Reaching compressible flow has usually forced lattice Boltzmann methods to abandon the compact stencil or the strict locality that make them efficient. We give up neither, solving the compressible Navier--Stokes--Fourier equations with a scheme that transports only the conserved mass, momentum and energy. The viscous stress and heat flux depend on gradients of the velocity and temperature. To capture them, existing compressible schemes go beyond a single lattice of the conserved fields. They enlarge the velocity set, carry the stress and heat flux as extra transported fields, place the energy on a separate grid, or give up exact streaming for an off-lattice advection. We instead recover the gradients from the non-equilibrium part of the distributions already present, inside the collision. No field beyond the conserved state is transported, no neighbour is read, streaming stays exact, and every computation runs in single precision. The method carries five fields where a transported-flux scheme carries fourteen, at a fraction of the memory and $4.9$ times the throughput. Verified against the exact Sod and Becker shock solutions and a supersonic Taylor--Green vortex, it tracks the direct-numerical-simulation reference on the kinetic energy and stays within the reference-solver spread on both dissipation rates.

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