SPECTRAL AND CYCLOTOMIC CONSTRAINTS ON ODD PERFECT NUMBERS

This paper presents a proof asserting the nonexistence of odd perfect numbers. The author develops a cyclotomic sieve based on the identity $\sigma(q^{2f})=\prod_{d|2f+1}\Phi_{d}(q)$ and quadratic reciprocity for 3 to partition the non-special primes of a hypothetical odd perfect number by their residue class modulo 3. Through exhaustive case analysis divided into distinct configurations, the work systematically eliminates all possible structures for an odd perfect number. Key mechanisms introduced in the methodology include a forced cascade originating from $\sigma(3^{2})=13$ that determines a rigid chain of primes constraining the hypothetical number. The paper introduces a break-point dichotomy, which eliminates early breaks via cascade overflow. An a-squeeze is utilized to eliminate the special prime 5. Furthermore, the Covering Lemma demonstrates that cyclotomic factors produce cascade triggers that lead to either budget starvation or overflow. The paper also includes a section detailing the computational verification required for the Covering Lemma, which involves checking cyclotomic divisibility conditions for specific break primes.