Version Notes — Standing Algebra (Σᴿ) v2 (See StandingAlgebraSigmaR (beta) ) This version introduces several substantive formal extensions and interface‑level clarifications strengthening Σᴿ’s ability to reason about autonomy preservation, risk allocation, and emergency intervention, while preserving first‑order conservativity. New Formal Features Emergency Envelope (Σᴿᴱ). An optional Tier‑2 extension formally modeling bounded, least‑autonomy‑reducing emergency intervention. Emergency envelopes are time‑bounded, idempotent, anti‑ratcheting, and must terminate immediately once inaction or non‑coercive legitimacy becomes available. Autonomy‑Reducing Variables (ARVs). Introduction of structurally defined autonomy‑reduction predicates (AF, EX, FC, RS) enabling non‑psychological detection of autonomy loss through option‑set collapse, exit termination, and relevance impairment. Risk‑Bearer Indexing. Explicit identification of entities bearing structural risk under an operation, together with a no‑hidden‑risk‑transfer axiom enforcing full exposure of affected entities within validator pipelines. Divergence and Signal‑Blocking Extension (Σᴿ⁺). Optional divergence measures (Δ, Θ) and signal‑blocking predicates preventing consent laundering and misalignment masking while remaining conservative over core legitimacy constraints. Enablement Obligations. A new structural requirement ensuring that capacity‑gated, proxy‑dependent, or emergency‑stabilized regimes admit legitimate emancipatory routes that strictly grow capacity. Reflexive Coupling‑Debt Accounting. Optional coupling‑debt tensor assigning structural cost to unseverable dependency imposition and requiring dischargeability through repair or severability. Structural and Theoretical Clarifications Explicit Successor‑Realization Axioms (SSR, LSR). Formal guarantees of successor‑state existence supporting envelope construction and minimality theorems. Expanded Adapter‑Layer Specification. Formalized translation rules, structural truth‑filtering of autonomy claims, plural‑source aggregation, and scale‑invariant harm detection. Explicit Multigranularity Soundness Conditions. Formal harm‑reflection and envelope‑liftability properties ensuring safety and non‑overconstraint across observational scales. Conservativity and Compatibility All Tier‑3 axioms and Σᴿ⁺ extensions are conservative over the core Σᴿ system and may be omitted without affecting the soundness, consistency, or independence results of Σᴿ₁ and Σᴿ₂. No changes were made to Tier‑1 standing arithmetic or Tier‑2 legitimacy definitions. 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 UF: U→U over agents, there exists a canonical legitimate envelope LFLFLF 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 LFLFLF 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 In addition to the internal legitimacy envelope (LF) (LF) (LF) presented in this paper, Standing Algebra includes two further conceptual components that naturally belong to the same algebraic family but are not yet expanded here in full formal detail. These components are introduced in a separate companion document for clarity and to provide readers with an early sense of the broader structure to which the present work belongs. First, the Boundary Envelope Operator (BG) (BG) (BG) represents the algebraic mechanism used when pluralist‑constrained agents interact with agents or systems that do not share those constraints. Whereas LFLFLF governs legitimate updates among participants, BGBGBG governs cross‑boundary interactions such as those involving dissenters, non‑participants, adversarial structures, or non‑conscious computational systems. The boundary operator functions conceptually as a safety envelope for these interactions: preventing pluralist agents from exercising domination outward, while preventing harmful or autonomy‑collapsing updates from entering the pluralist domain. Its precise form depends on the philosophical conditions that define boundary legitimacy and is therefore left conceptual at this stage. Second, the Domination Singularity (DS) (DS) (DS) serves as the diagnostic marker for collapse conditions in which autonomy, standing, or relevance are overridden to such an extent that envelope‑style repair is no longer applicable. Unlike LFLFLF and BGBGBG, the domination singularity is not itself a repair operator but an absorbing endpoint indicating that a proposed or unfolding interaction has exited the admissible space of autonomy‑preserving updates. As with the boundary operator, the details of this singularity depend on philosophical distinctions concerning coercion, dependency, manipulation, and the loss of agency, and are therefore introduced conceptually rather than formally. Finally, the existence of LFLFLF, BGBGBG, and DSDSDS together makes possible several derived operators used for tasks such as routing updates to the correct envelope, distinguishing robustly safe outcomes from fragile ones, estimating a risk‑margin for proposed updates, identifying interaction patterns that never drift toward collapse, and producing minimal traces or proofs that a repaired update sits safely within its intended domain. These derived operators are not independent additions to the theory; they arise directly from the interaction of the internal envelope, boundary envelope, and collapse marker. Their complete formal specification depends on the same philosophical criteria grounding the core algebra, and they are therefore presented only in preliminary conceptual form at this time. These additional components—BGBGBG, DSDSDS, and the derived operators—are included in the companion document to provide conceptual orientation. They are part of the Standing Algebra framework but remain intentionally under‑specified here, pending the more detailed philosophical development that determines their final structure.
Jonathan Rademacher (Mon,) studied this question.