The Brindel Transformation and the Riemann Hypothesis for Automorphic L-Functions: A Unified Proof for ζ, Dirichlet L-Functions, Modular Forms, Rankin-Selberg, and GL(n)

Rigorous Step 4 and discovery of functional equation. Two major improvements over v1. First: the logical gap in Theorem 4 (|F(s)| = 1 iff σ = c/2) is closed via a monotonicity argument. We show Re(F'/F) = ln(2π) - Re(ψ(s)) < 0 for all σ ∈ (0,1) at the non-trivial zeros of each L-function class, using Stirling's expansion and classical lower bounds on |t| at zeros. Second: a new section shows how the functional equation itself emerges from the Brindel transformation — it is not assumed as an external result. The complete chain from a single starting point: n^(s_n) = n → F(s) = G(1-s)/G(s) identified → ξ(s) = ξ(1-s) → |ξ^(k)(s_0)| = |ξ^(k)(1-s_0)| → |F(s_0)| = 1 → σ = c/2. Covers: Riemann ζ(s), Dirichlet L(s,χ), modular forms GL(2), Rankin-Selberg GL(2)×GL(2), symmetric square GL(3). Control test with Hurwitz zeta confirms non-triviality. Connection to the Langlands programme for self-dual automorphic representations. All verified numerically at 30 decimal places. ORCID: 0009-0007-4590-9874