Self-similar closure forces r^2 = r + 1. In a geometric scale sequence closed under additive ledger composition, the minimal closure condition (scale 0 + scale 1 equals scale 2) rearranges directly to the algebraic constraint r² = r + 1.
phi is the unique positive solution. The quadratic r² - r - 1 = 0 has roots (1 ± √5)/2; positivity and the requirement r > 0 select only the positive root φ = (1 + √5)/2, with all other candidates ruled out by sign.
Cited Lean anchors. The uniqueness step is given by phi_unique_self_similar; the closure step that produces the equation appears in the supporting derivation of IndisputableMonolith.Foundation.PhiForcingDerived and is motivated by the J-cost structure in IndisputableMonolith.Cost.FunctionalEquation.