Mode Decomposition for Homogeneous Symmetric Operators

The paper’s adaptive p-flow sets ψ_t = G(ψ) with G(ψ) = λ_ψ^{-1} Δ_p ψ and explicitly defines λ_ψ^{-1} = −⟨Δ_pψ, ψ⟩/||Δ_pψ||^2; with this choice one indeed gets d/dt J_p(ψ) = −p J_p(ψ) and exponential energy decay, so the model’s claim of a mismatch stems from swapping λ with its inverse. Items (1) and the decay-on-an-eigenfunction ray in (2) are therefore correct under the paper’s G. However, the paper’s proof of item (3) (“every element from f decays exponentially”) relies on a loose ‘triangle inequality’ step and lacks precise hypotheses on what ‘elements’ are, so (3) as stated is not rigorously justified. The model’s fix uses the same effective multiplier as the paper (just a naming inversion) and does not rigorously establish (3) either. Hence, both are incomplete: the paper for (3) and a minor sign slip in (2), and the model for misreading λ and not resolving (3).