Electron Cloud Computation via Minimal Phase Closure Triangles: A Novel Approach Beyond Traditional Quantum Mechanics

This paper presents a novel computational framework for electron cloud deter-mination based on the Minimal Phase Closure Triangle (MPCT) method,fundamentally distinct from traditional quantum mechanical approaches. Unlikethe Schr¨odinger equation-based methods that rely on wavefunctions and probabil-ity amplitudes, our approach derives electron distributions directly from the geo-metric properties of phase space trajectories. We introduce a two-level theoreticalfoundation: at the abstract level, a relational matrix describes quantum systemsthrough their fundamental interrelationships; at the implementation level, MPCTnetworks provide concrete geometric realizations. The method successfully com-putes hydrogen atom electron clouds with remarkable accuracy: Bohr radius de-termination within 1.9% error, n2 law verification within 0.6% error, and spectralline prediction with 0.1% accuracy. This framework offers a geometric-first alter-native to conventional quantum computation, potentially opening new avenues forquantum system analysis and material design.Keywords: Electron cloud computation; Minimal Phase Closure Triangle; Phasespace geometry; Relational quantum mechanics; Geometric quantum computation;MPCT method