Closure–Residue Filtration and Non-Markovian Shadows: A Finite Generative Principle for Non-Exhaustive Observational Closure

We introduce a finite-dimensional closure–residue filtration for quantum dynamics relative to a chosen observational algebra. Let A_N = M_N(C), let D_N be the diagonal algebra, and let Δ_D and Q = I − Δ_D denote the diagonal and off-diagonal projections. For a finite dynamics Φ: A_N → A_N, the closure filtration M_k(D_N) records the linear expansion generated from the observational algebra, while H_k = Q M_k(D_N) records the hidden off-diagonal sector generated by the dynamical non-invariance of D_N. For every depth k, one has the exact decomposition M_k(D_N) = D_N ⊕ H_k, with H_k equal to the kernel of the restricted observational projection. Thus the observational projection closes exactly onto D_N, but need not exhaust the generated finite dynamics; we call this observational non-exhaustivity. The quadratic residue filtration R_k detects diagonal traceless returns of hidden off-diagonal directions through commutator residues and, together with linear closure growth, defines the Closure–Residue Profile CRP(Φ, D_N) = {(g_k, q_k)}. We prove Flow–Residue Factorization, identifying diagonal quadratic residues with divergences of hidden Hadamard flows, and Residue–Stabilizer Duality, identifying residue growth with the loss of silent diagonal stabilizers. For the dynamically generated support graph G_k, one always has R_k ⊆ B(G_k). In the graph-saturated regime, dim R_k = N − c(G_k), and at consecutive saturated depths q_k = c(G_{k−1}) − c(G_k). Thus residue growth counts mergers of hidden connected components, while failure of saturation can record dynamical obstructions such as symmetries, conservation laws, selection rules, block decompositions, or dark sectors. The construction is entirely finite-dimensional. It isolates a mechanism in which closure succeeds while residue survives. The term non-Markovian shadow is used only for this structural persistence and does not, by itself, assert CP-indivisibility or a specific memory kernel.