SPECTRAL AND CYCLOTOMIC CONSTRAINTS ON ODD PERFECT NUMBERS

This paper presents a proof of the non-existence of Odd Perfect Numbers (OPNs) by synthesizing spectral and cyclotomic frameworks, assuming Nielsen's bound of at least nine distinct prime factors. The spectral approach utilizes the Ramanujan-Fourier expansion of the abundancy index to establish the Master Equation, the Spectral Divisibility Requirement, and the Exponent Parity Identity. The cyclotomic framework leverages the identity $\sigma(q^{2f})=\prod_{d|2f+1}\Phi_{d}(q)$ and quadratic reciprocity to partition non-special primes based on their residue class modulo 3. By resolving norm-form obstructions via the Nagell-Ljunggren theorem, the author first proves that no OPN can exist with a uniform even exponent for all non-special primes. The analysis is then extended to mixed exponents through the formalization of a factor digraph, the importation of the Weil packing bound, and the introduction of the General Cascade Elimination and Absorption Uniqueness Theorem. The synthesis of these algebraic and analytic mechanisms systematically limits the total number of surviving non-special primes, leading to the conclusion that no odd perfect number exists.