completed_zeta_functional_equation_mathlib
plain-language theorem explainer
The completed Riemann zeta function satisfies the reflection symmetry relating its value at any complex argument to the value at one minus that argument. Researchers in analytic number theory and the Recognition Science zeta program cite it as the Mathlib anchor for the theta-Mellin bridge. The proof reduces directly to the symmetry property of the completed zeta via a one-line term application.
Claim. For every complex number $s$, the completed Riemann zeta function satisfies $ξ(s) = ξ(1-s)$, where $ξ$ denotes the completed zeta function.
background
The ZetaFromTheta module forms phase 4 of the RS-native zeta program. It connects the Recognition Theta program to the completed zeta functional equation by isolating a theta-style Mellin transform bridge without claiming a full analytic derivation. The module records Mathlib's unconditional result as the current analytic source under the recovered-complex substrate.
proof idea
The proof is a term-mode one-liner. It invokes the symmetry lemma for the completed Riemann zeta function and takes the symmetric form.
why it matters
This declaration supplies the unconditional functional equation used by the completed zeta functional equation certificate, the logic version on recovered-complex inputs, and the phase 4 zeta-from-theta certificate. It anchors the transition from the theta kernel to the completed zeta via the Mellin transform in the Recognition framework while the native derivation remains open.
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