Computer-assisted proof of the zero-free property

Establish by a rigorous computer-assisted proof that the Fourier transform \(F=\widehat{\mu}\) of the golden-mean self-similar measure has no real zeros, thereby converting the numerical zero-detection results into a formal verification of the zero-free property.

Background

The numerical study scans the interval $[1,10^7]$ using a Riccati-function formulation, interpolation, and golden-section minimization, and reports no detected real zeros. Together with the analytically verified zero-free interval $[-1.08,1.08]$, this provides evidence—but not a proof—that the Fourier transform has no real zeros.

The authors explicitly state that the numerical work does not constitute a formal proof and identify computer-assisted verification as an unresolved follow-up problem.