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 Φq with q prime. Specifically, for every odd prime q ≥ 5 and integer b ≥ 2 we establish Φq (b-1) < Φ₂q (b) < Φq (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.
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