LMC-induced Perturbations in the Milky Way Halo II: Bridging Field-level Inference and Summary-level Simulation-Based Inference

The gravitational interaction between the Milky Way (MW) and the Large Magellanic Cloud (LMC) drives the outer halo into dynamical disequilibrium, imprinting the masses and structural parameters of both galaxies onto the 6D phase-space distribution of halo tracers. This signal has been characterised with summary statistics ranging from low-order velocity moments to basis function expansions, yet how much information these summaries discard, and whether they are complementary, remains unclear. We address these questions by comparing a field-level likelihood benchmark with physically interpretable summaries for constraining (M MW, M LMC, c, q), where c and q are the MW halo concentration and flattening. A Conditional Flow Matching (CFM) model trained on the HaloDance N-body suite provides an exact likelihood at a held-out fiducial point; for 5,000 tracers in 30 -- 120 kpc it tightens marginal constraints by factors of 2.5 -- 9.9 over an all-sky velocity-moment forecast. We then expand the halo density and velocity fields in a multipole basis-function expansion (BFE) and compress the coefficients with the Massive Optimised Parameter Estimation and Data compression (MOPED) algorithm into four parameter-sensitive summaries that preserve their Fisher information. A variational mutual-information analysis shows that the BFE+MOPED summaries and the velocity moments are complementary, so we combine them into a joint 19-dimensional vector as our primary inference pipeline: it tightens the marginal constraints by up to 15 per cent over BFE+MOPED alone and by 30 -- 71 per cent over velocity moments alone, reaching within a factor of 1.3 -- 2.9 of the field-level benchmark. We thus establish a physically interpretable summary-level route to MW--LMC inference alongside the field-level benchmark that bounds its information content.