Quantum π-Unification II: Definition, Mathematical Structure, and Foundational Properties of the Quantum π for Molecular Systems - Barack Ndenga (Publisher, 2025)
The second article in the Quantum π-Unification Series establishes a fully defined, operational, and mathematically rigorous formulation of the Quantum π for molecular systems. Unlike classical π-electron theory—which only describes delocalized electrons—the Quantum π introduced here represents a phase-information invariant governing chemical stability, resonance, symmetry, and reactivity. This work develops: 1. the conceptual foundations of the Quantum π, 2. its mathematical structure (phase operator, symmetry factor, information contribution), 3. the connection with chemical resonance, electronegativity flow, and energy minimization, 4. prediction rules for molecular stability and reactivity. The article also introduces the π-Stability Index (PSI) and the Quantum π-Symmetry Number, two new descriptors that unify chemical information, electronic delocalization, and energetic behavior.
This work presents the second article of the Quantum-π Series, introducing a complete mathematical and numerical demonstration of the Quantum-π invariant (π_q) for molecular π-systems. I formally define the Quantum-π value for discrete and continuous electronic amplitudes, propose the general computation rule π_q = π (1 + W_eff), and derive the effective winding W_eff from amplitude-weighted phase evolution along a molecular contour. Using model systems such as benzene, cyclobutadiene, and pyridine, I compute π_q numerically through phase unwrapping and discrete sampling of Ψ_eff, showing clear correlations with aromaticity, heteroatom perturbation, and electronic instability. The results confirm that π_q behaves as a robust phase-information invariant capable of distinguishing aromatic, antiaromatic, and perturbed π-structures. The manuscript includes a fully reproducible methodology, a computation workflow, proposed figure prompts, and a detailed theoretical framework supporting future extensions toward biomolecules, polymers, quantum devices, and machine-learning–augmented chemical prediction. This article continues the scientific program dedicated to unifying geometry, electronic structure, and phase topology through the Quantum-π concept.