Post Riemann Hypothesis Helson-Blur Program

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PRH | Essay | 7.21 • Fractal Dimension as a Mellin Slope under Positive Blur

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We formalize a blur-native route to fractal dimensions. A positive, normalized blur (e.g. a Gaussian heat kernel) acts as a lawful switch between additive and multiplicative descriptions; exponents then emerge as log-log slopes of blurred energies. We present two complementary routes: an $L^2$-correlation route (for measures) and a Minkowski-perimeter route (for sets). Both turn dimension into a Mellin slope read directly off a monotone functional, coupled to a single legal switch at the end. Finally, we give a concrete overlapping, graph-directed self similar example in one dimension where coverings are awkward, yet the blur method certifies the correlation dimension by solving a one-parameter spectral radius equation.

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A Blur–Native Framework with a Certified Overlap Case Study

References

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