AbstractHydrodynamic projections, the projection onto conserved charges representing ballistic propagation of fluid waves, give exact transport results in many-body systems, such as the exact Drude weights. Focussing one one-dimensional systems, I show that this principle can be extended beyond the Euler scale, in particular to the diffusive and superdiffusive scales. By hydrodynamic reduction, Hilbert spaces of observables are constructed that generalise the standard space of conserved densities and describe the finer scales of hydrodynamics. The Green–Kubo formula for the Onsager matrix has a natural expression within the diffusive space. This space is associated with quadratically extensive charges, and projections onto any such charge give generic lower bounds for diffusion. In particular, bilinear expressions in linearly extensive charges lead to explicit diffusion lower bounds calculable from the thermodynamics, and applicable for instance to generic momentum-conserving one-dimensional systems. Bilinear charges are interpreted as covariant derivatives on the manifold of maximal entropy states, and represent the contribution to diffusion from scattering of ballistic waves. An analysis of fractionally extensive charges, combined with clustering properties from the superdiffusion phenomenology, gives lower bounds for superdiffusion exponents. These bounds reproduce the predictions of nonlinear fluctuating hydrodynamics, including the Kardar–Parisi–Zhang exponent 2/3 for sound-like modes, the Levy-distribution exponent 3/5 for heat-like modes, and the full Fibonacci sequence.