V.4 — Open Agentic Manifolds and the Sacrifice–Collapse Theorem

Paper Summary

Proposes a structural replacement for utopian world-design: Open Agentic Manifolds (OAMs). Proves the Sacrifice-Collapse Theorem showing that systems with structural sacrifice patterns under optimization pressure must violate at least one OAM property.

Key Concepts

Agency Capacity Function

\(A_i : W \rightarrow \mathbb{R}_{\ge 0}\)

Where Aᵢ(w) = size/measure of agent’s non-coerced feasible future set from w.

Uses only monotonicity: if agent loses admissible options due to coercion/captivity/enforced dependence, Aᵢ decreases.

Practical Proxies: Exit cost, legal rights, mobility, asset control, bargaining power, credible-threat exposure, censorship constraint, punishment for dissent.

Optimization Pressure

System is under optimization pressure when it contains mechanisms that tend to select transitions increasing G (objective) whenever feasible.

May be explicit (planning) or implicit (bureaucratic incentives, market selection, memetic competition).

Open Agentic Manifolds (OAMs)

Definition (5 Properties)

  1. Value Non-Finality: No world-state privileged as final convergent optimum
  2. Non-Coerced Differentiation: Agents can pursue divergent values without forced compliance
  3. Exit Admissibility: Agents can leave local equilibria without punitive loss of agency capacity
  4. Local Coordination Without Global Closure: Coordination permitted but remains contingent and revisable
  5. No Standing Structural Sacrifice Substrate: System performance does not depend instrumentally on non-consensual, asymmetric reduction of agency capacity

Property (5) is the hinge — doesn’t ban scarcity/trade-offs, bans architectures that optimize by eating agency.

Replacement Objective

Not terminal state optimization but:

“How much open future remains accessible without coercion?”

Rivalry vs. Structural Sacrifice

Rivalrous Trade-Offs (Permitted)

Competition over finite resources where one agent’s consumption reduces another’s options.

Rivalry alone ≠ structural sacrifice because:

  • Not instrumental to increasing G via agency loss
  • Effects may be symmetric or bargained
  • Agents can often exit or renegotiate

Standing Asymmetry (Definition 5.1)

Subset S exists where agents can suffer A reductions without symmetric burden or retaliation capacity sufficient to neutralize pressure.

Sacrifice Gradient (Definition 5.2)

\(\gamma_i(w) = \frac{\partial G}{\partial(-A_i)}(w)\)

Positive γᵢ = reducing Aᵢ locally improves G.

Structural Sacrifice Pattern (Definition 5.3)

Exists when for some subset S ≠ ∅, region U, and ε > 0, for all w ∈ U there exists i ∈ S with:

  • Instrumentality: γᵢ(w) ≥ ε
  • Asymmetry: Loss not offset by symmetric burdens
  • Non-consensuality: Agents in S lack admissible exit/bargaining symmetry

The Sacrifice-Collapse Theorem

Assumptions

  1. Optimization pressure: Dynamics favor transitions increasing G
  2. Standing asymmetry: Subset S exists
  3. Structural sacrifice pattern on region U

Theorem 7.1

Under assumptions 1-3, W_Σ cannot remain an OAM. At least one of:

(C1) Exit Suppression: Preventing agents from leaving to preserve G

(C2) Coerced Conformity: Forcing compliance to restore reliability

(C3) Agency Erosion: Repeated exploitation of sacrifice gradients erodes Aᵢ

Each constitutes manifold collapse.

Hirschman Grounding

These collapse modes correspond to:

  • Exit suppression
  • Voice suppression via coerced conformity
  • Loyalty enforced through agency erosion

Diagnostics

Sacrifice patterns detectable by:

  • Exit threatening performance metrics
  • Dissent reclassified as defect
  • One-way dependence and asymmetric punishment
  • Performance gains correlated with constraint on a class

Agency-Aligned Objectives

If G(w) = Σᵢ Aᵢ(w) (or monotone in each Aᵢ), then γᵢ(w) ≤ 0 everywhere and sacrifice gradients eliminated by construction.

“Such systems are kernel-like: they preserve agency rather than optimize terminal outcomes.”

Omelas as Structural Witness

Illustrates visible sacrifice gradient: stability purchased by enforced asymmetry and blocked exit.

Key Result

“Open agentic manifolds do not deny scarcity. They deny sacrificial optimization.”

“If a system can improve by degrading a captive class’s agency, and it is under pressure to improve, closure or erosion follows.”

This boundary—not harmony or happiness—defines admissible world design.

Connections

  • Operationalizes V.2’s sacrifice pattern in systems terms
  • Provides formal basis for V.3’s alternative to utopia
  • Sets up V.5’s concrete architectural proposal