What question did this study set out to answer?

This work aims to resolve the fractional quantum hall effect through a geometrical perspective using spectral methods.

June 29, 2026Open Access

Exact Spectral Geometry and Topological Derivation of the Fractional Quantum Hall Effect via the Kaleidoscopic Filter Theorem

Key Points

This work aims to resolve the fractional quantum hall effect through a geometrical perspective using spectral methods.
Utilized strict spectral geometry within a discrete combinatorial framework to analyze anyonic quasiparticles.
Applied the Kaleidoscopic Filter Theorem to explore the relationship between integer partitions and fractional quantum phenomena.
Presented rigorous proofs of spectral gap generation and the odd-denominator rule.
Confirmed that the filling factor nu represents the exact topological rational winding number of the partition trajectory.
Established the geometric origin for fractional conductance through absolute topological truncation.
Demonstrated the cancellation of lower-dimensional geometry via Weyl reflections in the partitioning process.

Abstract

The Fractional Quantum Hall Effect (FQHE) is a defining phenomenon of modern condensed matter physics, traditionally rationalized through the heuristic postulation of anyonic quasiparticles carrying fractional charges. This work demonstrates that such continuous macroscopic models are emergent approximations. By utilizing strict spectral geometry, we analytically resolve the FQHE entirely within the discrete combinatorial framework of integer partitions. The foundation of this resolution is the Teorema del Filtro Caleidoscopico di Bonelli. Derived from the Kaleidoscopic Polynomials of Section 9. 1, this theorem states that applying the coefficients of Weyl reflections of dimension k to the infinite sequence of unrestricted partitions p (n) cancels out all lower-dimensional geometry. The result of the equation is exactly the number of partitions of n formed exclusively by pieces strictly greater than k. Under strong transverse magnetic confinement, the filling factor is proven to be the exact topological rational winding number of the purified partition trajectory traversing the restricted A₊-₁ root boundaries. Furthermore, we present rigorous, full-length proofs for the spectral gap generation and the empirical odd-denominator rule. By replacing arbitrary quasiparticle parameterization with absolute topological truncation, this paper establishes a definitive geometric origin for fractional conductance.

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