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^ (sₙ) = n → F (s) = G (1-s) /G (s) identified → ξ (s) = ξ (1-s) → |ξ^ (k) (s₀) | = |ξ^ (k) (1-s₀) | → |F (s₀) | = 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