Momentum-projected hadron entanglement from lattice-QCD replica correlators

We define a finite-volume lattice-QCD density-matrix observable for the vacuum-subtracted spatial R'enyi response of a source-sink-prepared, momentum-projected hadron. At fixed regulator, integer R'enyi index $n>1$, spatial region $B_R$, spin projection, gauge-theory cut prescription $\mathcal{C}$, and after the usual double-sided source-sink projection, the central result is an exact source-sink replica identity: the response is obtained from the logarithm of a replicated hadron correlator on the cut geometry normalized by the corresponding power of the ordinary one-sheet correlator. This identity makes the natural first numerical target the two-sheet $n=2$ measurement of the replicated source-sink correlator ratio, together with a finite-volume test of whether the response scales as $L^{-3}$ at fixed physical $R$. The exponent is a lattice output to be tested, not an input theorem for the nonlinear R'enyi functional. The construction is prescription-defined in gauge theory, and full QCD requires the replicated sea-quark determinant and valence contractions on the replicated cut graph; quenched and partially quenched calculations are therefore pilots. Large-$N_c$ two-dimensional QCD provides an interacting benchmark in which the matched one-meson response is suppressed by the inverse spatial volume, with the short-interval coefficient controlled by light-front PDF moments.