ICON₁₀: Non-local Kernel Verification Suite

Rigorous symbolic asymptotic analysis of non-local Gaussian convolution operators on bounded domains, with analytical verification of coercivity bounds and eigenvalue splitting induced by geometric truncation. Mathematical Verification Scope - Distributional Convergence: Verifies weak convergence of the integral operator to the local identity in the macroscopic limit ($\xi/L \to 0$), recovering the Dirac delta kernel in the distributional sense.- Geometric Symmetry Breaking: Characterises the emergence of rank-2 spectral anisotropy induced strictly by domain truncation ($\partial \Omega$), independent of material constitutive parameters.- Exact Arithmetic: Validates well-posedness of the constitutive definition using symbolic integration (SymPy) to preclude numerical discretization errors, with cross-validation against adaptive quadrature (error $< 10^{-15}$). Principal Analytical Result For the z-marginal of a 3D isotropic Gaussian kernel, $K(z) = (2\pi\xi^2)^{-1/2} \exp(-z^2/(2\xi^2))$, integrated over the slit domain $z \in [-L/2, L/2]$, the effective eigenvalues satisfy: - $\lambda_\parallel = 1$ — unbounded in-plane integration- $\lambda_\perp = \mathrm{erf}\left( \frac{L}{2\sqrt{2}\,\xi} \right)$ — truncated out-of-plane integration where $L$ denotes the confinement length [nm], $\xi$ the molecular correlation length [nm], and $\mathrm{erf}$ the Gauss error function. Asymptotic Limits (Symbolically Verified) - Local Recovery: $\lim_{\xi \to 0} \lambda_\perp = 1$ (kernel $\to \delta$, identity operator)- Bulk Limit: $\lim_{L \to \infty} \lambda_\perp = 1$ (full integration domain)- Strong Confinement: $\lim_{L \to 0} \lambda_\perp = 0$ (vanishing measure) Coercivity and Well-posedness The effective diffusion tensor $\mathbf{K}_{\mathrm{eff}} = \mathrm{diag}(1, 1, \lambda_\perp)$ admits Cholesky factorisation $\mathbf{K}_{\mathrm{eff}} = \mathbf{L}\mathbf{L}^\top$ with $\mathbf{L} = \mathrm{diag}(1, 1, \sqrt{\lambda_\perp})$, confirming: - Symmetry: $\mathbf{K} = \mathbf{K}^\top$ (Onsager reciprocity)- Positive Definiteness: $\mathbf{x}^\top \mathbf{K} \mathbf{x} > 0$ for all $\mathbf{x} \neq \mathbf{0}$- Coercivity Bound: $\lambda_K := \min(\lambda_\parallel, \lambda_\perp) = \lambda_\perp > 0$ for all $L > 0$ For the associated bilinear form $a(u,v) = \int_\Omega (\mathbf{K}_{\mathrm{eff}} \nabla u) \cdot \nabla v \, dx$, Gårding's inequality yields: $$a(u,u) \geq \lambda_K \|\nabla u\|^2_{L^2}$$ guaranteeing strict ellipticity and unique weak solution in $H^1(\Omega)$ via the Lax–Milgram theorem. Quantitative Benchmark The anisotropy ratio $\lambda_\parallel / \lambda_\perp = [\mathrm{erf}(L/(2\sqrt{2}\xi))]^{-1}$ yields approximately 1.1 for $L = 1$ nm, $\xi = 0.3$ nm, compared to experimental observations of ~500 in confined nanochannel systems. This ~400× discrepancy demonstrates that Gaussian kernels capture qualitative symmetry breaking but require refinement (e.g., non-Gaussian tails, multi-scale correlation functions) for quantitative accuracy in strongly confined geometries. Verification Pipeline 1. Symbolic Derivation: Analytical computation of $\lambda_\perp$ via SymPy with formal limit evaluation2. Numerical Cross-validation: Adaptive Gauss–Kronrod quadrature (scipy.integrate.quad) with tolerance $10^{-12}$3. Matrix Decomposition: Cholesky factorisation with reconstruction error $< 10^{-15}$4. Coercivity Certification: Eigenvalue positivity and Gårding bound verification5. Publication Figures: Two-panel analysis: (a) eigenvalue evolution vs $L/\xi$; (b) anisotropy ratio in logarithmic scale with experimental comparison Repository Contents - nonlocal_kernel_asymptotics.py — Main verification script (Python 3.8+)- README.md — Full mathematical documentation- requirements.txt — Dependencies: NumPy, SciPy, SymPy, Matplotlib- CITATION.cff — Citation metadata (CFF format)- LICENSE — MIT License Keywords Non-local operators, asymptotic analysis, coercivity, Gaussian convolution, bounded domains, eigenvalue splitting, symbolic computation, Lax–Milgram theorem, Gårding inequality.