  The classification of 4d $\mathcal{N}=2$ SCFTs boils down to the
classification of conical special geometries with closed Reeb orbits (CSG).
Under mild assumptions, one shows that the underlying complex space of a CSG is
(birational to) an affine cone over a simply-connected $\mathbb{Q}$-factorial
log-Fano variety with Hodge numbers $h^{p,q}=\delta_{p,q}$. With some plausible
restrictions, this means that the Coulomb branch chiral ring $\mathscr{R}$ is a
graded polynomial ring generated by global holomorphic functions $u_i$ of
dimension $\Delta_i$. The coarse-grained classification of the CSG consists in
listing the (finitely many) dimension $k$-tuples
$\{\Delta_1,\Delta_2,\cdots,\Delta_k\}$ which are realized as Coulomb branch
dimensions of some rank-$k$ CSG: this is the problem we address in this paper.
Our sheaf-theoretical analysis leads to an Universal Dimension Formula for the
possible $\{\Delta_1,\cdots,\Delta_k\}$'s. For Lagrangian SCFTs the Universal
Formula reduces to the fundamental theorem of Springer Theory.
  The number $\boldsymbol{N}(k)$ of dimensions allowed in rank $k$ is given by
a certain sum of the Erd\"os-Bateman Number-Theoretic function (sequence
A070243 in OEIS) so that for large $k$ $$
\boldsymbol{N}(k)=\frac{2\,\zeta(2)\,\zeta(3)}{\zeta(6)}\,k^2+o(k^2). $$ In the
special case $k=2$ our dimension formula reproduces a recent result by Argyres
et al.
  Class Field Theory implies a subtlety: certain dimension $k$-tuples
$\{\Delta_1,\cdots,\Delta_k\}$ are consistent only if supplemented by
additional selection rules on the electro-magnetic charges, that is, for a SCFT
with these Coulomb dimensions not all charges/fluxes consistent with Dirac
quantization are permitted.
  We illustrate the various aspects with several examples and perform a number
of explicit checks. We include tables of dimensions for the first few $k$'s.
