Proof of Global Inequality F(τ) ≥ χsym(ϕa + ϕc): Unique Interior Critical Point via the Master Identity

The main note R45 reduced the proof of bmin(a, c) = χsym(ϕa + ϕc) to showing F(τ) ≥ F(ϕ) for all τ ∈ [0, ϕ], where ϕ = ϕa + ϕc and F(τ) = maxx∈Iτ χ(T(τ, x)) is the reduced one-dimensional functional (formula ⋆). The previous addendum proved F(0) ≥ F(ϕ) algebraically (sealed, Theorem R45A-T1). This note completes the proof by establishing that F has exactly one interior critical point in (0, ϕ), which is a local maximum. The key is a Master Identity for the function u: u(ϕa/2) = 23 ϕa for all a ∈ (1/6, 1). From this identity, together with an algebraic computation of ka(ϕa/2), we identify the unique interior minimum of F at τ1 = ϕ/2, prove F(ϕ/2) = F(ϕ), and deduce F ≥ F (ϕ) globally.