Epistemic status: Early‑stage theoretical work. Seeking technical critique, counterexamples, and feedback on edge cases of the envelope construction. This post shares a preprint, a reference validator (Python), and test cases for reproducibility.
Hi everyone — this is a working paper introducing Standing Algebra (Σᴿ), a formal system for autonomy‑preserving update rules in multi‑agent settings. Σᴿ defines structural constraints on standing, capacity, and dependency, and provides a “Legitimate Envelope” that converts arbitrary updates into norm‑preserving ones.
This paper is one component of a broader research program I’m developing. It accompanies a philosophical monograph, On Relevance: The Social Physics of Pluralism, which analyzes legitimacy, standing, and non‑dominating coordination across domains. It also pairs with a shorter companion piece, Why Being Right Isn’t Enough, and a quick‑start guide/flowchart meant to make the framework easier to apply.
The math in this paper is the formal extension of the philosophical structure: if a framework for standing and legitimacy is genuinely cross‑domain, then the same constraints should apply at the mathematical and computational level. Formalizing those constraints led to the algebra and envelope operator presented here, which also turn out to have direct implications for AI systems that modify multi‑agent states.
The motivating question is:
How do we constrain system‑level updates so that no agent’s standing is reduced, no unfair asymmetries are introduced, and updates remain structurally safe by construction?
To explore that, the framework treats an update F : U→U as operating on agents who each carry three numerical invariants:
standing σ(i)∈N
capacity cap(i)∈N
dependency degree deg(i)∈N
For each agent, the algebra examines the induced changes:
Δσi=σ(F(i))−σ(i),Δdegi=deg(F(i))−deg(i)
and applies a set of structural constraints, including:
Standing monotonicity (no decreases, ALRP)
Successor‑consistency (Δσ∈{0,+1})
Drift bounds ( ∣Δdeg∣≤1)
Standing‑class uniformity (all agents with equal (σ,cap) get identical Δσ)
Capacity ceilings (σ(F(i))≤B(cap(i)))
Symmetric dependency treatment (NRPP)
When an update FFF violates any of these constraints, it is “repaired” into a Legitimate Envelope LFL_FLF. The envelope is:
idempotent
standing‑monotone
class‑uniform
successor‑consistent
drift‑bounded
minimal in the σ‑order among all legitimate updates with the same increment signature
Informally, this turns arbitrary updates into the closest autonomy‑preserving version consistent with the axioms.
The envelopes (modulo increment signatures) form a join‑semilattice under classwise OR, yielding an algebra of safe update policies. This can serve as:
a validator layer for MAS,
a safety normalizer for agentic LLM scaffolding, or
a structural constraint framework for institutional or organizational governance.
The preprint includes:
a many‑sorted signature
Tier‑1 and Tier‑2 axioms for standing, capacity, dependency, drift, uniformity, and repair
a constructive model showing consistency
independence proofs for all axioms
the Legitimate Envelope theorem
the resulting semilattice structure
examples of integrating Σᴿ into multi‑agent update pipelines
I’d appreciate feedback on any of the following:
edge‑case deltas
minimality of envelopes
whether the closure operator is well‑positioned relative to other safety formalisms
connections to prior work in MAS governance, closure operators, or formal safety mechanisms
Thanks for taking a look — I’m especially interested in critique, places it’s underspecified, or directions I should explore next.
A significant update has been made for anyone interested