  The remarkably high energies of the Large Hadron Collider (LHC) have allowed
for the first measurements of the shapes and scalings of multi-point
correlators of energy flow operators, $\langle \Psi | \mathcal{E}(\vec n_1)
\mathcal{E}(\vec n_2) \cdots \mathcal{E}(\vec n_k) |\Psi \rangle$, providing
new insights into the Lorentzian dynamics of quantum chromodynamics (QCD). In
this Letter, we use recent advances in effective field theory to derive a
rigorous factorization theorem for the light-ray density matrix, $\rho=
|\Psi\rangle \langle \Psi |$, inside high transverse momentum jets at the LHC.
Using the light-ray operator product expansion, the scaling behavior of
multi-point correlators can be computed from the expectation value of the
twist-2 spin-$J$ light-ray operators, $\mathbb{O}^{[J]}$, in this state,
$\text{Tr}[ \rho ~\mathbb{O}^{[J]} ]$. We compute the light-ray density matrix
at next-to-leading order, and combine this with results for the next-to-leading
logarithmic scaling behavior of the correlators up to six-points, comparing
with CMS Open Data. This theoretical accuracy allows us to resolve the quantum
scaling dimensions of QCD light-ray operators inside jets at the LHC. Our
factorization theorem for the light-ray density matrix at the LHC completes the
link between recent developments in the study of energy correlators and LHC
phenomenology, opening the door to a wide variety of precision jet substructure
studies.
