Obstruction pack (all proved): (a) if a is a totient value with n_a/a≥K then every n with φ(n)=a satisfies ∏_{p|n}p/(p−1)≥K; (b) fully elementary: ω_odd(n)≥(K²−4)/8 and v₂(a)≥(K²−4)/8, equivalently f(a)/a≤2√(2v₂(a)+1) for every totient value a — in particular a≡2 (mod 4) forces f(a)/a≤2√3, and any family witnessing #51 must have v₂(a)→∞; (c) with Mertens (asymptotic) and Rosser–Schoenfeld (explicit): ω_odd(n) ≥ exp((e^{−γ}−o(1))K), explicitly ≥ e^(0.5526K)/(0.5526K)−1 for K>10.226; (d) f(a)/a≤(e^γ+o(1))loglog a (known bound, proof included for completeness); combining (c) with v₂(a)≤log₂a recovers (d) with constant 1.8094 in place of e^γ=1.7811.