identical toric boundaries, we highlight the limitation of standard boundary approaches and formulate the KDP as a conjectural variational gauge. Note: The author is an independent enthusiast, offering this preliminary observation for the GCT community to evaluate.