Grand GRH (τ-effective)

Grand GRH at adelic level: for all adelic boundary characters on 𝔸_τ, the corresponding L-function has all non-trivial zeros on Re(s) = ½. Scaling chain: ζ → Dirichlet → Hecke → automorphic. Conjectural at adelic extension.

Grand GRH (τ-effective)

Summary

Grand GRH at adelic level: for all adelic boundary characters on 𝔸_τ, the corresponding L-function has all non-trivial zeros on Re(s) = ½. Scaling chain: ζ → Dirichlet → Hecke → automorphic. Conjectural at adelic extension.

Statement

\label{def:grand-grh}
Let $\pi$ be an automorphic representation of $\mathrm{GL}_m(\mathbb{A}_\tau)$ for any $m \geq 1$. The \textbf{Grand Generalized Riemann Hypothesis} asserts:
\begin{equation}
\text{All non-trivial zeros of } L(s, \pi) \text{ lie on the critical line } \Re(s) = \frac{1}{2}.
\end{equation}
Equivalently, the spectral operator $H_\pi$ on the adelic boundary Hilbert space $H_L^{\text{ad}}$ has all eigenvalues real.

Proof / Justification

This item is definitional. No manuscript proof is required.

Source Context

• Registry source: book-03.jsonl line 74
• Manuscript source: 2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part04/ch27-the-grand-grh.tex lines 146-152

Lean / Formalization Notes

• Formalization: formalized
• Module: TauLib.BookIII.Doors.GrandGRH
• Name: grand_grh_adelic

Dependencies

• Canonical: III.D26, III.T19, III.D22, III.D16

Related Results

Generated by later projection phases.

Related Publications

Generated by later projection phases.

Revision Notes

• 2026-04-24: Initial pilot migration.