Sharp thresholds for the Escobar functional: the Escobar-Willmore mass, geometric selection, and compactness trichotomy

We study the hemisphere threshold for the conformally covariant Escobar functional on compact Riemannian manifolds $(M^n,g)$ with boundary. The near-threshold landscape is organized by boundary invariants: the first-order coefficient $\rho_n^{\mathrm{conf}}H_g$ vanishes identically, so the leading obstruction is a renormalized boundary mass $\mathfrak R_g$ (second order, $n\ge5$), followed by a cubic invariant $\Theta_g$ (third order, $n\ge6$), with a Green kernel interaction $\mathsf G_\partial$ in the multi-bubble regime. Exact evaluation of weighted profile moments yields $\kappa_1=\kappa_2=0$: the coefficients of $\operatorname{Ric}_g(\nu,\nu)$ and $\mathrm{Scal}_{\bar g}$ in the bare mass vanish. On $\{H_g=0\}$ the mass reduces to $\mathfrak R_g^{\mathrm{bare}}=\frac{6-n}{2(n-1)(n-3)(n-4)}|\mathring{\mathrm{II}}|^2$. The Lyapunov--Schmidt correction gives $\mathfrak R_g^{\mathrm{red}}\le\mathfrak R_g^{\mathrm{bare}}\le0$ for $n\ge6$; for $n=5$ the nonlocal back-reaction overcomes the positive bare coefficient. In every dimension $n\ge5$, non-umbilic boundaries are automatically subcritical: $C^*_{\mathrm{Esc}}<S_\ast$ whenever $\mathring{\mathrm{II}}\neq0$. At threshold, on manifolds not conformally diffeomorphic to the hemisphere, every blow-up of positive constrained critical points is one-bubble and concentrates at an umbilic point with $\mathfrak R_g=0$, $\nabla_\partial\mathfrak R_g=0$. Since $\mathfrak R_g^{\mathrm{red}}<0$ at every non-umbilic point, threshold concentration occurs only on the umbilic stratum $\{\mathring{\mathrm{II}}=0\}$. There $\Theta_g$ governs the next bifurcation: $\Theta_g<0$ forces subcriticality; for $n\ge7$, $\Theta_g>0$ yields compactness and hemispherical rigidity. In the multi-bubble regime we establish global compactness at Escobar multiples with equal-mass quantization and conditional exclusion of pure multi-bubbling.