Minimal Structures for Consistent Evolution

Minimal Structures for Consistent Evolution is the first volume of the Foundations of Physical Evolution research monograph series. It develops a minimal mathematical framework for representing evolution before introducing metric time, geometry, probability, dynamics, or physical interpretation. The construction begins with a directed multigraph of admissible elementary transitions. Its free category encodes finite compositional histories, while reachability induces a preorder on the state space. Mutual reachability identifies recurrently connected states, and the corresponding quotient yields a partially ordered set of structural progression classes. This sequence is formulated through free-category construction, reachability, strongly connected components, thin-category reflection, and posetal reflection. The monograph examines deterministic, nondeterministic, reversible, recurrent, and quotient-progressive systems; establishes the relevant universal properties and structural results; and provides finite examples, countermodels, realization criteria, and comparisons with transition systems, quivers, causal sets, dynamical systems, and process theories. Its principal contribution is a dependency-controlled synthesis showing which mathematical structures are necessary—and which are not yet justified—when passing from elementary transitions to compositional evolution and structural order. The framework is intended as a theory-independent foundation for subsequent mathematical and physical models of evolution. This is a prepublication research monograph and has not undergone external peer review.