We propose and demonstrate an adaptive scale-similar closure approach that can represent subgrid terms accurately and stably even at or near the smallest resolved scales of a simulation. This dynamic approach is based on scale similarity and generalized representations of subgrid terms from the complete and minimal tensor representation theory of Smith (1971). The tensor polynomial coefficients in the generalized representation are adapted to the local turbulence state by solving a local system identification problem at a test-filter scale. The resulting test-scale coefficients are rescaled to the LES-scale and used in the generalized representation to evaluate the local subgrid term. In this Part 1 paper, subgrid stress and production fields from this adaptive scale-similar closure are seen to be nearly indistinguishable from corresponding true fields, and to be far more accurate than corresponding results from traditional subgrid closures. Even when implemented in a low-dissipation pseudo-spectral code, this new approach to subgrid closure is stable with only minor added dissipation, and with only slightly more dissipation shows E(k) ∼ k−5/3 scaling that extends to the smallest resolved scales. Results from a posteriori tests show greatly improved accuracy in inner-scale statistics compared to traditional closure with a prescribed subgrid model. Evaluating the subgrid stress takes only about three times longer than by traditional closure with the dynamic Smagorinsky model. For LES in which accuracy is needed across all simulated scales, including the smallest resolved scales, this slightly longer time may be acceptable, and this new closure approach can provide stable simulations while representing subgrid terms accurately across all resolved scales.