Grand GRH (adelic) axiom
The Grand GRH (adelic) axiom (III.D31) is the third custom axiom in TauLib. It asserts that the Prime Polarity Scaling Theorem (III.T20) extends to all automorphic L-functions in the adelic sense — not just the finite-rank classes verifiable in Lean. The finite envelope (bounded rank, bounded conductor) is checked via native_decide; the universal extension to unbounded rank and conductor is the conjectural step.
τ-Definition
The Grand GRH (adelic) axiom (III.D31) is the third custom axiom in TauLib. It asserts that the Prime Polarity Scaling Theorem (III.T20) extends to all automorphic L-functions in the adelic sense — not just the finite-rank classes verifiable in Lean. The finite envelope (bounded rank, bounded conductor) is checked via native_decide; the universal extension to unbounded rank and conductor is the conjectural step.
Categorical invariant. Universal spectral purity of τ-admissible automorphic data across the adelic class — asserted as the axiomatized extension of finite-rank purity (T20).
Primary registry anchor:
III.D31
τ-Derivation Chain
-
I.K0— Universe Postulate -
II.T40— Central theorem at rank (3, 15) -
III.T20— Prime Polarity Scaling Theorem — finite-rank automorphic spectral purity proved -
III.D31— Grand GRH adelic axiom — universal extension to all automorphic L-functions (this entry)
Lean modules referenced:
TauLib.BookIII.Spectral.Adeles
Mathematical content
For every automorphic L-function L(s, π) where π is an automorphic representation of GL_n over the adèles of a number field, the τ-internal spectral trichotomy (III.T14) places L's non-trivial zeros on the critical line ℜ(s) = 1/2.
What is finite-checked. Grand GRH spectral purity has been verified via native_decide for all τ-admissible automorphic data with bounded rank (n ≤ N_bound) and bounded conductor (cond(π) ≤ C_bound). The bounds are stated in TauLib BookIII/Spectral/Adeles.lean.
What is axiomatized. The universal extension to all automorphic L-functions (unbounded rank and conductor — the full adelic class).
What closes the gap. A rigorous argument that the τ-internal spectral trichotomy (III.T14) applies uniformly across the full adelic class. This is the most ambitious of the three Book III axioms and the least likely to be closed in the near term.
Load-bearing for. Grand GRH claims for higher-rank classes (not classical GRH for Dirichlet L-functions, which is finite-checked separately at rank n=1). Any 'internally addressed' GRH claim transitively dependent on this axiom is re-typed to Partial.
Relationship to classical GRH. Classical GRH for Dirichlet L-functions is the finite-rank case (n=1) and is proved separately within TauLib without depending on A03. The Grand GRH (adelic) axiom is strictly stronger: it covers all automorphic L-functions of all ranks, including non-Dirichlet cases.
Lean Coverage
Status: Formalized
Module: TauLib.BookIII.Spectral.Adeles
Lean kind: axiom
Lean symbol: Tau.BookIII.Spectral.grandGrhAdelic