Golden Unit Zeta Periodicity

Abstract

The golden-unit lattice zeta is periodic along the regulator flow.

Theorem 1.1 (The golden unit gives the regulator period).

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

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

Source. Repository-derived.

Commentary.

A coefficient pair (a,b) represents the quadratic integer a+b phi. The statement exposes both real embeddings, the anisotropic form, and the zeta sum over the nonzero coefficient lattice.

Multiplication by phi is the integral bijection (a,b) maps to (b,a+b). Its two embeddings scale by phi and its conjugate, so reindexing the totalized sum shifts the flow parameter by twice log(phi) without changing the value.

References

• Truth anchor: D5/S3/Analytic/Dilation/GoldenUnitZetaPeriodicity.golden_unit_zeta_periodicity