Sharp joint Carathéodory compatibility and fifth-order lifting obstructions in bi-univalent function theory

Let $ \Psi(z) = 1+L_1z+L_2z^2+L_3z^3+L_4z^4+\cdots $ be analytic in $ \mathbb{D} $ with $ L_1, \dots, L_4\in\mathbb R $ and $ L_1 > 0 $, and let $ \mathcal{M} $ consist of the bi-univalent functions $ f $ for which a prescribed linear operator with coefficients $ \Phi_2, \dots, \Phi_5 > 0 $, applied to $ f $ and to $ f^{-1} $, yields two functions subordinate to $ \Psi $. Put $ \kappa = L_1\Phi_3/\Phi_2^2-L_2/L_1 $. Coupling the two subordinations through the second-order Carathéodory parametrization gives $ (4-t^2)(x+y) = 2\kappa t^2 $, and hence $ |c_1|^2\le 4/(1+|\kappa|) $ on the second-order compatible coefficient body $ V_2 $. This envelope is sharp on $ V_2 $ and it is not asserted to be sharp for $ \mathcal{M} $. Its equality set is a single rotation orbit of an explicit finite Blaschke pair; for $ \kappa\ge0 $, the envelope coincides with the bound produced by the usual summation argument; for $ \kappa < 0 $, it improves it. We then ask whether this equality orbit can be the low-order truncation of a member of $ \mathcal{M} $. The inverse-compatibility residuals $ R_1 $, $ R_2 $, and $ R_3 $ vanish identically on it, and the residual $ R_4 $, attached to the fifth coefficient, is $ R_4 = -2P_\sigma\big[\Phi_4\big(L_1\Phi_2^2+\sigma(L_1^2\Phi_3-L_2\Phi_2^2)\big)^2\big]^{-1} $, $ \sigma = \operatorname{sgn}\kappa $, where $ P_\sigma $ is an explicit polynomial that is affine in $ L_4 $ with positive leading coefficient. Thus $ P_\sigma\neq0 $ means precisely that the equality orbit violates fifth-order inverse compatibility; it gives no information about other points of $ V_2 $. Combining this with the rigidity of the equality orbit and normality of the Schwarz class, we prove that $ P_\sigma\neq0 $ implies the strict separation $ \sup_{f\in \mathcal{M}}|c_1|^2 < 4/(1+|\kappa|) $. The gap is not quantified, and the exact value of the supremum is left open. For the Janowski generator $ (1+Az)/(1+Bz) $, one has $ \kappa = A $ and $ P_\sigma = A(A-B)^3(A+\sigma)(2A+\sigma-B) $, which is nonzero for $ A\neq0 $; for $ A = 0 $, the corresponding polynomial vanishes and no separation follows. For $ \Psi(z) = e^z $, one has $ P_{+1} = 23/12 $, whence $ \sup_{f\in \mathcal{M}}|a_2| < \sqrt{2/3} $.