SPECTRAL AND CYCLOTOMIC CONSTRAINTS ON ODD PERFECT NUMBERS

This paper presents two complementary frameworks—spectral and cyclotomic—to investigate the existence of Odd Perfect Numbers (OPNs). The spectral approach utilizes the Ramanujan-Fourier expansion 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 the non-special primes. By analyzing norm-form obstructions and resolving them via the Nagell-Ljunggren theorem, the author proves that no OPN can exist with a common even exponent for all non-special primes. Extending the analysis beyond uniform exponents, the paper synthesizes multiple mechanisms, including doubly-exponential factor chain growth, the General Cascade Elimination, and the Absorption Uniqueness Theorem. Ultimately, assuming Nielsen's bound of at least nine distinct prime factors, the synthesis of these constraints leads to the conclusion that no odd perfect number exists