Opening — Why this matters now
We have built systems that write code, trade assets, drive robots, and negotiate with humans. They act. They learn. They optimize.
And yet, when the environment shifts—even slightly—they drift.
The dominant narrative says: scale more data, more parameters, more compute. But the paper A Mathematical Theory of Agency and Intelligence fileciteturn0file0 suggests something more uncomfortable: reliability is not primarily a training problem. It is an architectural one.
At the center of this argument is a deceptively simple quantity: bi-predictability (P) — the fraction of total information in an interaction that is genuinely shared between observation, action, and outcome.
In other words: how much of what the system “knows” is actually coupling it to reality?
Background — From Feedback to Information Bounds
Classical cybernetics already warned us. Ashby’s Law of Requisite Variety, Wiener’s feedback loops—reliable regulation requires continuous information coupling.
But modern AI reliability tools tend to monitor fragments:
- Benchmark performance
- Reward trends
- Input drift
- Confidence estimates
They rarely measure the full observation–action–outcome loop.
This paper reframes the loop in strict information-theoretic terms.
For passive systems (no action channel), predictive coherence is defined as:
$$ P = \frac{MI(S; S’)}{H(S) + H(S’)} $$
For agentic systems, with action $A$:
$$ P = \frac{MI(S, A; S’)}{H(S) + H(A) + H(S’)} $$
Where:
- $S$ = internal state / observation
- $A$ = action
- $S’$ = next state / outcome
Crucially, this is not “how much information flows.” It is how efficiently the interaction uses its informational budget.
And here’s the part executives should pause at:
| Regime | Upper Bound on P |
|---|---|
| Quantum systems | 1 |
| Classical systems | 0.5 |
| Agentic classical systems | < 0.5 (in practice) |
Freedom has an information cost.
Analysis — Agency vs. Intelligence
The paper draws a sharp line:
Agency requires:
- Choice: $H(A|S) > 0$
- Effect: $MI(A; S’ | S) > 0$
- Predictive asymmetry: $\Delta H \neq 0$
Agency means the system can intervene and those interventions matter.
But intelligence requires more.
Intelligence requires:
- Increasing coupling ($MI(S,A;S’)$ grows through learning)
- Monitoring $P$ over time
- Adapting the structure of ${S}, {A}, {S’}$ when coupling degrades
This is the architectural leap.
Today’s systems optimize reward. They do not monitor their own coupling integrity.
They can win the game while losing grip on the environment.
Findings — Physics, RL, and LLMs Under the Same Lens
1. Physical Baseline: Double Pendulum
In a deterministic system without agency:
- $P \approx 0.48$ (near classical ceiling 0.5)
- $\Delta H \approx 0$
Chaos did not reduce coupling symmetry.
This establishes the calibration point: predictability loss is not the same as randomness.
2. Reinforcement Learning Agents
HalfCheetah (SAC/PPO):
| System | P | ΔH | Interpretation |
|---|---|---|---|
| Double Pendulum | 0.48 | ≈ 0 | Symmetric physics |
| HalfCheetah | 0.33 | -0.56 | Asymmetric agency |
Introducing action reduces coherence and breaks symmetry.
More interestingly, when perturbations were introduced:
| Detection Method | Perturbation Detection Rate | Median Latency |
|---|---|---|
| Reward-based | 44% | 184 windows |
| P/ΔH (IDT) | 89% | 42 windows |
Reward lags. Coupling degradation appears first.
3. Large Language Models
Multi-turn dialogue experiments showed:
- $P$ strongly correlates with structural coherence (85% of cases)
- $P$ detects contradictions, topic shifts, and non-sequiturs with 100% sensitivity
- Detection occurs immediately at injection points
And critically:
LLMs satisfy agency (choice, effect, asymmetry). They satisfy learning (next-token training). They do not compute $P$ internally. They cannot restructure their interface.
By this framework, they are agentic. Not intelligent.
The Architectural Proposal — Information Digital Twin (IDT)
The authors propose a Coupled Agency Architecture.
An auxiliary module—Information Digital Twin (IDT)—monitors:
- $P$
- $H_f$ (forward uncertainty)
- $H_b$ (backward uncertainty)
- $\Delta H$
In real time.
Instead of retraining, the system performs reflexive modulation:
- Dampening actions
- Gating inputs
- Adjusting bandwidth
This mirrors thalamocortical regulation in biological brains: monitor statistics, not semantics.
Separation of concerns:
| Layer | Function |
|---|---|
| Agent | Optimize task objective |
| IDT | Monitor coupling integrity |
| Controller | Modulate interface when drift occurs |
Performance and structural stability become distinct variables.
That distinction may be foundational.
Implications — For AI Builders and Operators
1. Reliability is an architectural problem
Scaling alone will not create intelligence. You need a self-monitoring coupling layer.
2. Reward ≠ Grip
A system can maintain reward while losing bidirectional constraint. This is operationally dangerous.
3. Attribution matters
If $P$ drops, is the world opaque (high $H_f$)? Or is the agent illegible (high $H_b$)?
Without this decomposition, adaptation is blind.
4. Governance angle
A first-person coupling metric offers:
- Model-agnostic oversight
- Drift detection without semantic judges
- A quantitative boundary between agency and intelligence
For regulators, this is more tractable than defining “alignment.”
Conclusion — The Cost of Freedom
The paper demonstrates a provable constraint:
- Classical systems: $P \leq 0.5$
- Agency reduces attainable coherence
- Intelligence is the management of that reduction
Agency introduces freedom. Freedom reduces raw predictability. Intelligence is not eliminating that trade-off. It is regulating it.
Current AI systems optimize objectives. They do not monitor their own informational grip.
Until they do, they remain powerful agents. Not intelligent systems.
And that distinction is no longer philosophical. It is measurable.
Cognaptus: Automate the Present, Incubate the Future.