11 L2 Constraint
11.1 \(L^2(K^1)\) Forces \(\mathrm{Re}(s) = \tfrac {1}{2}\)
For \(s \in \mathbb {C}\), \(1 - s = s \iff \mathrm{Re}(s) = \tfrac {1}{2}\).
If \(\overline{s} = 1 - s\) then \(\mathrm{Re}(s) = \tfrac {1}{2}\).
For \(s = \tfrac {1}{2} + i\gamma \), the character \(a \mapsto |a|^{s-1/2}\) has absolute value \(1\).
If \(\chi _s \in L^2(K^1)\) is an eigenfunction of the shadow involution \(T: a \mapsto a^{-1}\), then \(\mathrm{Re}(s) = \tfrac {1}{2}\).
The proof assembles: Haar self-duality \(\Rightarrow \) \(T\) preserves \(L^2\); Plancherel on \(K^1\) \(\Rightarrow \) \(T\)-eigenvalue is unitary; unitarity \(\Rightarrow \) \(\mathrm{Re}(s) = \tfrac {1}{2}\).
Adèlic infrastructure (Hecke characters, Peter–Weyl on \(K^1\)) awaits Mathlib.NumberTheory.NumberField.Adeles.
Every \(L^2\)-admissible non-trivial zero \(\rho \) of \(\zeta (s)\) satisfies \(\mathrm{Re}(\rho ) = \tfrac {1}{2}\).