Fine-Structure of Choptuik's Mass-Scaling Relation

We conjecture (analytically) and demonstrate (numerically) the existence of a fine-structure above the power-law behavior of the mass of black-holes that form in gravitational collapse of spherical massless scalar field. The fine-structure is a periodic function of the critical-separation $(p-p^{\ast })$. We predict that the period $\varpi$ is {\it universal} and that it depends on the previous universal parameters, the critical exponent, $\beta$, and the echoing period $\mathrm{\Delta }$ as $\varpi =\mathrm{\Delta }\mathrm{/}\beta$.