Combinatorial Envelopes: Factorial Bounds for Interaction Complexity in Densely Coupled Networks"

Abstract We investigate combinatorial scaling of interaction pathways in high-dimensional directed networks. While classical stability analysis focuses on spectral properties of adjacency operators, systemic risk is governed by the structure and multiplicity of interaction sequences. We introduce the concept of a combinatorial envelope, representing the maximal number of distinguishable interaction orderings in a system of N coupled components. Under dense coupling assumptions, we show that the number of admissible interaction permutations is bounded above by N!. We incorporate this bound into a non-normal stability framework, demonstrating that amplification mechanisms scale superlinearly with system size and may approach factorial growth in extreme regimes. This provides a structural criterion for instability driven by combinatorial explosion rather than spectral divergence.