Golden Unit Zeta Reflection

Abstract

Conjugation reflects the golden-unit flow, while unit multiplication supplies its period.

Theorem 1.1 (Conjugation and unit translation generate the flow symmetries).

$$\begin{array}{r}sigmaPlus:\mathbb{Z}\times \mathbb{Z}\to \mathbb{R},\mathrm{sigmaPlus}((a,b)):=a+b\times \phi ,\\ sigmaMinus:\mathbb{Z}\times \mathbb{Z}\to \mathbb{R},\mathrm{sigmaMinus}((a,b)):=a+b\times \psi ,\\ anisotropicForm:\mathbb{R}\to \mathbb{Z}\times \mathbb{Z}\to \mathbb{R},\mathrm{anisotropicForm}(eta,(a,b)):=\exp (eta)\times \mathrm{sigmaPlus}((a,b){)}^2+\exp (-eta)\times \mathrm{sigmaMinus}((a,b){)}^2,\\ goldenUnitZeta:\mathbb{C}\to \mathbb{R}\to \mathbb{C},\mathrm{goldenUnitZeta}(s,eta):=\sum _{alpha\in \mathbb{Z}\times \mathbb{Z}\setminus \{(0,0)\}}\mathrm{anisotropicForm}(eta,alpha{)}^{-s},\\ (\forall s\in \mathbb{C},\forall eta\in \mathbb{R},\mathrm{goldenUnitZeta}(s,eta)=\mathrm{goldenUnitZeta}(s,-eta))\wedge (\forall s\in \mathbb{C},\forall eta\in \mathbb{R},\mathrm{goldenUnitZeta}(s,eta+2\cdot \log (\phi ))=\mathrm{goldenUnitZeta}(s,eta)).\end{array}$$

Proof. Machine-checked in Lean as D5/S3/Analytic/Dilation/GoldenUnitZetaReflection.golden_unit_zeta_reflection (✓ std3). ∎

Source. Repository-derived.

Commentary.

A coefficient pair (a,b) represents the quadratic integer a+b phi. Both real embeddings, the anisotropic form, and the zeta sum over the nonzero coefficient lattice are exposed in the statement.

Quadratic conjugation is the integral involution (a,b) maps to (a+b,-b). It exchanges the two real embeddings and therefore reindexes the zeta at eta as the zeta at minus eta. The second public conjunct imports the regulator-period theorem on exactly the same carrier, exposing both symmetry generators.

Current D5 and pinned-Mathlib searches found no exact reflection theorem. The proof applies the canonical subtype equivalence and total-sum reindexing machinery; it does not define the zeta by its target symmetry or replace the coefficient lattice with a surrogate.

References

• Truth anchor: D5/S3/Analytic/Dilation/GoldenUnitZetaReflection.golden_unit_zeta_reflection
• Dependency: D5/S3/Analytic/Dilation/GoldenUnitZetaPeriodicity