Construct accurate multi-continuum micromorphic homogenisations in multi-D space-time with computer algebra

The paper’s Proposition 20 (Forward Theory) states exactly the three items under audit—existence of an M-mode centre invariant manifold tangent to M_X, exponential attraction at rate ≈ e^{-β_ℓ t}, and residual→error equivalence under the gap N+1<β_ℓ/β_L—within the embedded PDE (4.4) framework and under Assumptions 15, 18 and Definition 19.1; its proof is by reduction to X-local analysis (Bunder & Roberts 2021) and application of existing invariant–manifold theorems (Haragus & Iooss 2011; Aulbach & Wanner 2000; Potzsche & Rasmussen 2006) with the spatial-gradient remainder controlled to order N+1, and with a bounded-L proviso for non-autonomous cases . The candidate solution reconstructs the same three conclusions via a standard semigroup–based centre-manifold construction (graph transform/Lyapunov–Perron), explicit dichotomy estimates, and a residual→error contraction argument on the invariant graph, including the precise gap constraint N+1<β_ℓ/β_L and the non-autonomous bounded-L condition—fully consistent with the paper’s setting (4.4) and reduction to the macroscale model (4.2) . A minor divergence is that the model claims broad coverage of fractional-in-time cases via Volterra resolvent families, whereas the paper explicitly cautions that general invariant-manifold theory for nonlinear, non-autonomous fractional systems (fde) is not currently available and suggests a backward-theory route instead; thus, the model slightly overreaches in that specific regime . Net: for the semigroup settings asserted in Proposition 20 (α=1, and α=2 via first-order lifting; non-autonomous with bounded L), both arguments are correct; they follow different but compatible proof routes, and they agree on the essential gap/residual assumptions (Definition 19) .