Spectral Correspondence Theorem

CONDITIONAL on O3 (determinant representation): zeros of ζ_τ(s) correspond to spectral values of H_L via the spectral parameter Λ(s). This is the Hilbert–Pólya realization within the τ framework. O3 is the honest conjectural gap.

Spectral Correspondence Theorem

Summary

CONDITIONAL on O3 (determinant representation): zeros of ζ_τ(s) correspond to spectral values of H_L via the spectral parameter Λ(s). This is the Hilbert–Pólya realization within the τ framework. O3 is the honest conjectural gap.

Statement

\label{thm:spectral-correspondence}
Assume Conjecture~\ref{conj:determinant-representation} holds. Then:
\begin{enumerate}
\item $\zeta_\tau(\rho) = 0$ if and only if $\Lambda(\rho) \in \operatorname{Spec}^*(H_L)$, where $\operatorname{Spec}^*(H_L)$ denotes the point spectrum (eigenvalues) of $H_L$.
\item Every non-trivial zero of $\zeta_\tau(s)$ corresponds to a unique eigenvalue of $H_L$ via $\lambda = \Lambda(\rho)$.
\item The multiplicities of zeros and eigenvalues match: $\operatorname{ord}_{s=\rho} \zeta_\tau(s) = \dim \ker(H_L - \Lambda(\rho))$.
\item If $H_L$ is self-adjoint with real spectrum, then all non-trivial zeros satisfy $\Lambda(\rho) \in \mathbb{R}$.
\end{enumerate}

Proof / Justification

No immediate manuscript proof block was extracted in this pilot run.

Source Context

• Registry source: book-03.jsonl line 68
• Manuscript source: 2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part04/ch24-the-spectral-correspondence.tex lines 112-120

Lean / Formalization Notes

• Formalization: formalized
• Module: TauLib.BookIII.Doors.SpectralCorrespondence
• Name: spectral_correspondence_O3

Dependencies

• Canonical: III.D29, III.D26, III.T17, III.P09

Related Results

Generated by later projection phases.

Related Publications

Generated by later projection phases.

Revision Notes

• 2026-04-24: Initial pilot migration.