Spectral and Cyclotomic Constraints on Odd Perfect Numbers

This paper is a corrected and substantially narrowed version of a previous manuscript that claimed to prove the nonexistence of odd perfect numbers. That claim is explicitly withdrawn. The present version records only what the underlying cyclotomic framework can rigorously establish, and provides a precise account of where the prior global argument fails. The valid results include: elementary congruence lemmas for the cyclotomic polynomial Φ3₃ Φ3 using lifting-the-exponent and quadratic reciprocity; a corrected starvation–overflow lemma for the abundancy index σ (N) /N (N) /N σ (N) /N that replaces an overreaching earlier version; the genuinely forced conditional divisibility 13∣N13 N 13∣N under the hypotheses 3∣N3 N 3∣N and e3=2e₃ = 2 e3=2; and an explicit arithmetic description of the initial 13-chain extension at exponent 2. The paper also provides a fully rigorous conditional overflow contradiction in one explicit branch. Three specific fatal defects in the withdrawn proof are identified and explained: the case 3∤N3 N 3∤N fails because mixed exponents are ignored; the uniform-exponent case misapplies a one-directional congruence lemma; and the claimed absolute starvation bound for large break index relies on an unproved lower bound on new prime factors. The paper is intended as an honest record of partial progress and as a precise statement of what remains open.