Rigidity of Spectral Encodings under Weyl Growth Conditions

Abstract

We prove that the geometric Weyl bulk-density exponent $(d-2)/2$ rigidifies spectral encodings $C=\pi-\phi(\lambda)$ in the O-regularly varying class: the bulk power law forces $\phi\in\mathrm{RV}_1$ (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 $\phi\in\mathrm{RV}_k$, the induced map scales asymptotic spectral dimension as $d_{\mathrm{as}}\mapsto d_{\mathrm{as}}/k$; hence dimension preservation is equivalent to $\phi\in\mathrm{RV}_1$, with strict affine normalization at first order when $L(\lambda)\to1$.