We construct a microstructure-based constitutive model that successfully predicts experimental rheology signatures that no other model has previously described. The experimental observations are the low-dimensional descriptions of asymptotically nonlinear oscillatory shear [Ewoldt and Bharadwaj, Rheol. Acta 52, 201-209 (2013)], also known as medium-amplitude oscillatory shear, characterized by four frequency-dependent material measures: [e1](omega), [e(3)](omega), [v(1)](omega) and [v(3)](omega). These slightly nonlinear rheological measurements are the systematic step beyond linear viscoelastic characterization. The material is a transiently crosslinked polymeric hydrogel of aqueous polyvinyl alcohol cross-linked by sodium tetraborate (borax) [Bharadwaj and Ewoldt, J. Rheol. 59, 557-592 (2015)], which shows nonlinear elastic stiffening inferred from [e(1)](omega) > 0. Here, we hypothesize that the appropriate physical model is a transient network of strain-stiffening elastic elements. We rationalize that all nonlinearities are driven by the instantaneous stretch magnitude Q between junctions, either through strain-stiffening of network elements or through deformation-assisted network structuring. These two physical aspects are embedded into a single nonlinear parameter that successfully captures both elastic energy storage ([e(1)](omega) and [e(3)](omega)]) and viscous energy dissipation ([v(1)](omega) and [v(3)](omega)), including frequency-dependent sign changes. Analytical results are derived for all four asymptotic nonlinearities. The quantitative agreement provides fit parameters that are related to molecular features and network architecture. While the work here is focused on a specific polymeric system, it represents the broad potential contribution of asymptotic, leading-order nonlinearities to enable structure-rheology insight, constitutive model development, and model selection for soft materials in general. (C) 2017 The Society of Rheology.