I will present a first attempt at a foundation of inference for the assessment of semiparametric and other high-dimensional models, seeking reconciliation with the low-dimensional foundations and replacing out-of-sample prediction error as a criterion for semiparametric model assessment by a type of within-sample prediction error. The work highlights the possibility, in some contexts, of avoiding the usual semiparametric considerations, typically requiring estimation of unparametrised nuisance components through kernel smoothing or basis expansion, with the associated difficulties of tuning-parameter choice that blur the distinction between estimation and model assessment. A key aspect is the inducement of replication under the postulated model, the route to which is context-specific, as illustrated through examples, but based on foundational principles and structural properties of the model.