Rigidity of Spectral Encodings under Weyl Growth Conditions

Abstract

We prove that the geometric Weyl bulk-density exponent $(d-2)/2$ rigidifies spectral encodings $C=\pi -\varphi (\lambda )$ in the O-regularly varying class: the bulk power law forces $\varphi \in {\mathrm{RV}}_1$ (asymptotic linearity). For polynomial-type encodings $C=\pi -\epsilon {\lambda }^{k}L(\lambda )$ with $L\in {\mathrm{RV}}_0$, this yields the unique admissible exponent $k=1$. The affine encoding then gives $N_{{\mu }_C}(C)\sim {\gamma }_d{\epsilon }^{-d/2}(\pi -C{)}^{d/2}$ as $C\to -\infty$, allowing recovery of $d$ and ${\gamma }_d$ from bulk encoded data. This transfer is stable under perturbations $\delta (\lambda )=o(\lambda )$, with explicit slowly varying error control. We further formalize asymptotic spectral equivalence classes: if $\varphi \in {\mathrm{RV}}_k$, the induced map scales asymptotic spectral dimension as $d_{\mathrm{as}}\mapsto d_{\mathrm{as}}/k$; hence dimension preservation is equivalent to $\varphi \in {\mathrm{RV}}_1$, with strict affine normalization at first order when $L(\lambda )\to 1$.

32 pages, The manuscript has undergone a substantial revision. The introduction and abstract have been rewritten, and all sections have been carefully reviewed