  In this work, we develop a wavelet-inspired, $L^3$-based convex integration
framework for constructing weak solutions to the three-dimensional
incompressible Euler equations. The main innovations include a new multi-scale
building block, which we call an intermittent Mikado bundle; a wavelet-inspired
inductive set-up which includes assumptions on spatial and temporal support, in
addition to $L^p$ and pointwise estimates for Eulerian and Lagrangian
derivatives; and sharp decoupling lemmas, inverse divergence estimates, and
space-frequency localization technology which is well-adapted to functions
satisfying $L^p$ estimates for $p$ other than $1$, $2$, or $\infty$. We develop
these tools in the context of the Euler-Reynolds system, enabling us to give
both a new proof of the intermittent Onsager theorem (An Intermittent Onsager
Theorem, Inventiones Mathematicae, (2023), 233) in this paper, and a proof of
the $L^3$-based strong Onsager conjecture in a companion paper
(arXiv:2305.18509).
