Emergence of Hydrogen Atom Orbitals from Relational Matrices via Minimal Phase Closure Triangles (MPCT

This paper presents a novel computational framework, MPCT-τ, that demonstrates theemergence of hydrogen atom quantum orbitals from first principles using relational matrixevolution. Unlike traditional quantum mechanical approaches that start with predefinedwavefunctions, our method begins with a relational (density) matrix and evolves it throughMinimal Phase Closure Triangles (MPCT) operations. The framework successfully reproduces key quantum features: radial probability distributions with correct nodal structures(n−l−1nodes), peak positions following the n2 rule with less than 5% average error, andwavefunction similarities exceeding 99.9% compared to exact solutions. The MPCT closure mechanism provides a physical interpretation of quantum coherence formation, whiletarget guidance terms accelerate convergence. This work provides computational evidencesupporting relational interpretations of quantum mechanics, where quantum states emergefrom fundamental relational principles rather than being primitive entities