One Equation Selects the Golden Split and Makes the Sum Rule a Corollary: the product condition on a one-rate channel pair has the single root a = 1/φ on the C60 adjacency, the sum rule holds exactly when a² + a - 1 = 0, and no gradient flow of the c

One equation picks the pair (4 - φ, φ) out of a one-parameter family. Write the two factors of the attractor Ψ∞ = 3φ - 1 of the icosahedral coherence flow as functions of a single positive rate a: a bandwidth G(a) = λ_max - a and a reciprocal L(a) = λ₉/a, built from the largest eigenvalue of the C60 adjacency and from its nine-fold level. Then G(a)L(a) = Ψ∞ has exactly one solution. Cleared of denominators the condition is linear, 3 = 3φa, so a = 1/φ and there is no second branch. The split (4 - φ, φ) follows, and so does the sum rule G + L = λ_max + λ₉ = 4, which is a corollary here and not an input. The sum rule holds if and only if the rate satisfies a² + λ₉a - λ₉ = 0, which at λ₉ = 1 is a² + a - 1 = 0. The entire golden content of the construction sits in that one quadratic. The product condition by itself selects nothing. Its solution set is a hyperbola. In logarithmic channel coordinates the two partial derivatives of the cubic potential are equal, exactly, so ln(G/L) is a constant of motion under every constant metric and the point reached on the hyperbola is the point you started above. Zero of 1001 starting ratios reach the golden split. What carries the selection is the one-rate family, and the family is an assumption. This 1/φ is not the φ of the character table. The eigenspace at 1/φ is five-dimensional; φ is the character of a 72° rotation on the two three-dimensional irreducibles, and 1/φ is not a character value of A₅ anywhere. Changing the pin value by one percent moves the rate by -0.78773%. The quadratic is sharp: under the measured bond alternation, where the frontier level no longer solves it, the sum rule misses λ_max + λ₉ by 4.61% and 5.18%. A script recomputes all 99 checkable statements from the adjacency matrix; all 99 pass.