This work reports on a new and effective method that simplifies the many-electron problem. Instead of a wave function, an operator is used to describe the symmetry of the many-electron system through a transformation. It is proven that the two approaches are physically equivalent, but the operator formalism works much more effectively than the wave function formalism. In the operator formalism, the wave function of an n-electron system can be separated into its spatial and spin parts, and the symmetries of identical particles can be correctly described. For example, it can provide the correct spin and spatial symmetry of many-electron atoms or molecules. For many-electron atoms, one can obtain the simultaneous eigenfunction of L̂2 and Ŝ2. Consequently, the spectroscopic terms of a configuration can be obtained, for example, the hydrogen atom in an energy formula with four quantum numbers (n, l, ml, and ms) for each electron in an atom, including the Hartree–Fock theory. We theoretically analyze and discuss the atomic structure of many-electron atoms using this formula.

The conventional framework for quantum statistics is built upon gauge theory, where particle exchanges generate path-dependent phases. However, the apparent consistency of this approach masks a deeper question: is gauge invariance truly sufficient to satisfy the physical requirement of indistinguishability? We demonstrate that gauge transformations, while preserving probabilities in a formal sense, are inadequate to capture the full constraints of identical particles, thereby allowing for unphysical statistical outcomes. This critical limitation necessitates a reconstruction of the theory by strictly enforcing indistinguishability as the foundational principle, thus moving beyond the conventional topological paradigm. This shift yields a radically simplified framework in which the statistical phase emerges as a path-independent quantity, \( \alpha = e^{\pm i\theta} \), unifying bosons, fermions, and anyons within a single consistent description. Building upon the operator-based formalism of Series I and the dual-phase theory of Series II, we further present an exact and computationally tractable approach for solving N-anyon systems.

This work uncovers an essential dual-phase structure of identical particles, revealing a fundamental limitation in existing quantum statistical theories. For a considerable period, it was commonly assumed that an interchange of two particles would be accompanied by a fixed phase factor. Here, the author discovers that the exchange process inherently involves two distinct phases: a relative phase α and an exchange phase β, rigorously related through α = β^-1 = e^±iθ. Specifically, α = β = 1 characterizes bosons, while α = β = -1 describes fermions. This finding necessitates a radical change in handling identical particles. Within this new framework, clockwise exchange yields α = e^iθ and β = e^-iθ, while counter-clockwise exchange gives α = e^-iθ and β = e^iθ. Crucially, both scenarios, despite their topological differences, are physically equivalent, governed by the universal constraint αβ = 1 for all identical particles. This framework not only fundamentally changes the conceptual basis of quantum statistics but also provides critical theoretical constraints on the existence conditions of non-Abelian anyons in two-dimensional topological systems.