Quadratic Failure Scaling in Pairwise-Coupled Systems

This note proves that quadratic failure scaling in pairwise-coupled systems is not an empirical regularity but a necessary consequence of coupling topology. Two layers of result are established. Static layer. For any dense pairwise-coupled family, the number of potentially interacting failure channels scales as Θ(n²) while node-local resolution scales at most linearly — a combinatorial asymmetry embedded in the graph structure itself (Theorems 1–3, Corollary 3.1). Dynamic layer. For the complete graph Kₙ, a two-round independent cascade process on the line graph produces expected cascade size Θ(n²) from a single seed edge at any fixed transmission probability q > 0 (Theorem 4A). This quadratic scaling is robust to correlation: under arbitrary cross-route dependence with independent within-route transmission, the expected cascade size remains Θ(n²) with a weakened constant (Theorem 4A′). In the opposite regime, a branching-style upper bound shows that when qₙ ≤ (1−ε)/(2n−5), the expected total cascade remains O(1) (Proposition 4B). Together, these identify qₙ ~ 1/n as the natural phase transition scale on Kₙ. Substrate independence. Every result is substrate-independent within its own hypothesis class: the proofs depend only on the abstract combinatorial and probabilistic hypotheses stated in each theorem, not on the physical nature of the nodes (Theorem 6). Application to any concrete domain requires independent verification that the domain satisfies the relevant hypothesis class. Generalization. Dense k-wise coupling produces a failure surface of order Θ(nᵏ) (Proposition 5). The paper includes explicit falsifiability conditions for every result and identifies six open problems, including the sharp threshold on Kₙ, extension to dense non-complete graphs, within-route correlation boundaries, state-dependent transmission, empirical pairwise dominance, and prospective predictive testing. A companion Logic Key (included as supplementary material) defines every symbol, operator, and notational convention used in the paper. This paper provides the mathematical foundation for the Our Gods Research Program's substrate-independent analysis of correlated failure, previously demonstrated empirically across orbital debris (Kessler Syndrome), autonomous vehicle fleets, and computational infrastructure. Keywords quadratic failure scaling pairwise coupling cascade process correlated failure substrate independence graph theory line graph phase transition combinatorial asymmetry Kessler Syndrome coupled systems Related identifiers 10.5281/zenodo.17835722 — Our Gods Haven't Fallen, Yet (orbital debris / N² foundation) 10.5281/zenodo.17851041 — Our Gods Haven't Crashed, Yet (autonomous vehicle fleets) 10.5281/zenodo.17925255 — Our Gods Haven't Computed, Yet (computational infrastructure) 10.5281/zenodo.17926796 — An Epi-Phenomenological Series in Thrice (meta-structural synthesis) 10.5281/zenodo.17942325 — Five Beasts: One Resurrection (cosmological extension) All related identifiers: relationship = "is supplemented by this upload" or "this upload supplements" License Creative Commons Attribution 4.0 International (CC BY 4.0) Part of the Our Gods Research Program — historiesrawtruth.com | tonyoconnor17.substack.com