Theorem: Budyko warm-branch linearisation: F'(T) + 4 sigma T^3 = 0 (Planck feedback)

Theorem (Budyko warm-branch stability canonical): on the warm equilibrium branch the albedo is locally constant (alpha = alpha_o), so F(T) = S(1 - alpha_o)/4 - sigma T^4. Differentiating yields F'(T) = -4 sigma T^3, identical in magnitude…