It's a little easier to see what's happening if you fully write out the central flow:

    -1/η * dw/dt = ∇L - ∇S * ⟨∇L, ∇S⟩/‖∇S‖²

We're projecting the loss gradient onto the sharpness gradient, and subtracting it off. If you didn't read the article, the sharpness S is the sum of the eigenvalues of the Hessian of the loss that are larger than 2/η, a measure of how unstable the learning dynamics are.

This is almost Sobolev preconditioning:

    -1/η * dw/dt = ∇L - ∇S = ∇(I - Δ)L

where this time S is the sum of all the eigenvalues (so, the Laplacian of L).