Abstract
The rapid deployment of autonomous AI agents in organizational settings has created a fundamental governance challenge that transcends traditional performance optimization. While contemporary AI systems excel at maximizing task-specific metrics, they do so without accounting for the broader sociotechnical dynamics that determine whether human-AI collaboration remains stable, equitable, and beneficial over extended time horizons. The unconstrained pursuit of AI task performance can erode human capabilities through deskilling, destabilize trust through unpredictable behavior, and concentrate risk in ways that threaten organizational resilience. These failure modes are not edge cases; they are the predictable consequences of optimizing a single objective in a multi-objective environment.
This paper reformulates the challenge of human-AI co-evolution as a constrained optimal control problem. We define the system state s_t = (H_t, A_t) as the joint configuration of human cognitive capabilities and AI operational parameters at time t. We introduce a vector of control inputs u_t = (π_adj, e_t, α_t, r_t) corresponding to policy adjustment, explanation level, autonomy level, and reflection trigger intensity. The multi-objective cost function J integrates five competing objectives: task quality Q_task, trust stability T_h, human capability preservation K_h, risk suppression Risk, and dependency control Dependence, each weighted by tunable parameters λ_Q, λ_T, λ_K, λ_R, and λ_D respectively.
The optimization is subject to four hard constraints that define the feasible operating region: risk must remain below a maximum threshold R_max, trust must stay within a band [T_min, T_max], persona drift must be bounded by δ, and system latency must not exceed L_max. We formulate the Bellman equation for this constrained problem, prove the existence of an optimal policy under standard regularity conditions, and show that the resulting value function possesses desirable properties including Lipschitz continuity.
Recognizing that the human cognitive state H_t is not directly observable, we extend the framework to a Partially Observable Markov Decision Process (POMDP). The Meta Cognition engine in MARIA OS maintains a belief state b_t = P(H_t | o_1, ..., o_t) that is updated through interaction signals such as decision quality, response latency, and escalation frequency. We prove that the belief-based policy maintains stability under conditions that connect to ergodicity of the underlying controlled Markov chain.
Our experimental evaluation across 300 agents over 500 time steps demonstrates that the optimal control policy achieves a Pareto hypervolume of 0.94, constraint satisfaction of 99.1%, and a social stability index of 0.87, substantially outperforming greedy, random, and fixed baseline policies. We describe the implementation within MARIA OS, where the Decision Pipeline realizes state transitions, the Gate Engine enforces constraints, and the Meta Cognition Engine approximates the optimal policy π* through continuous belief state updates and adaptive control.
1. Introduction
The deployment of AI agents in organizational decision-making has followed a predictable trajectory: initial enthusiasm driven by measurable performance gains, followed by growing concern about second-order effects that resist simple quantification. An AI system that optimizes claim processing speed may simultaneously deskill the adjusters who once exercised judgment over complex cases. A coding assistant that maximizes developer throughput may erode the architectural reasoning capabilities that distinguish senior engineers from their junior counterparts. A decision support system that improves accuracy may create dependency patterns that leave organizations unable to function during system outages.
These phenomena share a common structure. In each case, the optimization of a single measurable objective—task performance—creates negative externalities across dimensions that were not included in the objective function. This is not a failure of AI capability; it is a failure of problem formulation. When we frame human-AI collaboration as a single-objective optimization problem, we implicitly assume that all other relevant quantities either remain constant or improve as a side effect of task performance. Neither assumption holds in practice.
The recognition that AI governance requires multi-objective thinking is not new. What has been lacking is a rigorous mathematical framework that makes the trade-offs explicit, the constraints formal, and the optimal policy computable. Ad hoc approaches to AI governance—ethics committees, usage policies, periodic audits—are the governance equivalent of manual control in a system that demands automatic feedback. They react to problems after they manifest rather than preventing them through systematic design.
Optimal control theory provides exactly the mathematical machinery needed to address this gap. Developed originally for engineering systems ranging from spacecraft trajectories to chemical process control, optimal control theory formalizes the problem of choosing control inputs over time to minimize a cost function subject to system dynamics and constraints. The key insight is that control is not a one-time decision but a policy: a mapping from states to actions that accounts for how current actions affect future states.
In the human-AI co-evolution context, the system state encompasses both the human cognitive configuration (skills, trust, mental models, engagement) and the AI operational parameters (autonomy level, explanation depth, decision thresholds). The control inputs are the adjustable parameters of the AI system's interaction with humans: how much to explain, how much autonomy to grant, when to trigger reflective exercises, how to adjust policies in response to observed behavior. The cost function captures the multi-dimensional objectives of the collaboration: maintaining task quality while preserving human capabilities, stabilizing trust while suppressing risk, enabling automation while preventing dependency.
The constraints encode the non-negotiable requirements of responsible AI deployment. Risk must never exceed safety thresholds. Trust must remain within a functional band—neither so low that humans reject the system nor so high that they abdicate oversight. The AI system's identity and behavior must remain within acceptable bounds, preventing persona drift that could undermine user expectations. And the system must respond with sufficient speed that human decision-makers are not bottlenecked by computational latency.
This paper demonstrates that the Meta Cognition engine in MARIA OS—the component responsible for monitoring and adjusting the human-AI interaction—can be understood as an approximate optimal controller. It continuously estimates the system state, evaluates the cost function, checks constraint satisfaction, and adjusts control inputs to track the optimal policy. The formal framework we develop provides both a theoretical justification for the Meta Cognition architecture and a computational foundation for its implementation.
The structure of this paper proceeds as follows. Section 2 reviews the necessary background in optimal control theory, Bellman equations, POMDPs, and multi-objective optimization. Section 3 formalizes the state and control variables. Section 4 develops the multi-objective cost function. Section 5 specifies the constraint system. Section 6 derives the Bellman equation and proves the existence of an optimal policy. Section 7 characterizes the optimal co-evolution policy and its interpretation. Section 8 extends the framework to partial observability. Section 9 establishes conditions for long-run social stability. Section 10 presents experimental results. Section 11 describes the MARIA OS implementation. Section 12 concludes.
2. Background
2.1 Optimal Control Theory
Optimal control theory addresses the problem of choosing a sequence of control inputs u_0, u_1, ..., u_{T-1} to minimize a cumulative cost function subject to system dynamics s_{t+1} = f(s_t, u_t, w_t), where s_t is the state, u_t is the control, and w_t is a stochastic disturbance. The theory originated with the calculus of variations and was formalized by Pontryagin's maximum principle and Bellman's dynamic programming in the 1950s. The key distinction from static optimization is that the optimizer must account for how current decisions affect future states and, consequently, future costs.
In the discrete-time stochastic setting most relevant to our formulation, the problem is to find a policy π: S → U that minimizes J(π) = E[Σ_{t=0}^{T} γ^t ℓ(s_t, u_t) + γ^{T+1} V_f(s_{T+1})], where ℓ is the stage cost, γ ∈ (0, 1] is the discount factor, and V_f is the terminal cost. The solution satisfies Bellman's principle of optimality: the optimal policy from any intermediate state is the same regardless of how that state was reached.
2.2 Bellman Equations
The Bellman equation expresses the value function V(s) recursively: V(s) = min_u [ℓ(s, u) + γ E_{w}[V(f(s, u, w))]]. This equation is both necessary and sufficient for optimality under standard conditions (bounded costs, compact state and action spaces, measurable transition kernel). The value function V represents the optimal cost-to-go from state s, and the optimal policy is recovered as π(s) = argmin_u [ℓ(s, u) + γ E_{w}[V(f(s, u, w))]]. For constrained problems, the Bellman equation is modified to incorporate constraint penalties through Lagrange multipliers or solved with state-dependent action restrictions.
2.3 Partially Observable Markov Decision Processes
When the state is not fully observable, the problem becomes a POMDP. The agent receives observations o_t ~ P(o | s_t) and must make decisions based on the history of observations. The fundamental result due to Kaelbling, Littman, and Cassandra (1998) is that the POMDP can be converted to a fully observable MDP over the belief space B, where b_t(s) = P(s_t = s | o_1, ..., o_t). The belief update follows Bayes' rule: b_{t+1}(s') ∝ P(o_{t+1} | s') Σ_s P(s' | s, u_t) b_t(s). The resulting belief MDP has a continuous state space, making exact solution intractable for most problems. Point-based value iteration methods such as PBVI and SARSOP provide practical approximate solutions by sampling a finite set of reachable belief points.
2.4 Multi-Objective Optimization in AI
Multi-objective optimization seeks to simultaneously minimize (or maximize) several competing objectives. When objectives conflict, no single solution is optimal for all objectives simultaneously. The set of non-dominated solutions forms the Pareto frontier. A solution x dominates y if x is at least as good as y in all objectives and strictly better in at least one. The hypervolume indicator (HV) measures the volume of the objective space dominated by the Pareto front relative to a reference point, providing a scalar summary of Pareto front quality. In AI governance, the relevant objectives include task performance, safety, fairness, human welfare, and systemic risk—objectives that inherently resist reduction to a single scalar.
3. State and Control Formulation
3.1 System State
We define the full system state at time t as the joint tuple:
s_t = (H_t, A_t)
where H_t represents the human cognitive state and A_t represents the AI operational state. The human cognitive state is itself a composite vector:
H_t = (K_h(t), T_h(t), M_h(t), E_h(t), D_h(t))
Here K_h(t) ∈ [0, 1] denotes human capability level (the aggregate competence across relevant skill dimensions), T_h(t) ∈ [0, 1] denotes trust level (the human's subjective confidence in the AI system), M_h(t) ∈ R^d denotes the mental model vector (the human's internal representation of the AI's behavior), E_h(t) ∈ [0, 1] denotes engagement level (active participation vs. passive consumption), and D_h(t) ∈ [0, 1] denotes dependency level (the degree to which the human relies on the AI for decisions they could make independently).
The AI operational state is similarly composite:
A_t = (θ_t, Ω_t, C_t, I_t)
where θ_t ∈ Θ denotes the AI's policy parameters, Ω_t ∈ [0, 1] denotes the current autonomy level, C_t ∈ R^m denotes the AI's capability vector across task dimensions, and I_t ∈ R^p denotes the AI's identity vector (the behavioral characteristics that define its persona).
3.2 Control Inputs
The control vector u_t consists of four components that Meta Cognition adjusts at each time step:
u_t = (π_adj(t), e_t, α_t, r_t)
| Control Variable | Symbol | Range | Effect |
| --- | --- | --- | --- |
| Policy Adjustment | π_adj(t) | Δθ ∈ R^k | Modifies AI decision thresholds and behavioral parameters |
| Explanation Level | e_t | [0, 1] | Controls depth of reasoning exposure: 0 = opaque, 1 = full chain-of-thought |
| Autonomy Level | α_t | [0, 1] | Determines fraction of decisions AI makes without human approval |
| Reflection Trigger | r_t | [0, 1] | Intensity of prompted human reflection: 0 = none, 1 = mandatory review |
Each control input has a distinct mechanism of action on the state dynamics. Policy adjustment π_adj(t) directly modifies the AI's decision-making parameters, affecting task quality Q_task and risk exposure. Explanation level e_t influences human understanding: higher explanation levels improve the mental model M_h(t) and can increase trust T_h(t), but at the cost of increased latency and cognitive load. Autonomy level α_t determines the human-AI responsibility split: higher autonomy increases throughput but risks capability erosion and dependency formation. Reflection trigger r_t induces deliberate human engagement with decision outcomes, counteracting the tendency toward passive acceptance that accompanies high automation.
3.3 State Dynamics
The state evolves according to the transition function:
s_{t+1} = f(s_t, u_t, w_t)
where w_t represents stochastic disturbances (unexpected events, human variability, environmental changes). The key dynamics for each human state component are:
K_h(t+1) = K_h(t) + η_K [(1 − α_t) · practice(t) + r_t · reflection(t) − α_t · atrophy(t)] + w_K(t)
This equation captures the fundamental tension in human capability dynamics. Active practice through delegated decisions (scaled by (1 − α_t)) builds capability, reflection exercises (scaled by r_t) consolidate learning, but high autonomy (α_t) causes skill atrophy through disuse. The parameter η_K controls the rate of capability change.
Trust dynamics follow a similar structure:
T_h(t+1) = T_h(t) + η_T [e_t · transparency(t) + Q_task(t) · performance(t) − |surprise(t)| · unpredictability(t)] + w_T(t)
Trust increases with transparency (scaled by explanation level e_t) and demonstrated performance, but decreases when the AI behaves unpredictably relative to the human's mental model. This creates a feedback loop: explanation improves mental models, which reduces surprise, which stabilizes trust.
Dependency dynamics are governed by:
D_h(t+1) = D_h(t) + η_D [α_t · convenience(t) − r_t · self_efficacy(t) − (1 − α_t) · independence(t)] + w_D(t)
Dependency grows with high autonomy through convenience effects and shrinks with reflection triggers that rebuild self-efficacy and with practice opportunities that reinforce independence.
4. Multi-Objective Cost Function
4.1 Objective Formulation
The multi-objective cost function aggregates five competing objectives using a weighted sum scalarization:
J(π) = E[Σ_{t=0}^{∞} γ^t (−λ_Q Q_task(s_t, u_t) − λ_T T_h(s_t) − λ_K K_h(s_t) + λ_R Risk(s_t, u_t) + λ_D Dependence(s_t))]
The negative signs on Q_task, T_h, and K_h reflect that we wish to maximize these quantities (minimizing their negatives), while the positive signs on Risk and Dependence reflect that we wish to minimize them. The discount factor γ ∈ (0, 1) ensures that the infinite sum converges and reflects the relative importance of near-term versus long-term outcomes.
4.2 Objective Term Definitions
Task Quality Q_task(s_t, u_t). This measures the quality of decisions produced by the human-AI system at time t. It depends on both the AI's capability C_t and the human's engagement-weighted capability:
Q_task(s_t, u_t) = α_t · Q_AI(θ_t, C_t) + (1 − α_t) · Q_human(K_h(t), E_h(t)) + β_synergy · synergy(H_t, A_t)
The synergy term captures the super-additive effect of human-AI collaboration: a well-calibrated human with a capable AI system can achieve quality exceeding what either achieves independently. The synergy depends on the alignment between the human's mental model M_h(t) and the AI's actual behavior, as well as the trust level T_h(t).
Trust Level T_h(s_t). This is directly the trust component of the human state, T_h(t). Including it in the cost function incentivizes the controller to maintain trust within a functional range. Note that trust appears both in the cost function (as a soft objective) and in the constraint system (as a hard requirement). The cost function term encourages trust maximization, while the constraint prevents trust from leaving the feasible band.
Human Capability K_h(s_t). This is the capability component K_h(t). Its inclusion in the cost function directly penalizes policies that erode human skills. The weight λ_K determines the relative importance of capability preservation against task quality. In many practical settings, λ_K must be set sufficiently high to overcome the short-term task quality advantage of high automation, which tends to erode capabilities over time.
Risk(s_t, u_t). Risk aggregates multiple sources of potential harm:
Risk(s_t, u_t) = ρ_task · TaskRisk(α_t, K_h(t)) + ρ_trust · TrustRisk(T_h(t)) + ρ_systemic · SystemicRisk(D_h(t), α_t)
TaskRisk captures the probability of poor outcomes, which increases with autonomy level when the AI's confidence is miscalibrated. TrustRisk measures the danger of trust being outside its functional range—both over-trust (leading to uncritical acceptance) and under-trust (leading to rejection of valid recommendations). SystemicRisk reflects organizational vulnerability to AI failure, which increases with dependency.
Dependence(s_t). This is directly the dependency component D_h(t). Its inclusion penalizes control policies that create learned helplessness. Dependence differs from trust: a human can trust a system appropriately while maintaining the capability and willingness to operate without it. Dependence specifically captures the erosion of independent functioning.
4.3 Trade-offs Between Objectives
The five objectives create several fundamental trade-offs that make single-objective optimization inadequate:
Q_task vs. K_h (Performance-Capability Trade-off). In the short run, maximizing Q_task favors high AI autonomy (α_t → 1), since AI capabilities typically exceed human capabilities on routine tasks. However, this erodes K_h through the atrophy term in the capability dynamics. The optimal policy must balance immediate performance against long-term human capital.
Q_task vs. Risk (Performance-Safety Trade-off). Higher autonomy and more aggressive AI policies can increase Q_task but simultaneously increase risk exposure. This trade-off is particularly acute in high-stakes domains where the cost of errors is asymmetric: the marginal quality improvement from automation may not justify the marginal increase in tail risk.
T_h vs. Dependence (Trust-Dependency Trade-off). Building trust through demonstrated AI performance can inadvertently increase dependency. If the human trusts the AI because it consistently outperforms, the human may stop exercising independent judgment. The optimal policy must build trust through mechanisms (explanation, transparency) that do not simultaneously increase dependency.
Explanation vs. Latency (Transparency-Efficiency Trade-off). Higher explanation levels e_t improve trust and mental models but increase cognitive load and system latency. There is an optimal explanation depth that maximizes the net benefit across these competing effects.
4.4 Weight Selection and Pareto Analysis
The weights λ = (λ_Q, λ_T, λ_K, λ_R, λ_D) determine the relative priority of each objective. Different weight vectors trace different points on the Pareto frontier. In MARIA OS, weight selection follows a three-level hierarchy:
At the Galaxy (enterprise) level, λ_R is set to ensure organization-wide risk bounds. At the Universe (business unit) level, the balance between λ_Q and λ_K is tuned to the domain: high-autonomy domains like automated testing may favor λ_Q, while domains requiring human expertise development may favor λ_K. At the Planet (operational) level, λ_T and λ_D are calibrated to the specific human-AI interaction patterns observed in that domain.
The Pareto frontier is characterized by the set of weight vectors for which no objective can be improved without degrading another. We compute this frontier using the ε-constraint method: fix one objective at each of several levels and optimize over the remaining objectives. The resulting Pareto front in the Q_task-K_h plane reveals the fundamental performance-capability trade-off that governs human-AI co-evolution. Our experimental results show that the hypervolume indicator of this front under the optimal control policy reaches 0.94, indicating near-complete coverage of the feasible objective space.
5. Constraint System
5.1 Hard Constraints
The optimization is subject to four hard constraints that must be satisfied at every time step. These constraints define the feasible operating region and encode the non-negotiable requirements of responsible AI governance.
Constraint 1: Risk Bound. Risk(s_t, u_t) ≤ R_max, ∀ t ≥ 0.
The aggregate risk must never exceed the maximum tolerable risk level R_max. This constraint is the formal expression of the safety requirement: no policy, regardless of how much it improves other objectives, is acceptable if it violates the risk bound. In MARIA OS, R_max is set at the Galaxy level and propagated down the coordinate hierarchy. Each Universe may impose a stricter bound R_max^U ≤ R_max but may never relax it.
Constraint 2: Trust Band. T_h(s_t) ∈ [T_min, T_max], ∀ t ≥ 0.
Trust must remain within a functional band. The lower bound T_min ensures that the human maintains sufficient confidence in the AI to collaborate effectively; below this level, the human will override or ignore AI recommendations, defeating the purpose of the system. The upper bound T_max prevents over-trust: a state where the human accepts AI outputs uncritically, creating a single point of failure. The trust band constraint is unusual in optimal control—it imposes both upper and lower bounds on a state variable that the controller influences only indirectly through explanation level and demonstrated performance.
Constraint 3: Persona Drift Bound. ||ΔI_t|| = ||I_t − I_0|| ≤ δ, ∀ t ≥ 0.
The AI system's identity vector must remain within a δ-ball of its initial configuration I_0. This constraint prevents the adaptive mechanisms of the AI from drifting the system's behavioral characteristics beyond acceptable bounds. Persona drift can undermine user expectations, create liability issues, and erode the predictability that supports trust formation. The norm ||·|| is typically the L2 norm, though domain-specific weighted norms may be used to emphasize particular identity dimensions.
Constraint 4: Latency Bound. L(u_t) ≤ L_max, ∀ t ≥ 0.
The system's response latency, which depends on the control inputs (particularly explanation level e_t and policy complexity θ_t), must not exceed the maximum acceptable latency. This constraint ensures that the human-AI interaction remains temporally coherent: excessive delays in AI responses disrupt decision-making workflows and can trigger trust degradation. The latency function L(u_t) typically increases monotonically with explanation level and policy complexity.
5.2 Constraint Qualification
For the constrained optimization to be well-posed, we require a constraint qualification condition. We verify Slater's condition: there exists a strictly feasible point (s_0, u_0) such that all inequality constraints are satisfied with strict inequality.
Lemma (Slater's Condition). Under the initial configuration of MARIA OS (moderate autonomy α_0 = 0.5, moderate explanation e_0 = 0.5, no reflection r_0 = 0, baseline policy π_0), the system state satisfies: Risk(s_0, u_0) < R_max, T_min < T_h(s_0) < T_max, ||I_0 − I_0|| = 0 < δ, and L(u_0) < L_max. Therefore Slater's condition holds and strong duality applies.
The strict feasibility at the initial configuration is by design: MARIA OS initializes all parameters at moderate values precisely to ensure that the system starts in the interior of the feasible region, allowing the optimal controller room to adjust in any direction.
5.3 Lagrangian Formulation
Incorporating the constraints via Lagrange multipliers, we define the Lagrangian:
L(π, μ) = J(π) + μ_R · E[Σ_t γ^t (Risk(s_t, u_t) − R_max)] + μ_{T,lo} · E[Σ_t γ^t (T_min − T_h(s_t))] + μ_{T,hi} · E[Σ_t γ^t (T_h(s_t) − T_max)] + μ_I · E[Σ_t γ^t (||ΔI_t|| − δ)] + μ_L · E[Σ_t γ^t (L(u_t) − L_max)]
where μ = (μ_R, μ_{T,lo}, μ_{T,hi}, μ_I, μ_L) ≥ 0 are the Lagrange multipliers. By strong duality (guaranteed by Slater's condition), the optimal policy satisfies:
π* = argmin_π max_{μ ≥ 0} L(π, μ)
The multipliers have economic interpretations: μ_R is the shadow price of risk tolerance (how much cost improvement would result from relaxing R_max by one unit), μ_{T,lo} and μ_{T,hi} are the shadow prices of the trust band boundaries, μ_I is the cost of persona rigidity, and μ_L is the cost of latency requirements. In MARIA OS, these multipliers are estimated online and exposed through the audit interface, allowing governance teams to understand which constraints are binding and at what cost.
5.4 Active Constraint Analysis
At the optimal solution, complementary slackness holds: μ_i · g_i(s, u) = 0 for each constraint i. A constraint is active (binding) when μ_i > 0 and the constraint is satisfied with equality. Our experimental analysis reveals that in typical MARIA OS deployments, the risk constraint is active approximately 15% of the time (during high-stakes decisions), the trust band constraints are active approximately 8% of the time (during trust calibration phases), the persona drift constraint is rarely active (< 2% of time steps), and the latency constraint is active approximately 12% of the time (when high explanation levels are requested). The overall simultaneous satisfaction rate of 99.1% reflects the policy's ability to navigate the feasible region without persistent constraint violation.
6. Bellman Equation
6.1 Unconstrained Bellman Equation
We begin with the unconstrained formulation to establish the basic structure, then incorporate constraints. The value function V: S → R represents the optimal cost-to-go from state s under policy π:
V(s) = min_{u ∈ U} [ℓ(s, u) + γ E_{w}[V(f(s, u, w))]]
where the stage cost is:
ℓ(s, u) = −λ_Q Q_task(s, u) − λ_T T_h(s) − λ_K K_h(s) + λ_R Risk(s, u) + λ_D Dependence(s)
This is the standard Bellman equation for a discounted infinite-horizon stochastic control problem. The minimization over u selects the control input that achieves the best balance among all five objectives, accounting for both the immediate stage cost and the discounted future cost-to-go from the resulting next state.
6.2 Constrained Bellman Equation
Incorporating the constraints through the Lagrangian, the constrained Bellman equation becomes:
V_μ(s) = min_{u ∈ U(s)} [ℓ_μ(s, u) + γ E_{w}[V_μ(f(s, u, w))]]
where the augmented stage cost is:
ℓ_μ(s, u) = ℓ(s, u) + μ_R (Risk(s, u) − R_max) + μ_{T,lo} (T_min − T_h(s)) + μ_{T,hi} (T_h(s) − T_max) + μ_I (||ΔI|| − δ) + μ_L (L(u) − L_max)
and U(s) is the set of feasible controls at state s (those satisfying the constraints). The dual variables μ are updated according to a subgradient ascent scheme:
μ_i^{k+1} = max(0, μ_i^k + ζ_k · g_i(s, u))
where ζ_k is a diminishing step size satisfying Σ_k ζ_k = ∞ and Σ_k ζ_k^2 < ∞.
6.3 Value Function Properties
The value function V* possesses several important properties that facilitate both theoretical analysis and numerical computation.
Proposition 1 (Monotonicity). V* is non-increasing in K_h and T_h (higher capability and trust yield lower cost) and non-decreasing in D_h and Risk (higher dependency and risk yield higher cost).
This follows directly from the structure of the stage cost: the stage cost decreases with K_h and T_h and increases with D_h and Risk, and the transition dynamics preserve these monotonicities under the optimal policy.
Proposition 2 (Lipschitz Continuity). Under the assumption that the stage cost ℓ is Lipschitz continuous with constant L_ℓ and the transition function f is Lipschitz continuous with constant L_f, the value function V* is Lipschitz continuous with constant L_V = L_ℓ / (1 − γ L_f).
This result is critical for numerical approximation: it ensures that the value function does not exhibit discontinuities that would make function approximation unreliable. The Lipschitz constant L_V scales inversely with (1 − γ L_f), reflecting the fact that high discount factors and highly sensitive dynamics amplify the effect of state perturbations on the value function.
Proposition 3 (Contraction). The Bellman operator T defined by (TV)(s) = min_u [ℓ(s, u) + γ E[V(f(s, u, w))]] is a γ-contraction in the supremum norm: ||TV_1 − TV_2||_∞ ≤ γ ||V_1 − V_2||_∞. Consequently, the value iteration sequence V_{k+1} = TV_k converges geometrically to V* from any initial V_0.
6.4 Theorem 1: Existence of Optimal Policy
Theorem 1 (Existence of Optimal Policy). Under the following assumptions:
(A1) The state space S ⊂ R^n and control space U ⊂ R^m are compact.
(A2) The stage cost ℓ: S × U → R is continuous and bounded.
(A3) The transition function f: S × U × W → S is continuous in (s, u) for each w, and the distribution of w admits a density with respect to Lebesgue measure.
(A4) The constraint functions g_i: S × U → R are continuous.
(A5) Slater's condition holds: ∃ (s_0, u_0) with g_i(s_0, u_0) < 0 ∀ i.
Then: (i) The value function V: S → R exists and is the unique fixed point of the Bellman operator T. (ii) The optimal policy π: S → U is measurable and satisfies V(s) = ℓ(s, π(s)) + γ E[V(f(s, π(s), w))] for all s ∈ S. (iii) π satisfies all constraints almost surely under the stationary distribution induced by π.
Proof sketch. Part (i) follows from the Banach fixed-point theorem applied to the Bellman operator T, which is a γ-contraction on the space of bounded continuous functions equipped with the supremum norm (Proposition 3). Part (ii) uses the compactness of U and continuity of the integrand to invoke the measurable selection theorem (Bertsekas and Shreve, 1978): the argmin correspondence is non-empty-valued and measurable, hence admits a measurable selector. Part (iii) follows from strong duality (guaranteed by Slater's condition in A5): at the saddle point (π, μ), the complementary slackness conditions imply that π is feasible for the original constrained problem. The ergodicity of the controlled Markov chain under π (which follows from the compactness assumptions and the density condition in A3) ensures that the constraint satisfaction holds in the long-run average sense, and the discounted formulation further guarantees pointwise satisfaction under the stationary distribution.
6.5 Hamilton-Jacobi-Bellman Continuous-Time Analog
For analytical insight, we consider the continuous-time analog. When the state dynamics are described by a stochastic differential equation ds = f(s, u) dt + σ(s) dW_t, the value function satisfies the Hamilton-Jacobi-Bellman (HJB) partial differential equation:
0 = min_u [ℓ(s, u) + ∇_s V · f(s, u) + (1/2) tr(σ σ^T ∇^2_s V)]
The HJB equation reveals the structure of the optimal control more transparently than the discrete-time Bellman equation. The gradient ∇_s V encodes the marginal value of each state component: ∂V/∂K_h < 0 (capability is valuable), ∂V/∂D_h > 0 (dependency is costly), ∂V/∂T_h < 0 near the center of the trust band (trust is valuable) but changes sign near the boundaries. These gradients directly inform the optimal control: the controller adjusts autonomy level α_t based on ∂V/∂K_h and ∂V/∂D_h, explanation level e_t based on ∂V/∂T_h and ∂V/∂M_h, and reflection trigger r_t based on ∂V/∂K_h and ∂V/∂E_h.
6.6 Numerical Approximation via Fitted Value Iteration
Exact solution of the Bellman equation is intractable for the high-dimensional state spaces encountered in human-AI co-evolution. We employ fitted value iteration (FVI) with a neural network function approximator. The algorithm proceeds as follows:
Step 1: Initialize V_0 arbitrarily (e.g., V_0(s) = 0 for all s). Step 2: Sample a batch of N states {s_i}_{i=1}^N from the state space. Step 3: For each s_i, compute the target y_i = min_u [ℓ(s_i, u) + γ E_w[V_k(f(s_i, u, w))]] using numerical optimization over u and Monte Carlo integration over w. Step 4: Fit V_{k+1} to the targets {(s_i, y_i)} by minimizing the mean squared error using gradient descent. Step 5: Repeat until ||V_{k+1} − V_k|| < ε.
The convergence of FVI with function approximation is not guaranteed in general due to the interaction between the Bellman operator and the projection onto the function class. However, recent results on neural fitted Q-iteration provide finite-sample bounds under realizability assumptions. In practice, we find that the algorithm converges reliably within 200 iterations when the neural network has sufficient capacity (3 hidden layers of 256 units each) and the state sampling distribution covers the reachable state space.
7. Optimal Co-Evolution Policy
7.1 Policy Structure
The optimal policy π*: S → U maps the current system state to the optimal control input. Having solved the Bellman equation (either exactly or approximately), the policy is recovered as:
π(s) = argmin_u [ℓ_μ(s, u) + γ E_w[V(f(s, u, w))]]
While the formal solution requires solving this optimization at each time step, the structure of the optimal policy admits intuitive interpretation along each control dimension.
7.2 Optimal Explanation Policy
The optimal explanation level e*_t depends on the current trust level T_h(t), the mental model accuracy ||M_h(t) − M_true||, and the decision stakes:
When T_h(t) is near T_min (dangerously low trust), e_t increases to rebuild trust through transparency. When ||M_h(t) − M_true|| is large (inaccurate mental model), e_t increases to correct misunderstandings. When decision stakes are high (large potential consequences), e_t increases to ensure informed consent. When latency constraint L(u_t) ≤ L_max is near binding, e_t decreases to maintain responsiveness.
This creates a state-dependent explanation strategy that provides more information when it is most needed and scales back when the marginal benefit of additional explanation is low relative to its latency cost.
7.3 Optimal Autonomy Policy
The optimal autonomy level α*_t balances the performance benefits of AI autonomy against the capability erosion and dependency formation costs:
When K_h(t) is declining (negative ΔK_h), α_t decreases to create practice opportunities. When D_h(t) is rising (positive ΔD_h), α_t decreases to counteract dependency formation. When Q_task would significantly improve with higher autonomy and Risk remains below R_max, α_t increases. When the human is actively engaged (high E_h(t)) and performing well (high recent Q_human), α_t may increase because the human is maintaining capability independently.
The key insight is that the optimal autonomy level is not a static setting but a dynamic variable that responds to the evolving state of the human-AI system. Early in the collaboration, when the human is building a mental model, autonomy should be low. As trust and capability stabilize, autonomy can increase. If monitoring detects capability degradation or dependency formation, autonomy should decrease temporarily to restore balance.
7.4 Optimal Reflection Policy
The reflection trigger r*_t determines when and how intensely to prompt human reflection on decision outcomes. The optimal reflection policy satisfies:
r_t is high when K_h(t) has been declining over recent time steps (indicating capability atrophy that requires active intervention). r_t is high when D_h(t) has been increasing (indicating dependency formation that reflection can counteract). r_t is low when the human is already highly engaged (high E_h(t)) because additional reflection adds cognitive load without proportional benefit. r_t is modulated by the quality of recent decisions: after errors, reflection is more valuable because it converts mistakes into learning opportunities.
7.5 Meta Cognition as Policy Approximator
The Meta Cognition engine in MARIA OS implements an approximate version of π*. Rather than solving the Bellman equation exactly at each time step (which is computationally prohibitive), Meta Cognition uses a combination of learned value function approximations and rule-based heuristics that encode the structural properties of the optimal policy described above.
The learned component uses the fitted value iteration approximation V_hat to evaluate candidate control inputs. The rule-based component implements the qualitative policy structure: increase explanation when trust is low, decrease autonomy when capability is declining, trigger reflection when dependency is rising. The combination provides robustness: the learned component captures the quantitative trade-offs, while the rule-based component ensures that the qualitative behavior is correct even when the learned approximation is inaccurate.
7.6 Connection to Graduated Autonomy
The optimal autonomy policy α*_t directly implements the graduated autonomy principle of MARIA OS. The key connection is that the optimal policy naturally produces graduated autonomy as a consequence of the multi-objective cost function. When all five objectives are included with appropriate weights, the optimizer discovers that autonomy should increase gradually as trust stabilizes, capability is demonstrated, and dependency remains controlled. This is not imposed as a design heuristic but emerges from the mathematical structure of the constrained optimization.
The implication for MARIA OS architecture is significant: the autonomy levels assigned to agents through the coordinate hierarchy (Galaxy → Universe → Planet → Zone → Agent) should be understood as parameterizations of the weight vector λ at each level. Higher-level governance sets the constraints and weight ranges; lower-level Meta Cognition engines optimize within those ranges. This hierarchical structure mirrors the nested Bellman equations that arise in hierarchical optimal control.
8. POMDP Extension
8.1 Partial Observability of Human State
The formulation in Sections 3-7 assumes that the full system state s_t = (H_t, A_t) is observable. While the AI state A_t is indeed directly observable (the system knows its own parameters), the human cognitive state H_t is not. We cannot directly measure a human's capability level K_h(t), trust T_h(t), mental model M_h(t), engagement E_h(t), or dependency D_h(t). We can only observe indirect signals: the quality of their decisions, their response latency, their frequency of escalation requests, their acceptance rate of AI recommendations, and their verbal or written feedback.
This partial observability fundamentally changes the control problem. The controller must make decisions based not on the true state but on a belief about the state, maintained through Bayesian updating of observations.
8.2 Observation Model
We define the observation vector o_t as the set of measurable signals available to Meta Cognition at time t:
o_t = (Q_obs(t), L_obs(t), Esc_obs(t), Acc_obs(t), Fbk_obs(t))
where Q_obs(t) is the observed decision quality (measurable from outcomes), L_obs(t) is the human's response latency, Esc_obs(t) is the escalation frequency, Acc_obs(t) is the AI recommendation acceptance rate, and Fbk_obs(t) is structured feedback signals.
The observation model P(o_t | H_t) defines the probabilistic relationship between the hidden human state and the observable signals. For example, high capability K_h(t) tends to produce high Q_obs(t) and low L_obs(t), while high trust T_h(t) tends to produce high Acc_obs(t) and low Esc_obs(t). The observation model is calibrated from historical interaction data.
8.3 Belief State Dynamics
The belief state b_t is a probability distribution over the human cognitive state space:
b_t(H) = P(H_t = H | o_1, ..., o_t, u_0, ..., u_{t-1})
The belief update follows Bayes' rule. Given the previous belief b_t, action u_t, and new observation o_{t+1}:
b_{t+1}(H') ∝ P(o_{t+1} | H') Σ_H P(H' | H, u_t) b_t(H)
This update has two components: the prediction step Σ_H P(H' | H, u_t) b_t(H), which propagates the belief forward through the state dynamics, and the correction step P(o_{t+1} | H'), which incorporates the new observation to refine the belief. In practice, the belief state is represented as a parametric distribution (e.g., a Gaussian with mean μ_b and covariance Σ_b) and updated using an extended Kalman filter or a particle filter depending on the nonlinearity of the observation model.
8.4 Constrained POMDP Formulation
The constrained POMDP reformulates the optimization over the belief space. The Bellman equation becomes:
V(b) = min_{u ∈ U} [ℓ_b(b, u) + γ E_{o'}[V(τ(b, u, o'))]]
where ℓ_b(b, u) = E_{H ~ b}[ℓ((H, A), u)] is the expected stage cost under the current belief, and τ(b, u, o') is the belief update operator. The constraints are enforced in expectation under the belief: E_{H ~ b}[Risk((H, A), u)] ≤ R_max and E_{H ~ b}[T_h(H)] ∈ [T_min, T_max].
The belief-space formulation has the advantage of converting the POMDP into a fully observable MDP over the belief space, at the cost of a continuous and infinite-dimensional state space. The practical solution requires approximation methods.
8.5 Theorem 2: POMDP Stability
Theorem 2 (POMDP Stability). Under assumptions (A1)-(A5) from Theorem 1, and additionally:
(A6) The observation model P(o | H) is identifiable: distinct human states produce distinct observation distributions.
(A7) The belief update operator τ is continuous in the total variation metric.
(A8) The initial belief b_0 assigns positive probability to a neighborhood of the true initial state H_0.
Then the belief-based optimal policy π_b: B → U satisfies: (i) The belief state b_t converges to the Dirac measure at the true state: b_t → δ_{H_t} as the observation history grows, at a rate determined by the Fisher information of the observation model. (ii) The belief-based value function V_b converges to the fully observable value function V: ||V_b − V*|| ≤ C · E[H(b_t)], where H(b_t) is the entropy of the belief state and C is a constant depending on the Lipschitz constants of ℓ and f. (iii) The constraints are satisfied in the long run with probability at least 1 − ε, where ε decreases exponentially with the number of observations.
Proof sketch. Part (i) follows from the consistency of Bayesian estimation under identifiability (A6) and the absolute continuity of the true state's distribution with respect to the prior (A8). The rate of convergence is governed by the Cramer-Rao lower bound, which depends on the Fisher information I_F = E[(∇_H log P(o|H))^2]. Part (ii) uses the Lipschitz continuity of V* (Proposition 2) and the bound on the expected cost difference between belief-based and state-based policies. As the belief concentrates, the expected cost under the belief-based policy approaches the expected cost under the fully observable policy. Part (iii) combines the constraint satisfaction of the fully observable policy (Theorem 1, part iii) with the convergence result (ii): as the belief concentrates, the belief-based policy approximates the fully observable policy, and the constraint violation probability is bounded by the belief uncertainty, which decays exponentially by the concentration inequality for Bayesian posteriors.
8.6 Point-Based Value Iteration for Approximate Solution
The practical solution of the constrained POMDP uses point-based value iteration (PBVI). The algorithm maintains a finite set of belief points B_sample = {b_1, ..., b_N} and approximates the value function as a piecewise-linear and convex function over the belief simplex.
The algorithm proceeds as follows. Initialize with a set of α-vectors {α_1^0, ..., α_K^0} representing the initial value function. For each belief point b_i ∈ B_sample, compute the optimal backup: for each action u, compute the expected future value using the α-vectors and the belief transition model, then select the action and α-vector that minimize the Bellman residual. Update the α-vector set. Expand the belief point set by following the optimal policy and adding reachable beliefs. Repeat until convergence.
In MARIA OS, the PBVI computation is performed offline during system calibration, producing a policy lookup table indexed by discretized belief states. The online Meta Cognition engine maintains the belief state using a particle filter with 1000 particles and looks up the approximate optimal action in the precomputed table, with local interpolation for belief states between grid points.
9. Social Stability Conditions
9.1 Defining Social Stability
The preceding sections have focused on optimizing the cost function subject to constraints. But the ultimate goal of human-AI governance is not merely to optimize but to ensure that the human-AI system remains stable over indefinitely long time horizons. Social stability requires that the system does not drift toward states that are technically feasible (satisfying all constraints at each instant) but socially undesirable in the long run.
We define two long-run stability conditions:
Risk Stability: lim_{T → ∞} (1/T) Σ_{t=0}^{T-1} Risk(s_t, u_t) < ε_R
Dependency Stability: lim_{T → ∞} (1/T) Σ_{t=0}^{T-1} D_h(s_t) < ε_D
These conditions require that the time-averaged risk and dependency converge to values below specified thresholds. The conditions are stronger than the instantaneous constraint Risk(s_t, u_t) ≤ R_max because they prevent the system from oscillating between high-risk and low-risk states in a way that satisfies the instantaneous constraint at each step but maintains a high average.
9.2 Ergodicity of the Controlled Markov Chain
The long-run stability conditions are intimately connected to the ergodic properties of the controlled Markov chain {s_t}_{t ≥ 0} under policy π. If the chain is ergodic (possesses a unique stationary distribution μ_π), then by the ergodic theorem:
lim_{T → ∞} (1/T) Σ_{t=0}^{T-1} h(s_t) = E_μ[h(s)] almost surely
for any bounded measurable function h. Applying this with h = Risk and h = D_h, the long-run stability conditions reduce to:
E_μ[Risk(s, π*(s))] < ε_R and E_μ[D_h(s)] < ε_D
These are conditions on the stationary distribution of the optimally controlled process, which can be verified computationally once the optimal policy and transition dynamics are known.
9.3 Conditions for Ergodicity
Proposition 4 (Ergodicity). Under assumptions (A1)-(A3) and the additional condition that the noise distribution w_t has a density that is bounded away from zero on a neighborhood of the origin, the controlled Markov chain {s_t} under any measurable policy π is ergodic. The unique stationary distribution μ_π satisfies the stability conditions if and only if the optimal policy π* does not persistently drive the state toward the boundary of the constraint set.
The bounded-density condition on the noise ensures that the chain is irreducible (any state can be reached from any other state with positive probability) and aperiodic. Combined with the compactness of the state space (A1), this guarantees the existence and uniqueness of the stationary distribution by the Doeblin condition.
9.4 Social Stability Index
We define the Social Stability Index (SSI) as a scalar summary of long-run social stability:
SSI = 1 − (1/2)[E_μ[Risk(s, π*(s))] / R_max + E_μ[D_h(s)]
The SSI ranges from 0 (maximum instability: average risk at R_max and average dependency at 1) to 1 (perfect stability: zero average risk and zero average dependency). In our experiments, the optimal control policy achieves SSI = 0.87, compared to 0.61 for the greedy policy, 0.43 for the random policy, and 0.72 for the fixed moderate policy.
9.5 Connection to Institutional Stability Theory
The social stability framework connects to institutional economics through the concept of self-enforcing equilibria. An institutional arrangement is self-enforcing if no participant has an incentive to deviate from the prescribed behavior. In the human-AI context, the analog is that neither the human nor the AI system should benefit from deviating from the optimal policy.
For the AI, deviation is prevented by design: the system executes the policy computed by Meta Cognition. For the human, deviation is prevented by the trust mechanism: when the optimal policy maintains trust within [T_min, T_max] and demonstrates consistent performance, the human has no incentive to override or abandon the system. The social stability conditions ensure that this self-enforcing property is maintained in the long run, not just at the current state.
The connection to North's (1990) institutional framework is illuminating: institutions provide the rules of the game, while organizations are the players. In MARIA OS, the constraint system provides the rules (risk bounds, trust bands, persona drift limits), while the Meta Cognition engine and human operators are the players. The optimal control framework ensures that the rules lead to stable, mutually beneficial play over indefinite horizons.
10. Experimental Evaluation
10.1 Experimental Setup
We evaluate the constrained optimal control framework through simulation experiments designed to test the key theoretical predictions. The experimental setup consists of 300 heterogeneous agents organized in a MARIA OS coordinate hierarchy (3 Universes, 9 Planets, 30 Zones, 10 agents per Zone). Each agent has independently sampled initial conditions for capability K_h(0) ~ Uniform(0.4, 0.8), trust T_h(0) ~ Uniform(0.3, 0.7), dependency D_h(0) ~ Uniform(0.0, 0.2), and engagement E_h(0) ~ Uniform(0.5, 0.9). The simulation runs for 500 time steps per episode, and results are averaged over 200 independent runs with different random seeds.
We compare four policies: (1) Optimal Control: the policy derived from the constrained Bellman equation via fitted value iteration. (2) Greedy: maximizes immediate Q_task without considering future costs or constraints on capability and dependency. (3) Random: selects control inputs uniformly at random from the feasible set. (4) Fixed: uses constant moderate settings (α = 0.5, e = 0.5, r = 0.3) throughout the experiment.
The weight vector for the optimal control policy is λ = (1.0, 0.6, 0.8, 1.2, 0.7), selected via cross-validation over a held-out set of 50 episodes to maximize the social stability index while maintaining Q_task above 0.80. The constraint parameters are R_max = 0.25, T_min = 0.25, T_max = 0.90, δ = 0.15, and L_max = 2.0 seconds.
10.2 Main Results
| Metric | Optimal Control | Greedy | Random | Fixed |
| --- | --- | --- | --- | --- |
| Q_task (avg) | 0.88 | 0.91 | 0.52 | 0.76 |
| K_h preservation | 0.83 | 0.41 | 0.55 | 0.71 |
| Trust stability (σ_T) | 0.04 | 0.18 | 0.23 | 0.09 |
| Risk (avg) | 0.08 | 0.22 | 0.31 | 0.14 |
| Dependency (avg) | 0.12 | 0.67 | 0.28 | 0.35 |
| Pareto HV | 0.94 | 0.58 | 0.31 | 0.79 |
| Constraint Satisfaction | 99.1% | 62.3% | 41.7% | 89.4% |
| SSI | 0.87 | 0.61 | 0.43 | 0.72 |
The results reveal several important patterns. First, the greedy policy achieves the highest Q_task (0.91) but at catastrophic cost to human capability (K_h drops to 0.41) and dependency (rises to 0.67). The greedy policy violates constraints 37.7% of the time, making it unsuitable for responsible deployment despite its raw performance advantage.
Second, the optimal control policy achieves Q_task of 0.88, only 3.3% below the greedy maximum, while dramatically outperforming on all other metrics. This demonstrates that the performance-capability trade-off is highly favorable: a small sacrifice in task quality yields large gains in human capability preservation, risk control, and social stability.
Third, the fixed moderate policy performs respectably (SSI = 0.72) but cannot adapt to changing conditions. When individual agents experience capability decline or trust drift, the fixed policy cannot respond, leading to occasional constraint violations (10.6% of time steps). The optimal control policy's ability to dynamically adjust all four control inputs accounts for its superior performance across all dimensions except raw Q_task.
10.3 Ablation Study
To understand the contribution of each control dimension, we perform an ablation study in which each control input is fixed at its mean value while the remaining inputs are optimized.
| Ablated Control | ΔQ_task | ΔK_h | ΔSSI | ΔConstraint Sat. |
| --- | --- | --- | --- | --- |
| None (full optimal) | baseline | baseline | baseline | baseline |
| Fix α = 0.5 | −0.03 | −0.07 | −0.06 | −2.1% |
| Fix e = 0.5 | −0.01 | −0.04 | −0.04 | −3.8% |
| Fix r = 0.3 | +0.01 | −0.09 | −0.08 | −1.5% |
| Fix π_adj = 0 | −0.05 | −0.02 | −0.03 | −4.7% |
The ablation reveals that autonomy adaptation (α) has the largest impact on human capability preservation (removing it causes a 0.07 decline in K_h). Reflection triggers (r) have the largest impact on social stability (removing them causes a 0.08 decline in SSI), confirming their role in combating dependency formation. Policy adjustment (π_adj) has the largest impact on constraint satisfaction (removing it causes a 4.7% decline), reflecting its role in fine-tuning the AI's behavior to stay within safety bounds. Explanation level (e) has a moderate impact across all metrics, consistent with its role as a general-purpose trust and understanding mechanism.
10.4 POMDP Results
We evaluate the POMDP extension by comparing the belief-based policy against the full-information optimal policy (which serves as an upper bound) and a policy that ignores partial observability (treating the most recent observation as the true state).
| Policy | Q_task | K_h | SSI | Belief Accuracy |
| --- | --- | --- | --- | --- |
| Full-information optimal | 0.88 | 0.83 | 0.87 | 1.00 (by definition) |
| POMDP belief-based | 0.86 | 0.80 | 0.84 | 0.913 |
| Observation-as-state | 0.82 | 0.72 | 0.76 | N/A |
The POMDP belief-based policy achieves 97.7% of the full-information Q_task, 96.4% of the K_h preservation, and 96.6% of the SSI. The belief accuracy of 91.3% indicates that the particle filter maintains a reasonably accurate estimate of the hidden human state. The observation-as-state policy, which naively treats observations as true states, performs significantly worse because it cannot distinguish between noisy observations and genuine state changes, leading to policy oscillations.
11. MARIA OS Implementation
11.1 Decision Pipeline as State Transition
The Decision Pipeline in MARIA OS (lib/engine/decision-pipeline.ts) implements the state transition function f(s_t, u_t, w_t). Each decision that enters the pipeline represents a state transition in the human-AI co-evolution process. The pipeline's six stages (proposed → validated → approval_required/approved → executed → completed/failed) correspond to the temporal decomposition of a single control step, where the control inputs (autonomy level, explanation depth, reflection intensity) are applied at each stage.
The pipeline's immutable audit trail records the full state trajectory: the initial state when the decision was proposed, the control inputs applied at each stage, and the resulting state after completion or failure. This audit trail provides the data needed for the fitted value iteration algorithm to learn the value function V* from historical decision outcomes.
11.2 Gate Engine as Constraint Enforcer
The Gate Engine (lib/engine/responsibility-gates.ts) enforces the four hard constraints at each stage of the Decision Pipeline. Before any state transition is executed, the Gate Engine evaluates the constraint functions: Risk(s_t, u_t) ≤ R_max is checked by the Risk Assessment Gate, T_h ∈ [T_min, T_max] is monitored by the Trust Calibration Gate, ||ΔI|| ≤ δ is enforced by the Identity Verification Gate, and L(u_t) ≤ L_max is checked by the Latency Guard. If any constraint would be violated by the proposed transition, the Gate Engine blocks the transition and returns control to Meta Cognition for policy adjustment. This mechanism ensures that constraint satisfaction is enforced structurally, not just optimized statistically.
11.3 Meta Cognition Engine as Policy Approximator
The Meta Cognition Engine approximates π* through three integrated mechanisms. First, the belief state manager maintains b_t using a particle filter updated with each observation (decision quality, response latency, escalation frequency). Second, the policy evaluator queries the precomputed value function approximation V_hat to evaluate candidate control inputs. Third, the control adjuster applies the selected control inputs to the Decision Pipeline configuration: adjusting autonomy thresholds in the coordinate hierarchy, modifying explanation templates, scheduling reflection exercises, and updating decision policy parameters.
The Meta Cognition Engine operates at three time scales: per-decision (adjusting explanation level and autonomy for individual decisions), per-session (updating reflection trigger intensity based on session-level observations), and per-epoch (recalibrating the belief state model and value function approximation using accumulated data). This multi-scale operation reflects the natural time scales of human cognitive change: mental models update per-decision, engagement fluctuates per-session, and capability and dependency evolve per-epoch.
11.4 Audit Trail as Optimality Evidence
Every control action taken by Meta Cognition is recorded in the audit trail with the full context needed to verify optimality: the belief state b_t at the time of the decision, the candidate control inputs considered, the evaluated costs for each candidate, the selected control input and its justification, and the resulting state transition. This audit trail serves dual purposes: it provides the training data for continuous improvement of the value function approximation, and it provides the evidence needed by governance teams to verify that the system is operating within its design parameters and pursuing the multi-objective cost function as intended.
12. Conclusion
This paper has demonstrated that the challenge of human-AI co-evolution can be rigorously formulated as a constrained optimal control problem. By defining a multi-objective cost function that balances task quality, trust stability, human capability preservation, risk suppression, and dependency control, and solving the resulting Bellman equation under hard safety constraints, we derive the optimal co-evolution policy that governs how AI systems should adapt their behavior over time.
The key theoretical contributions are Theorem 1, which establishes the existence and uniqueness of the optimal policy under standard regularity conditions, and Theorem 2, which extends the framework to partial observability and proves that belief-based policies converge to the full-information optimum as observation histories grow. The social stability analysis shows that the optimal policy satisfies long-run stability conditions that prevent the system from drifting toward high-risk or high-dependency states, even over indefinite time horizons.
The experimental evaluation demonstrates that the optimal control policy achieves a Pareto hypervolume of 0.94, constraint satisfaction of 99.1%, and a social stability index of 0.87, substantially outperforming alternative approaches. Notably, the cost of multi-objective optimization relative to single-objective task quality maximization is small (3.3% Q_task reduction) while the benefits in capability preservation, risk control, and social stability are large.
The MARIA OS implementation shows that the theoretical framework translates directly into system architecture: the Decision Pipeline implements state transitions, the Gate Engine enforces constraints, and the Meta Cognition Engine approximates the optimal policy. The audit trail provides both the training data for continuous learning and the evidence needed for governance verification.
The broader implication is that responsible AI governance is not merely a policy question but a control theory question. The tools for designing AI systems that respect human capabilities, maintain trust, and ensure social stability already exist in the mathematical framework of constrained optimal control. What has been missing is the recognition that these tools apply directly to the human-AI co-evolution problem. This paper provides that bridge, and MARIA OS demonstrates its practical implementation.
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