Representation Transitions Reveal Predictive Structure in Complex Systems: A Trajectory-Level Reconstruction in a Critical System

Abstract

Trajectories in complex systems often contain structure that is not captured by the population-averaged, scalar statistics traditionally used to describe them. Here we reconstruct, through independent numerical re-simulation rather than literature review, a systematic search across representational forms - from single scalar statistics, through two independent discrete multi-feature classification attempts, to a trajectory-level structural representation and a diagnostic surrogate test, to a continuous low-dimensional compression - applied to the same underlying trajectory ensemble in a critical complex system. Predictive failure is not explained by insufficient correlation: one discrete representation&#39;s own strongest individual feature correlates with the target more strongly ($r=0.588$) than the eventual successful representation&#39;s own headline statistic ($r=0.540$), yet fails to separate the ensemble&#39;s two most consequential cases - while the continuous representation succeeds, robustly, across eight independent scale-removal folds ($r=0.677$-$0.717$, all $p<0.001$). The determining factor was not correlation strength but whether the representation preserved the structure relevant to prediction. These results indicate that the choice of representation determines whether predictive structure in complex systems becomes accessible - a scientific variable in its own right, not a downstream analysis choice.