Standing Algebra Σᴿ: A Domain-Agnostic Autonomy-Preserving Update Operator

Standing Algebra (Σᴿ) introduces a many‑sorted, first‑order algebraic framework for constraining state updates in multi‑agent systems so that autonomy, non‑domination, and standing symmetry are preserved under all legitimate operations. The framework formalizes agents as elements of a typed universe with associated standing (σ), capacity (cap), and dependency degree (deg), and defines a signature of operations and predicates that capture structural safety constraints. The core contribution is the Legitimate Envelope Theorem, which proves that for every admissible endofunction F:U→UF: U \to UF:U→U over agents, there exists a canonical legitimate envelope LFL_FLF that is: standing‑monotonic (no decreases in σ), successor‑consistent (standing may only remain constant or increment by +1), class‑uniform over STC‑5 standing/capacity equivalence classes, idempotent (rerunning LFL_FLF has no further effect), and minimal in the σ‑order among all legitimate operations that satisfy the same increment signature as FFF. This result yields a closure operator on proposed updates, turning arbitrary or unsafe state transitions into the closest safe version allowed by the axioms. The collection of envelopes, modulo increment signatures, forms a join‑semilattice under class‑wise OR, providing an algebra of safe update policies suitable for multi‑agent governance, AI action firewalls, and autonomy‑preserving coordination systems. The paper includes: A fully specified many‑sorted algebraic signature for agents, standing, capacity, and dependency relations Tier‑1 and Tier‑2 axiom systems governing admissibility, standing monotonicity (ALRP), capacity bounds (CIA), anti‑asymmetric dependencies (NRPP), class‑uniformity constraints (STC‑5), drift bounds, and repair requirements A constructive model demonstrating consistency of the axiom system Independence proofs for all axioms Proofs of the Legitimate Envelope Theorem and the resulting semilattice structure A practical interpretation of Σᴿ as a validator/normalizer layer for multi‑agent systems and AI decision pipelines Σᴿ is intended as a domain‑agnostic, structurally grounded framework for ensuring that updates in multi‑agent environments cannot decrease autonomy, create unfair asymmetries, or accumulate harmful drift. The system provides mathematically defined guarantees for safety‑preserving updates independent of any specific reward function, preference model, or optimization objective. Context This paper is one component of a larger research system on standing, legitimacy, and non-domination in coordinating pluralist systems. The conceptual foundation is developed in On Relevance: The Social Physics of Pluralism, alongside companion works such as Why Being Right Isn't Enough, an analytic essay Symmetry, Standing and Structural Stability (Episteme Submission), and an applied quick-start guide with flowchart. Building on that foundation, the present paper formalizes the structural constraints of autonomy-preserving updates in the form of Standing Algebra (Σᴿ). This includes core axioms, equations, and the definition of the Legitimate Envelope LF, which repairs arbitrary proposed changes into the closest permissible update. To make the formalism operational, this record also provides a Python Reference Validator, a JSON test suite, and practical tools for exploring the behavior of Σᴿ in both single-agent and batch multi-agent settings. Together these components form an integrated system:Conceptual Foundations -> Formal Framework -> Executable Semantics -> Practical Tools