  We calculate the contribution to the muon anomalous magnetic moment hadronic
vacuum polarization from {the} connected diagrams of up and down quarks,
omitting electromagnetism. We employ QCD gauge-field configurations with
dynamical $u$, $d$, $s$, and $c$ quarks and the physical pion mass, and analyze
five ensembles with lattice spacings ranging from $a \approx 0.06$ to~0.15~fm.
The up- and down-quark masses in our simulations have equal masses $m_l$. We
obtain, in this world where all pions have the mass of the $\pi^0$, $10^{10}
a_\mu^{ll}({\rm conn.}) = 637.8\,(8.8)$, in agreement with independent
lattice-QCD calculations. We then combine this value with published lattice-QCD
results for the connected contributions from strange, charm, and bottom quarks,
and an estimate of the uncertainty due to the fact that our calculation does
not include strong-isospin breaking, electromagnetism, or contributions from
quark-disconnected diagrams. Our final result for the total
$\mathcal{O}(\alpha^2)$ hadronic vacuum polarization to the muon's anomalous
magnetic moment is~$10^{10}a_\mu^{\rm HVP,LO} = 699(15)_{u,d}(1)_{s,c,b}$,
where the errors are from the light-quark and heavy-quark contributions,
respectively. Our result agrees with both {\it ab-initio} lattice-QCD
calculations and phenomenological determinations from experimental
$e^+e^-$-scattering data. It is $1.3\sigma$ below the "no new physics" value of
the hadronic-vacuum-polarization contribution inferred from combining the BNL
E821 measurement of $a_\mu$ with theoretical calculations of the other
contributions.
