  We present the analytic calculation of the leading order three-point energy
correlator (EEEC) in hadronic Higgs decays, including both gluon-initiated
channel $H\rightarrow g g+X$ and quark-initiated channel $H\rightarrow q\bar
q+X$. The phase space integration is evaluated directly using Mandelstam
variables $s_{ij}=(p_i+p_j)^2$, and the appearing square roots can be
rationalized by either conformal ratios or celestial coordinate variables.
Throughout the calculation, we observe the same transcendental function space
as in $\mathcal{N}=4$ super Yang-Mills (SYM) theory and $e^+e^-\rightarrow
\text{ hadrons}$. Different infrared limits are also explored using the full
analytic result, offering the fixed-order data for EEEC factorization and
resummation. Given its non-trivial shape dependence, the EEEC presents an
excellent opportunity to explore the dynamics of gluon jets originating from
the $H \to gg$ decay channel at future lepton colliders.
