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 eight 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 utilizes a break-point dichotomy to eliminate early breaks via cascade overflow, and an a-squeeze 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 details the computational verification required for the Covering Lemma and branch starvation bounds, involving cyclotomic divisibility checks for specific break primes