Rigidity of Spectral Encodings under Weyl Growth Conditions

We prove that the geometric Weyl bulk-density exponent $(d-2)\mathrm{/}2$ rigidifies spectral encodings $C=\pi -\phi (\lambda )$ in the O-regularly varying class: the bulk power law forces $\phi \in {\mathrm{R}\mathrm{V}}_1$ (asymptotic linearity). For polynomial-type encodings $C=\pi -\epsilon {\lambda }^{k}L(\lambda )$ with $L\in {\mathrm{R}\mathrm{V}}_0$, this yields the unique admissible exponent $k=1$. The affine encoding then gives $N_{{\mu }_C}(C)\sim {\gamma }_d{\epsilon }^{-d\mathrm{/}2}(\pi -C{)}^{d\mathrm{/}2}$ as $C\to -\mathrm{\infty }$, 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 $\phi \in {\mathrm{R}\mathrm{V}}_k$, the induced map scales asymptotic spectral dimension as $d_{\mathrm{a}\mathrm{s}}\mapsto d_{\mathrm{a}\mathrm{s}}\mathrm{/}k$; hence dimension preservation is equivalent to $\phi \in {\mathrm{R}\mathrm{V}}_1$, with strict affine normalization at first order when $L(\lambda )\to 1$.