Rigidity of Spectral Encodings under Weyl Growth Conditions

Abstract

We prove that the geometric Weyl bulk-density exponent $(d-2)/2$ rigidifies spectral encodings $C=π-ϕ(λ)$ in the O-regularly varying class: the bulk power law forces $ϕ\in\mathrm{RV}_1$ (asymptotic linearity). For polynomial-type encodings $C=π-ελ^k L(λ)$ with $L\in\mathrm{RV}_0$, this yields the unique admissible exponent $k=1$. The affine encoding then gives $N_{μ_C}(C)\simγ_d\,ε^{-d/2}(π-C)^{d/2}$ as $C\to-\infty$, allowing recovery of $d$ and $γ_d$ from bulk encoded data. This transfer is stable under perturbations $δ(λ)=o(λ)$, with explicit slowly varying error control. We further formalize asymptotic spectral equivalence classes: if $ϕ\in\mathrm{RV}_k$, the induced map scales asymptotic spectral dimension as $d_{\mathrm{as}}\mapsto d_{\mathrm{as}}/k$; hence dimension preservation is equivalent to $ϕ\in\mathrm{RV}_1$, with strict affine normalization at first order when $L(λ)\to1$.