For any regularity exponent $\beta<\frac 12$, we construct
non-conservative weak solutions to the 3D incompressible Euler
equations in the class $C^0_t (H^{\beta} \cap
L^{\frac{1}{(1-2\beta)}})$. By interpolation, such solutions
belong to $C^0_tB^{s}_{3,\infty}$ for $s$ approaching $\frac 13$ as
$\beta$ approaches $\frac 12$. Hence this result provides
a new proof of the flexible side of the Onsager
conjecture, which is independent from that of Isett. Of
equal importance is that the intermittent nature of our solutions
matches that of turbulent flows, which are observed to possess an
$L^2$-based regularity index exceeding $\frac 13$. The proof
employs an intermittent convex integration scheme for the 3D
incompressible Euler equations. We employ a scheme with
higher-order Reynolds stresses, which are corrected via a
combinatorial placement of intermittent pipe flows of optimal
relative intermittency.
