Cross-Substrate Evidence for Quadratic Scaling Between Relational Complexity and Energetic Residue

This paper reports a substrate-independent scaling signature relating energetic cost to relational density across three physically unrelated dynamical systems. Relational Complexity (C) is defined as the effective density of actively tracked pairwise interactions within a bounded system; Informational Residue (R) denotes the measurable energetic cost associated with maintaining bounded stability under increasing relational density. The scaling relation R = αCᵖ is evaluated using operational proxies appropriate to each substrate: a Random Field Ising Model (RFIM) exhibiting disorder-driven avalanche dynamics, an open quantum transport model inspired by the Fenna–Matthews–Olson (FMO) photosynthetic complex, and neuromorphic silicon hardware (Intel Loihi 2). Across all three regimes, log–log regression yields exponents converging near p ≈ 2, despite radically distinct underlying physical mechanisms. Negative-control experiments restricting effective interaction windows systematically reduce the observed exponent toward linear scaling, supporting a causal interpretation. The convergence of scaling behavior across substrates suggests that quadratic residue growth may represent a geometric consequence of bounded stabilization under combinatorially expanding relational density. Simulation code and data are included in the accompanying archive. Take advantage of the python toy and reproduce the results, make improvements but I especially encourage researchers to try analysis of of Markovian and non-Markovian substrates to see how the alignment to the present evidence of a pattern for a general constraint will present as consistent.