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We introduce $w_i{f}_i$ensembles as a novel framework to obtain asymptotic frequentist uncertainties on density ratios, with a particular focus on neural ratio estimation in the context of high-energy physics. When the density ratio of interest is a likelihood ratio conditioned on parameters, $w_i{f}_i$ensembles can be used to perform simulation-based inference on those parameters. After training the basis functions $f_i\left(x\right)$, uncertainties on the weights $w_i$can be straightforwardly propagated to the estimated parameters without requiring extraneous bootstraps. To demonstrate this approach, we present an application in quantum chromodynamics at the Large Hadron Collider, using $w_i{f}_i$ensembles to estimate the likelihood ratio between generated quark and gluon jets. We use this learned likelihood ratio to estimate the quark fraction in a synthetic mixed quark/gluon sample, showing that the resultant uncertainties empirically satisfy the desired coverage properties. 

Abstract We lay out a comprehensive physics case for a future high-energy muon collider, exploring a range of collision energies (from 1 to 100 TeV) and luminosities. We highlight the advantages of such a collider over proposed alternatives. We show how one can leverage both the point-like nature of the muons themselves as well as the cloud of electroweak radiation that surrounds the beam to blur the dichotomy between energy and precision in the search for new physics. The physics case is buttressed by a range of studies with applications to electroweak symmetry breaking, dark matter, and the naturalness of the weak scale. Furthermore, we make sharp connections with complementary experiments that are probing new physics effects using electric dipole moments, flavor violation, and gravitational waves. An extensive appendix provides cross section predictions as a function of the center-of-mass energy for many canonical simplified models.