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Abstract

We prove a Tauberian transfer principle showing that for any compact closed Riemannian manifold $(M^d,g)$ the affine spectral encoding $C=\pi-\epsilon\lambda$ transports Laplacian Weyl asymptotics to a Weyl law in the edge variable in the bulk regime $C\to-\infty$ (equivalently, $(\pi-C)/\epsilon\to\infty$): $N_{\mu_C}(C)\sim \gamma_d \epsilon^{-d/2}(\pi-C)^{d/2}$ and $\rho_{\mathrm{bulk}}(C)\sim \frac{d}{2}\gamma_d \epsilon^{-d/2}(\pi-C)^{(d-2)/2}$, so that $d$ and the Weyl constant $\gamma_d$ are recoverable from one-dimensional edge-variable data. Conversely, a bulk power law $N_{\mu_C}(C)\sim A(\pi-C)^\alpha$ as $C\to-\infty$ implies $d=2\alpha$ and $\gamma_d=A\epsilon^{d/2}$. We establish the uniqueness of the affine rule among polynomial-type encodings $g(\lambda)=a-b\lambda^{k}L(\lambda)$ (the edge-variable exponent forces $k=1$) and stability under small perturbations $C=\pi-\epsilon\lambda+\delta(\lambda)$ with $\delta(\lambda)=o(\lambda)$. For constant-curvature model spaces we record strengthened correspondences for heat traces and spectral zeta, $H_{\mathrm{edge}}(s)=\Theta_\Delta(\epsilon s)$ and $\zeta_{\mathrm{edge}}(u)=\epsilon^{-u}\zeta_\Delta(u)$, and we realize multiplicities via generalized one-dimensional models (Krein strings). When a Weyl remainder $O(\Lambda^{(d-1)/2})$ is available, it transfers to a bulk remainder $O((\pi-C)^{(d-1)/2})$ in the $C$-variable.