Cyclotomic Closure Defects in Higher-Order Divisor Sums

Hansen and Nielsen parameterized solutions to Φ₃(x) = Φ₃(a₁)···Φ₃(aₙ) with each Φ₃(aᵢ) prime, proving no triple threat exists. We extend this analysis to Φ₅ and Φ₇. We prove that the algebraic mechanism enabling double threats for Φ₃ — the identity Φ₃(b²) = Φ₃(b)·Φ₃(b-1) — is unique to Φ₃ among all Φ_q with q prime. Specifically, for every odd prime q ≥ 5 and integer b ≥ 2 we establish Φ_q(b-1) < Φ_2q(b) < Φ_q(b) which precludes the identity-based mechanism entirely. A computational search confirms the complete absence of double threats for Φ₅ (x ≤ 10⁵) and Φ₇ (x ≤ 10⁴). We discuss implications for the ratio Ω(N)/ω(N) in odd perfect numbers.