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. 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 text proves no OPN can exist with a uniform even exponent for all non-special primes. Extending the analysis to mixed exponents, the proof uses a direct $\sigma_{-1}$ budget elimination constraint alongside the General Cascade Elimination, the Cascade Closure Principle, and the Absorption Uniqueness Theorem. Crucially, this iteration introduces the Cyclotomic Incompatibility Theorem, which establishes algebraically that the equation $\Phi_{3}(x)=\Phi_{5}(y)$ has no integer solutions for $y \ge 3$. This proves that specific exponent slots cannot coexist, successfully strengthening the unconditional upper bound of distinct prime factors to $\omega(N) \le 6$. This upper bound directly contradicts Chein's established lower bound of $\omega(N) \ge 7$, culminating in the conclusion that no odd perfect number exists