Standing Algebra Σᴿ: A Solution to AI Violating Human Autonomy

By Jon Rademacher @ 2026-03-23T21:40 (–3)

This is a linkpost to https://zenodo.org/records/19186551

Hi everyone — this is a working paper I’m releasing that introduces Standing Algebra (Σᴿ), a formal system I’ve been developing to express autonomy‑preserving update rules in multi‑agent systems.

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.