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

We formalize a blur-native route to fractal dimension. A positive, normalized blur, such as the Gaussian heat kernel, gives a lawful scale probe: it keeps positivity, converts sharp singular information into a stable scale family, and postpones the singular limit until the final readout. Exponents then emerge as log-log slopes of blurred energies. We present two complementary routes: an $L^2$-correlation route for measures and a blurred-perimeter route for boundaries of sets. Both turn dimension into a Mellin slope read from a positive scale functional. Finally, we give two concrete one-dimensional overlap examples: an exact-overlap self-similar measure and a finite carry-overlap measure. In the exact-overlap case, a naive symbolic count gives the wrong value, while the blur calculation certifies the correlation dimension after the overlapping branches are positively merged. In the carry-overlap case, a finite positive transfer matrix records the relevant displacement states and the dimension is read from the Perron slope.