Equilibrium Validation for Triblock Copolymers via Inverse Norm Bounds for Fourth-Order Elliptic Operators

The paper splits L using a finite-dimensional projection and an infinite tail, builds an approximate inverse S = diag(L_N^{-1}, T), and proves ||I − SL|| ≤ τ := sqrt(A^2 + B^2) with explicit A, B (Lemma 4.8), yielding ||L^{-1}|| ≤ max(K_N, C_T)/(1 − τ) when τ < 1 (via Proposition 4.2 and Theorem 4.1). The candidate solution performs the same block-splitting and estimates but phrases it as preconditioning by M = diag(L_N, Q_N(−βΔ^2)) and bounding ||M^{-1}E||; since S = M^{-1}, the two arguments are equivalent. The constants A and B in the candidate match those in Lemma 4.8 exactly, including the key assumption a_{kj} ∈ P_N U_j used to annihilate the low-low coupling of the l_{kj}-terms and the same Sobolev/projection tail and Banach algebra bounds (Lemmas 3.2–3.4) used by the paper. The final Neumann-series invertibility condition and inverse norm estimate coincide with the paper’s conclusion. See the decomposition identity and I − SL representation (Lemma 4.7 and Eq. (57)), the explicit A, B bounds (Lemma 4.8), and the concluding inverse estimate (end of Section 4) in the paper . The supporting Sobolev, Laplacian isometry, and projection tail inequalities appear in Section 3 (Lemmas 3.2–3.4) .