Electron Orbitals as 4-D Rotor Projections: A Geometric Basis for Hydrogenic Structure and Chemical Bonding

Stephen Euin Cobb

Published Oct 6, 2025• 0 citations

Abstract

The hydrogen atom exhibits an exact SO(4) symmetry that has long been recognized but seldom explained. Conventional quantum mechanics reproduces the spectrum through algebraic methods, yet leaves unanswered why spin-½, orbital degeneracies, and the Pauli exclusion principle exist. In this work, we show that these features arise naturally if the electron is modeled as a four-dimensional rotor constrained to the three-sphere, S 3 . The two independent spin modes of the rotor generate the SU(2) + × SU(2) - structure, whose balanced irreducible representations project into three-dimensional orbitals with the familiar quantum numbers n, l, and m. The radial equation reduces to the associated Laguerre form, yielding exact hydrogenic wavefunctions and energies. Orbital filling, exclusion, and hybridization emerge as geometric necessities rather than axioms. The model also predicts small but definite deviations from ideal quantum mechanics, including corrections from finite rotor inertia, spin–spin couplings, and symmetry-breaking effects. These deviations are accessible to precision spectroscopy and condensed matter experiments, providing falsifiable tests. By grounding quantum mechanics in four-dimensional geometry, the rotor framework preserves all known successes while offering new explanatory clarity and predictive power.

DOI: 10.22541/au.175973067.76205913/v1

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Publisher

Abstract

DOI: 10.22541/au.175973067.76205913/v1

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\Psi \in \mathbb{C}^N

, whose observable properties arise only after projection into three-dimensional space. Within this framework, the familiar constraints of quantum chemistry---such as the two-electron occupancy limit of an orbital, the discrete shapes of atomic and hybrid orbitals, and the fixed angles of molecular bonds---are understood as consequences of information loss induced by dimensional projection. To formalize this idea, we introduce an analogy with operating-system (OS) state models, in which rich internal process states are compressed into limited external interfaces, producing apparent resource constraints. We show that the structure of electronic orbitals, Pauli exclusion, and molecular bonding can be interpreted as manifestations of a general internal--projection--observation architecture. This structure is closely related to that previously developed in the Twin-Complex Cosmology framework, where cosmological observables arise from the projection of internal dynamical degrees of freedom. The result is a unified conceptual model in which diverse physical and computational phenomena share a common pattern: high-dimensional internal dynamics, a projection map that discards information, and observable behavior that appears constrained or quantized as a result. This suggests that projection geometry may serve as a general explanatory mechanism across multiple domains of physics and information theory.