Global Minimisation of gid over Γa × Γc: Diagonal Case Exact Lower Bound

We continue the proof of the global inequality gid(λ, µ) ≥ χsym(ϕ(a) + ϕ(c)) for all (λ, µ) ∈ Γa × Γc with a, c ∈ (1/6, 1). R44 has three components. First: a partial retraction of R43-T3, whose monotonicity claim for u(x) is false; the sign of u′ and its consequences for the maximisation of Φ(x) = u(x) + v(τ−x) are corrected. Second: a complete analysis of the diagonal case a = c, establishing (i) strict concavity of u (Lemma R44-L1), (ii) strict monotonicity of A(x) = −u′(x) (Corollary R44-C1), (iii) uniqueness of the maximiser x∗(τ) = τ/2 (Corollary R44-C2), and (iv) the global lower bound F(τ) ≥ F (2ϕa) = χsym(2ϕa) for all τ ∈ [0, 2ϕa] (Lemma R44-L2). Third: the diagonal inequality gid(λ, λ) ≥ χsym(2ϕ(a)) = bmin(a, a) is formally sealed (Theorem R44-T1). The off-diagonal case a ̸= c remains open.