Bulk Weyl Asymptotics in the Edge Variable under Affine Spectral Encoding

Abstract: We prove a Tauberian transfer principle showing that for any compact closed Riemannian manifold $(M<sup>d,g)$ 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, $(\pi-C)/\epsilon\to\infty$): $$N_{\mu_C}(C)\sim \gamma_d \epsilon<sup>{-d/2}(\pi-C)<sup>{d/2}$$ and $$\rho_{\mathrm{bulk}}(C)\sim \frac{d}{2}\gamma_d \epsilon<sup>{-d/2}(\pi-C)<sup>{(d-2)/2}$$, so that $d$ and the Weyl constant $\gamma_d$ are recoverable from one-dimensional edge-variable data. Conversely, a bulk power law $N_{\mu_C}(C)\sim A(\pi-C)<sup>\alpha$ as $C\to-\infty$ implies $d=2\alpha$ and $\gamma_d=A\epsilon<sup>{d/2}$. We establish the uniqueness of the affine rule among polynomial-type encodings $g(\lambda)=a-b\lambda<sup>{k}L(\lambda)$ (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 $$\zeta_{\mathrm{edge}}(u)=\epsilon<sup>{-u}\zeta_\Delta(u)$$, and we realize multiplicities via generalized one-dimensional models (Krein strings). When a Weyl remainder $O(\Lambda<sup>{(d-1)/2})$ is available, it transfers to a bulk remainder $O((\pi-C)<sup>{(d-1)/2})$ in the $C$-variable.