A Quantitative Framework for Scientific Structure

This paper develops a formal representation of scientific theories as structured dynamical systems. A theory is modelled as a tuple comprising ontic and epistemic state manifolds, a dynamical operator, symmetry structure, regime specification, a scale/coarse-graining operator, and parameter space. From this representation we derive a minimal, non-redundant basis of five structural invariants measuring empirical correspondence, internal dynamical consistency, regime adequacy, symmetry realization, and multiscale coherence. Under admissibility conditions ensuring smooth structural variation, these invariants are unified in a bounded completeness functional defined on differentiable theory-space. We prove boundedness, independence, almost-everywhere differentiability, invariance under admissible reparameterization, and monotonicity under refinement operations. The framework shows that empirical adequacy alone does not guarantee structural completeness and provides a domain-neutral metric for evaluating scientific theories as structured mathematical objects rather than merely as sets of predictions. The subsequent paper establishes a Lyapunov-type interpretation governing rational theory refinement.