From Finite-Contact Hamiltonian Dynamics to an Emergent Hydrodynamic Propagator

How can a local hydrodynamic response emerge from reversible Hamiltonian many-body dynamics without viscosity, a constitutive law, or a diffusion operator being introduced microscopically? We address this question in the transverse shear sector of a frozen finite-contact Hamiltonian model. Starting from the microscopic dynamics, we construct the exact momentum-flux and shear-stress ledger and resolve the finite-contact contribution to transport. We then analyze periodic equilibrium transverse-current correlations through the causal projected operator M(k,s)=CT(k,s)−1−s, without imposing diffusive spatial scaling. In the resolved long-wavelength and low-frequency regime, the collision-dressed response crosses from a kinetic/free-streaming M∼∣k∣ sector to a hydrodynamic branch satisfying M(k,s)≃νbulkk2. Over the primary window 0.015≤s≤0.10, the inferred transport scale is ηbulk=0.09955 and νbulk=0.33183, with through-origin k2 fits giving R2≥0.9928. An independent microscopic-stress Green–Kubo calculation yields ηGK=0.09379, differing by 5.79% from the frozen response-based value. An independently measured equilibrium stress-memory kernel has first-moment memory time τmem=1.983 and quantitatively explains the failure of an instantaneous Newtonian law during abruptly prepared shear transients. Once the resolved shear becomes slow relative to this memory, the generalized constitutive response approaches the Newtonian normal solution. Finally, freezing ν=0.331833 before a distinct three-mode test predicts the full transverse field without mode-specific refitting, with a normalized spacetime RMS discrepancy of 2.05%. These results establish, within the audited transverse sector, a concrete route from finite-contact Hamiltonian dynamics to a predictive coarse-grained hydrodynamic propagator.