Radius-Flow Entanglement in Hadron States and Gravitational Form Factors

We define an operational integer-$n$ lattice observable for state-dependent spatial entanglement in QCD hadrons and a falsifiable local-window analysis of its radius dependence. The observable is the vacuum-subtracted logarithmic radius derivative of the ball R\'enyi entropy, $\mathfrak{s}_n(R;h)=R\partial_R\Delta S_n(B_R;h)$, evaluated in a rest-frame momentum-projected one-hadron state. At fixed replica/cut prescription, $\Delta S_n$ is a direct integer-$n$ replica ratio and $\mathfrak{s}_n$ is obtained as its correlated radius derivative; the intended first target is $n=2$, and spin averaging is performed only after forming the flow. Continuum trace-response identities motivate a trace-channel organization, but the proposed lattice observable and fits are defined directly at fixed integer $n$. We test boundary dominance by fitting local $R$ windows to a low-curvature remainder plus two candidate endpoint templates, $\mathfrak{t}_h^{(0)}(R)=R^3\rho_S(R)$ from the trace gravitational form factor $A^S(t)$ and $\mathfrak{t}_h^{(2)}(R)=R^3\rho_A(R)$ from the spin-2 gravitational form factor $A(t)$. A soft-wall AdS/QCD trace--energy calculation motivates this two-function basis and gives an optional model-dependent fixed-ratio benchmark; the lattice coefficients are left free. For representative nucleon dipole gravitational form factors, the pure endpoint-template peak scales are $R_{\rm peak}^{(0)}\simeq0.84~{\rm fm}$ and $R_{\rm peak}^{(2)}\simeq0.43~{\rm fm}$, so lattice data can distinguish scalar dominance, spin-2 dominance, genuine mixing, or failure of the boundary-dominance ansatz.