Structurally Separated DAG Learning with Multi-Scale Normalized Closure

Continuous DAG learners typically optimize a single weighted adjacency matrix, forcing each entry to represent both direct-edge support and signed effect magnitude. As a result, data fitting, sparsity, acyclicity, and any auxiliary structural information are all imposed on the same variables, even when the available information concerns higher-order relations such as reachability or ancestral order. We propose a structure-separated continuous DAG-learning framework that gives graph support its own soft representation. The effective adjacency is ( A=(W f(M)\circ G(Z))\circ(1-I), \qquad W f(M)=\beta\tanh(M), ) where (W_f(M)) is a bounded signed-magnitude branch and (G(Z)) is a structural gate. Data fitting, sparsity, and the DAGMA barrier are evaluated on (A), while cycle and relation constraints are applied to closure scores computed from (G). The bounded magnitude branch prevents strong effective edges from being hidden behind vanishing structural gates, making closure-side structural regularization meaningful. On this separated structural gate, we introduce SMNC, a scan-efficient multi-scale normalized closure module. SMNC computes dyadic closure scores over (G) and normalizes each scale by its path-length budget, producing scale-normalized matrix-valued logits for self-reachability and off-diagonal reachability. This design addresses a key limitation of unnormalized product-style closure scores, whose path evidence scales as (p^L) on a length-(L) path with gate value (p), causing long-cycle signals to decay exponentially. At the appropriate covering scale, SMNC routes gradients through an active length-(L) cycle with a polynomial (1/L)-type scale factor; in the all-scale soft readout used in practice, this should be interpreted as a local active-scale effect. The same relation space supports diagonal self-reachability losses for cycle suppression and masked off-diagonal losses for optional ancestry, backward-exclusion, and incomparability cues, without treating positive reachability labels as direct-edge labels. Across linear and nonlinear SEM benchmarks, with ablations over factorization and closure normalization, we show that structure-separated SMNC improves DAG recovery and provides a principled closure-space interface for higher-order structural supervision.