Version 1.0 of the Continuum Operator Axiom System presents a unified, operator‑centric formal framework for modeling reversible, identity‑preserving state evolution under admissibility constraints. The system is designed for domains where transformations must preserve structural invariants and support bidirectional reasoning, including reversible computation, graph rewriting, symbolic term rewriting, and conservation‑law–constrained dynamics. The framework separates and formalizes five core components: • syntax — raw operators and rewrite rules • semantics — a partial lifecycle operator governing state evolution • admissibility — predicates defining the valid state space • identity — equivalence‑class descriptors preserved by admissible operators • reversibility — designated inverse operators ensuring local invertibility The axiom system introduces locally invertible, identity‑preserving, and globally invertible operator classes, together with a designated operational set that defines the admissible dynamics. Lemmas establish identity preservation, two‑step reversibility, and path reversal, culminating in a general multi‑step reversibility theorem for all finite admissible evolutions. Two fully worked example models instantiate the abstract framework: Finite labelled graphs, using interface‑preserving graph rewrite rules Symbolic term rewriting, using typed term patterns and substitution‑based rewriting Both models satisfy the axioms and demonstrate how the abstract system applies across structurally different domains. A model‑based relative consistency argument is provided: by interpreting the abstract system over either finite labelled graphs or well‑typed symbolic terms, all definitions, axioms, and lemmas are satisfied. This establishes that the theory has at least one concrete model and is therefore relatively consistent. This document is intended as a precise, self‑contained, appendix‑style formal specification. It makes no claim about global consistency or the existence of nontrivial models beyond those exhibited. Keywords:reversible systems; reversible computation; identity‑preserving operators; operator semantics; partial evolution; admissible states; equivalence invariants; designated inverses; local invertibility; global invertibility; structural reversibility; lifecycle operator; state transition systems; rewrite systems; graph rewriting; symbolic term rewriting; interface‑preserving rewriting; invariant‑preserving dynamics; model‑based consistency; relative consistency; formal specification; axiomatic framework; algebraic rewriting; labelled graphs; term rewriting systems; structural invariants; bidirectional transformations; operational semantics; equivalence‑class identity; reversible graph transformations; reversible term rewriting; admissibility predicates; partial operators; invertible rewrite rules; formal methods; theoretical computer science; state evolution frameworks; reversible dynamics; identity continuity; multi‑step reversibility; formal appendix‑style specification
Matthew Arthur Carlo (Tue,) studied this question.