Minimal `2 Norm Discrete Multiplier Method

Both the paper and the candidate solution use the same core argument: define f^τ_MN = f^τ − (Λ^τ)^+(Λ^τ f^τ + ∂^τ_t ψ) and rely on full-row-rank persistence of Λ^τ and the pseudoinverse identity Λ^τ(Λ^τ)^+ = I_m to obtain Λ^τ f^τ_MN + ∂^τ_t ψ = 0 identically, which ensures exact discrete conservation (via the discrete chain rule and constant-compatibility) and preserves order-q consistency; see the MN-DMM definition and Theorem 3.4 in the paper, and its supporting lemmas on rank persistence and bounded pseudoinverses . The candidate also flags a sign typo in (3.7a): the natural identity from the discrete chain rule is Λ^τ D^τ_t x − D^τ_t ψ + ∂^τ_t ψ = 0, whereas the theorem statement shows a minus sign before ∂^τ_t ψ; the correct sign is implied by (2.7a) in the paper itself . Their consistency estimate follows the paper’s inequalities (3.8)–(3.10) verbatim .