Golden Observer-Light Spectral Zeta

Abstract

The golden massless observer tower has the scaled Riemann zeta shape spectrum.

Theorem 1.1 (The golden light tower has Riemann zeta shape).

$$\forall s\in \mathbb{C},1<\mathrm{re}(s)\Rightarrow \left(\mathrm{chiralSpectralZeta}(s)=goldenLightScal{e}^{-s}\times \mathrm{riemannZeta}(s)\wedge \left(\mathrm{fullSpectralZeta}(s)=2\times goldenLightScal{e}^{-s}\times \mathrm{riemannZeta}(s)\wedge \left(\left(\forall n\in \mathbb{N},\frac{\mathrm{chiralEnergy}(n)}{goldenLightScale}=n+1\right)\wedge \sum _{n\in \mathbb{N}}\mathrm{cpow}(\mathrm{ofReal}(\frac{\mathrm{chiralEnergy}(n)}{goldenLightScale}),-s)=\mathrm{riemannZeta}(s)\right)\right)\right)$$

Proof. Machine-checked in Lean as D5/S3/Analytic/Adelic/GoldenObserverLightSpectralZeta.golden_observer_light_spectral_zeta (✓ std3). ∎

Source. Repository-derived.

Commentary.

The level spacing is pi squared divided by twice log(phi), and the positive-mode energy is that spacing times n+1. The chiral and full spectral zeta functions are constructed as one-branch and two-branch totalized sums.

The displayed convergence premise is required by the Dirichlet-series representation. Factoring the positive scale gives the chiral identity, and the finite two-branch sum gives the factor of two.

Dividing each energy by the physical spacing yields n+1 at every mode. Consequently the normalized tower sum is exactly Riemann zeta, which states the dimensionless shape-spectrum clause directly.

References

• Truth anchor: D5/S3/Analytic/Adelic/GoldenObserverLightSpectralZeta.golden_observer_light_spectral_zeta