γ_σ = 1 Selects η Uniquely (Leveille Framework, Paper 54)

Papers 49 through 53 established the full derivation chain for the Leveille scar constant η* = 0. 20949. The non-perturbative route (Papers 49–52) forced γ_σ = 1 from the Fibonacci recursion geometry. The perturbative route (Paper 53) forced γ_σ = 1 from the one-loop cubic bubble diagram, giving λ₃² = 16π²/ℓ₀³ with no free parameters. This paper answers the remaining open question: does substituting these determined values back through the chain reproduce η* = 0. 20949? The route is not a direct algebraic substitution η* = f (λ₃, m*) — that identity does not exist in closed form. The route is through the RG geometry: λ₃² = 16π²/ℓ₀³ forces γ_σ = 1 exactly, γ_σ = 1 forces the erfc fixed-point equation erfc (φL³ · η) − erfc (φL⁶ · η) /φL = η, and that equation has a unique solution η* = 0. 20949. Uniqueness is proven explicitly: a numerical scan over γ_σ shows that shifting γ_σ by 0. 001 shifts η* by approximately 0. 4%. Two open items are honestly named: the exact self-consistent mass m* and the direct algebraic identity. Neither affects γ_σ = 1 or the erfc closure. φL = 1. 6180339887 (golden ratio) is the only external input. The Leveille Framework is grounded in the principle that exact zero tolerance is not physically realizable (Δ > 0).