Grand GRH Approach

Overview

The Grand Riemann Hypothesis (Grand GRH) extends the Riemann Hypothesis from the single zeta function to all Dirichlet L-functions, Hecke L-functions, and automorphic L-functions. The $\tau$-framework’s spectral purity approach scales from $\zeta$ to the full L-function family through the Label${}_n$ classifier (III.D23).

Detail

Every L-function decomposes into B-, C-, and X-sector contributions via the spectral trichotomy (III.T14). The Grand GRH asserts spectral purity in each sector: all zeros lie on the critical axis of the B/C classification. The scaling preserves polarity structure, and all L-functions are expressed as spectral determinants of operators on the boundary Hilbert space $H_L$. The framework establishes the scaling chain $\zeta \to$ Dirichlet $\to$ Hecke $\to$ automorphic, previewing the Langlands program approach. The tau-internal spectral purity is established at each level, but the orthodox bridge (identifying the tau-spectral decomposition with classical analytic continuation) remains conjectural at the highest levels.

Result Statement

Grand GRH: spectral purity scales from $\zeta$ to all L-functions via the label classifier. Tau-internal purity established; orthodox bridge at higher levels conjectural. Status: Partial (tau-effective for spectral framework; conjectural for full orthodox identification).