SPECTRAL AND CYCLOTOMIC CONSTRAINTS ON ODD PERFECT NUMBERS

We develop a cyclotomic sieve for odd perfect numbers (OPNs) based on the identity σ(q^{2f}) = ∏ Φ_d(q) and quadratic reciprocity for Φ₃. Partitioning the non-special primes of a hypothetical OPN by residue class modulo 3, we eliminate all eight configurations determined by divisibility by 3, the exponent e₃, the special prime p, and the break-point index along a forced chain of primes initiated by σ(3²) = 13. The proof combines four mechanisms: (1) the forced cascade from σ(3²) = 13, which determines a rigid chain 13 → 61 → 97 → 3169 → 3348577 → ··· constraining any hypothetical OPN; (2) the starvation–overflow dichotomy, which shows that for any pre-chain budget B < 2, no integer count of primes exceeding 59 achieves σ₋₁(N) = 2 exactly; (3) the α-squeeze, which eliminates the special prime p = 5 by showing σ₋₁ < 2 at α = 1 and σ₋₁ > 2 at α ≥ 5; and (4) the Covering Lemma, which verifies computationally that for every prime divisor d of the break exponent, the cyclotomic factor Φ_d(c_k) produces a cascade trigger via its Φ₃-closure tree or falls above the starvation threshold. Together these eliminate all configurations, proving that no odd perfect number exists. The computational component (669 cyclotomic divisibility checks per break prime, verified in the companion notebook) is modest in scale and fully reproducible using exact rational arithmetic. The paper was developed with editorial and computational assistance from Claude (Anthropic).