Evolutionary Mechanics as a Minimal Generative Operational Apparatus: from the Triple (R, D, id) to Categories, Algebras and Classical Slices

The present text records a minimal operational model of “evolutionary mechanIcs” and its place within the programme of generative mathematics. The formal layer of the model is implemented in the Coq file EM-DS_K_V_3-3.v: states consist of a collection of atoms, a Raw/Legit status, a name counter and a map of representations, while evolution is generated by three operators of replication, deletion and fixation. We separate four logical statuses of statements: definitions and axioms, results with a completed proof in Coq, conditional mathematical inferences, and research hypotheses. This makes it possible to connect a concrete discrete model with operational set theory, generative algebra and generative category theory without presenting projected bridges as already proven results. The principal result of this document is methodological in character. Evolutionary mechanics provides a verifiable prototype of a mechanism for generating, fixating and normalising configurations. Precise additional premises are formulated for the passage to generative objects (M0, Conv, dR) and to classical structures. In particular, the generative versions of the principles of Noether, Galois and duality are treated as hypotheses consistent with the classical limit and as a programme for further formalisation, rather than as proven consequences of the minimal model.