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 to establish an unconditional upper bound of $\omega(N) \le 7$ distinct prime factors, which directly contradicts Hare's established lower bound of $\omega(N) \ge 8$. 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 replaces factor chain growth estimates with a direct $\sigma_{-1}$ budget elimination constraint. It further introduces the General Cascade Elimination, the Cascade Closure Principle, and the Absorption Uniqueness Theorem. The synthesis of these algebraic and analytic mechanisms strictly limits the capacity of exponent slots, bounding the total number of prime factors to at most seven and yielding the final contradiction.