Second-Order Organization as Constraint Geometry Over Admissible Trajectories

Many biological and physical systems exhibit behavior that depends not only on instantaneous state, but on the ordering and history of interactions. Such effects—commonly described as hysteresis, gating, or sequence dependence—are typically modeled piecemeal, without a unified organizational framework. This paper introduces a formal notion of second-order organization, defined as a constraint geometry over admissible trajectories rather than stored states or representations. A second-order space, Σ₂(E), is defined as the set of trajectories viable under a specified embodiment, providing explicit criteria for when trajectory history becomes causally relevant and state-based (Markovian) descriptions fail. The framework identifies noncommutativity of transformations and holonomy as operational diagnostics of order relevance, and distinguishes proto-second-order systems—such as individual molecules and molecular machines—from systems that actively stabilize organizational modes through sustained energy throughput. A minimal computational toy model demonstrates how admissibility, ordering sensitivity, and maintenance costs can be quantified without invoking new physics or violating conservation laws. This work is pre-empirical and methodological in scope. It introduces no new biological claims, but provides a disciplined framework for analyzing history-dependent organization in biological, cognitive, and engineered systems where trajectories, rather than instantaneous states, are the correct objects of description.