functionalReflection
plain-language theorem explainer
The functional reflection map sends a complex number ρ to 1 − ρ, implementing reflection across the critical line Re(s) = 1/2. Researchers working on the completed xi function and zero-pairing symmetries cite this definition to establish invariance properties under the functional equation. The definition is a direct one-line assignment with no additional lemmas required.
Claim. The reflection map across the line Re(s) = 1/2 is given by ρ ↦ 1 − ρ.
background
In the Zero Location Cost module the dictionary between zeta-zero locations and defect costs is formalized via zeroDeviation ρ := 2 (Re ρ − 1/2) and zeroDefect ρ := defect (exp (zeroDeviation ρ)). The critical line Re ρ = 1/2 is exactly the zero-defect locus. This definition supplies the reflection operation used to relate zero locations under the functional equation of the completed xi surface.
proof idea
The definition is a one-line wrapper returning the expression 1 - ρ for input ρ : ℂ.
why it matters
This definition is invoked in multiple theorems establishing reflection and conjugation symmetries for the completed xi function, including functionalEquation_gives_pairing_invariants and zero_pairing_under_reflection. It fills the role of the reflection map in the RS dictionary for zero locations, enabling proofs that zeros come in pairs under the functional equation. It supports the zero-deviation set being closed under negation.
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