Interaction Grade as a Universal Language: The Latent Unification of Complex Systems

We survey the transdisciplinary connections enabled by the Latent framework's grade decomposition, establishing exact correspondences between complex systems in physics, game theory, biology, and machine learning. We identify the following structural isomorphisms: (1) partition function = game equilibrium (Nash-Boltzmann duality); (2) cluster expansion = ANOVA decomposition; (3) renormalization group = grade truncation; (4) phase transition = complexity transition (\(\rho = 1\)); (5) mean-field theory = mean-field games = grade-1 truncation; (6) Ising model = congestion game; (7) fitness landscape = payoff function (evolutionary game theory); (8) GAN = zero-sum game with spectral collapse at \(\rho \to 1\); (9) entanglement order = interaction grade (quantum games); (10) protein folding cooperativity = grade-3+ energy contributions. For each isomorphism, we state the mathematical correspondence, identify transferred theorems, and propose new predictions. The Latent Number \(\rho\) emerges as a universal diagnostic: \(\rho \gg 1\) signals pairwise-dominated systems across all domains; \(\rho = 1\) marks critical phenomena universally; \(\rho < 1\) identifies fundamentally irreducible complexity.