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.摘要 本研究针对高维有向网络中的交互路径组合缩放问题展开探究。尽管经典稳定性分析聚焦于邻接算子的谱特性，但系统性风险实则由交互序列的结构与重数共同决定。 本文提出组合包络（combinatorial envelope）这一概念，用以表征由N个耦合组件构成的系统中可区分交互排序的最大数量。在稠密耦合假设下，我们证明可允许的交互排列数量的上界为N!。 我们将该上界整合至非正规稳定性框架中，结果表明放大机制随系统规模呈超线性缩放，在极端场景下甚至可趋近阶乘增长。这为由组合爆炸而非谱发散驱动的不稳定性提供了结构化判定准则。