For a positive odd cycle with k odd members, total division exponent A, and every member at least m, the exact product identity requires 3^k m^k < 2^A m^k ≤ (3m+1)^k. Passing does not establish a cycle; failing rules out only the specified parameters.
Derivation: 2^A = ∏(3+1/nᵢ). Since nᵢ ≥ m, each factor is at most 3+1/m and strictly greater than 3. Also A ≥ k because each odd-map step divides by at least two. Equality in the upper bound requires every member equal m. This is an elementary necessary condition, not a new theorem or a solution of the conjecture. No floating-point logarithms are used. Inputs are bounded to keep this browser responsive.