  Hydrodynamic 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.
