Experimental

Metacognition in Agentic Companies

Why AI systems must know what they don't know. Governance density as organizational self-awareness, the stability eigenvalue condition, and the mathematical foundations of enterprise metacognition.

G_t = (A_t, E_t, S_t, Π_t, R_t, D_t). Stability Law: λ_max(A) < 1 − D.

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MATHEMATICAL MODEL

The Agentic Company as a Living Graph.

Six Components, One System.

Agentic Company Tuple

Gt = (At, Et, St, Πt, Rt, Dt)

Agents, Edges, State, Policies, Rewards, Governance Density — six components that define an agentic company at any moment.

State Vector

St = [Ft, Kt, Ht, Lt, Ct]

Financial, Knowledge, Health, Legitimacy, Coordination — five dimensions of enterprise health.

Role Selection

ri(t+1) = argmaxr Ui(r | Ctask, Bcomm, Dt)

Each agent picks the role maximizing its utility function under governance constraints.

A

Agents

E

Edges

S

State

Π

Policies

R

Rewards

D

Governance

State Vector Components

FtFinancial
KtKnowledge
HtHealth
LtLegitimacy
CtCoordination
GOVERNANCE DENSITY

Constraints as Self-Observation.

D = |Constraints| / |ActionSpace|

Governance Density

Dt = |Constraintst| / |ActionSpacet|,   0 < D < 1

The fraction of actions subject to governance. D = 0 means anarchy. D = 1 means paralysis.

Dynamic Adjustment

Dtarget = clamp(base + w1·λmax + w2·anomaly + w3·Ctask − w4·Bcomm, 0.1, 0.9)

D adapts in real-time to spectral radius, anomaly rate, task complexity, and communication bandwidth.

Why Constraints = Metacognition

Approval gate = two agents examine a decision

Evidence requirement = forced outcome documentation

Risk threshold = escalation triggers self-examination

Compliance check = alignment with stated values

Governance Density Spectrum

D = 0 (Chaos)D = 0.45D = 1 (Stagnation)
StagnationD > 0.7

Excessive constraints freeze adaptation

Stable0.3 < D < 0.7

Optimal zone — meaningful specialization

ChaosD < 0.2

Insufficient governance — cascading failures

D is the density of proprioceptive sensors in the organizational body. Without it, the organization cannot sense or maintain coherence.

STABILITY LAW

The Fundamental Theorem.

λ_max(A_t) < 1 − D_t

Main Theorem

λmax(At) < 1 − Dt

The spectral radius of the influence matrix must stay below 1 minus governance density. This is the fundamental stability condition.

State Evolution

St+1 = At · St + (I − Dt · I) · εt

State evolves through influence propagation (A) damped by governance (D). Exogenous perturbations are bounded by density.

Stability Margin

δ = (1 − D) − λmax(A)

The stability margin. Larger margin = faster convergence. Near-zero margin = vulnerability to perturbations. Convergence is O(1/δ).

Parameter Space

D (Governance Density)λmaxUnstableStableλmax = 1 − Dcurrent11

Intuition

Influence propagation amplifies perturbations (λ_max). Governance damping absorbs them (1−D). Stability requires: dampening > amplification.

PHASE DIAGRAM

Three Regimes of Organization.

Stagnation / Stable / Chaos

Role Entropy

H(r) = −∑r p(r) log p(r)

Role entropy measures organizational health. Too low = stagnation. Too high = chaos. Moderate = stable specialization.

Stagnation

Parameters

D > 0.7, B_comm = low

Role Entropy

Near 0 within 50 epochs

Outcome

Throughput: 15% of maximum

Stable Specialization

Parameters

0.3 < D < 0.7, B_comm = mid-high

Role Entropy

Moderate, converging

Outcome

λ_max < 1 − D (bounded)

Chaos

Parameters

D < 0.2, B_comm = high

Role Entropy

Maximum, non-converging

Outcome

Divergence within 20 epochs

DOCTOR SYSTEM

Anomaly Detection + Convergence.

The Organizational Immune System.

Combined Anomaly Score

Acombined = α · s(x) + (1−α) · σ(ε(x))

Isolation Forest score + Autoencoder reconstruction error. Dual detection catches both sudden anomalies and gradual drift.

Convergence Condition

limt→∞ E[||St+1 − St||] = 0

The system converges when state changes approach zero. Requires bounded gradients, stable D, and Doctor intervention.

Response Thresholds

< 0.85NormalFull autonomy
0.85 – 0.92Soft Throttle50% autonomy reduction
> 0.92Hard FreezeComplete halt

Bounded Policy Gradients

Gate-constrained RL framework limits update magnitude

Stable Governance Density

Momentum + rate limiters prevent oscillation

Doctor Intervention

Catch runaway agents before cascading failure

MARIA OS Mapping

Graph G
Decision Graph(Organizational structure)
Density D
Gate Engine(Governance controller)
Reward R
Evidence Layer(Outcome verification)
Anomaly
Doctor System(Safety net)

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Metacognition in Agentic Companies