GQL Safe Residual Layer

The paper evolves conservative variables using a low-order safe update and a limited antidiffusive update, as in the extracted formulas $\mathbf U^L_{ij}=\mathbf U^n_{ij}-\frac{\Delta t}{\Delta x}(\widehat{\mathcal F}^{L}_{i+1/2,j}-\widehat{\mathcal F}^{L}_{i-1/2,j})-\frac{\Delta t}{\Delta y}(\widehat{\mathcal G}^{L}_{i,j+1/2}-\widehat{\mathcal G}^{L}_{i,j-1/2})$ and $\mathbf U^{n+1}_{ij}=\mathbf U^L_{ij}-\frac{\Delta t}{\Delta x}(\theta_{i+1/2,j}\widehat{\mathcal F}^{A}_{i+1/2,j}-\theta_{i-1/2,j}\widehat{\mathcal F}^{A}_{i-1/2,j})-\frac{\Delta t}{\Delta y}(\theta_{i,j+1/2}\widehat{\mathcal G}^{A}_{i,j+1/2}-\theta_{i,j-1/2}\widehat{\mathcal G}^{A}_{i,j-1/2})$. Here $\mathbf U^L$ is the safe low-order state, $\widehat{\mathcal F}^{A},\widehat{\mathcal G}^{A}$ are antidiffusive fluxes, and $0\leq\theta\leq1$ limits them. The paper also represents coefficients as sign-index dot products, $c_\star=\mathbf s_0\cdot\mathbf V_{c_\star}=\sum_{i=1}^{4}s_i(\mathbf V_{c_\star})_i$, compactly enumerating sign combinations. For the neural adaptation, let $\mathbf U_0$ be a valid baseline and $\mathbf U_1$ the network output, and use $\mathbf U(\theta)=\mathbf U_0+\theta(\mathbf U_1-\mathbf U_0)$. The relativistic admissibility condition is represented by the GQL family $g_{\mathbf s}(\mathbf U)=E-s_DD-\mathbf s_m^\top\mathbf m\geq\varepsilon$ for every auxiliary vector $\mathbf s=(s_D,\mathbf s_m)$ with $\|\mathbf s\|_2\leq1$, together with $D\geq\varepsilon$. Since $E-\sqrt{D^2+\|\mathbf m\|_2^2}=\min_{\|\mathbf s\|\leq1}g_{\mathbf s}(\mathbf U)$, these inequalities enforce the relativistic admissibility condition. For each normal with a negative denominator, the maximal safe interpolation coefficient is $\theta_{\mathbf s}=\frac{g_{\mathbf s}(\mathbf U_0)-\varepsilon}{g_{\mathbf s}(\mathbf U_0)-g_{\mathbf s}(\mathbf U_1)}$; take the minimum over normals. In the basic relativistic case, the worst normal is $\mathbf s^\star=(D_1-D_0,\mathbf m_1-\mathbf m_0)/\|(D_1-D_0,\mathbf m_1-\mathbf m_0)\|$. For more general GQL constraints, the paper’s stereographic parameterization transforms the worst-case minimization into a generalized Rayleigh quotient and obtains the optimum from a small symmetric eigenvalue problem.