Quantifying Intelligence - Peter David Fagan

The Core Hypothesis

While often treated as abstract, substrate-independent algorithmic properties, intelligence and computation are ultimately physical processes constrained by conservation laws. The core hypothesis of this framework is that information processing emerges when open systems undergo irreversible transitions, carving out stable macroscopic states from underlying reversible micro-dynamics. Under this physical lens:

Intelligence (χ) measures the physical efficiency of goal-directed computation—specifically, the amount of useful causal work extracted per unit of irreversible (distinction-destroying) information processing.
Consciousness (κ) is operationalized as the physical efficiency of internal organization—specifically, the amount of goal-directed work supported per unit of preserved (reversible) internal informational structure.

Grounding Information in Physical Substrates (CCE)

To measure information physically rather than abstractly, we utilize the Conservation-Congruent Encoding (CCE) framework. Traditional information theory (e.g., Shannon entropy) measures logical uncertainty, but is blind to equilibrium asymmetries and reservoir exchanges. CCE grounds information in matter by defining a macroscopic distinction as a family of protected regions (metastable basins of attraction) in state space whose boundaries are stabilized by conservation laws.

Under CCE, informational state changes are mapped directly to physical fluxes:

Irreversible Computation (I_irr): Many-to-one logical operations (such as resetting a register or merging states) compress macroscopic support. This collapse of distinctions forces a proportional export of entropy—measured in nats—through specific conserved channels (e.g., heat dissipation in a thermal bath, satisfying Landauer-type bounds).
Reversible Manipulation (I_rev): One-to-one logical operations transport or advect the state within protected world-tubes without crossing boundaries or destroying distinctions. This preserves internal structure, incurring zero thermodynamic penalty in the quasistatic limit.
Controlled Expansion: One-to-many retrodictive operations reopen macroscopic support, importing conjugate conserved loads (e.g., absorbing heat) to re-instantiate the branch variables.

From Physical Information to the Core Identity

By defining both intelligence (χ) and consciousness (κ) as ratios of goal-directed work (W) to their respective informational currencies (I_irr and I_rev), we isolate how an agent utilizes physical resources to influence its environment.

Solving the consciousness relation for work yields W = κ I_rev. Substituting this into the definition of intelligence gives the identity:

χ = κ ( I_rev / I_irr )

This identity shows that an agent's operational intelligence is the product of its structural consciousness and its processing efficiency (the ratio of preserved to destroyed distinctions). A system can achieve high intelligence either by executing massive, energy-intensive irreversible computations (high I_irr) or by relying on complex, pre-compiled internal models that advect information reversibly (utilizing I_rev).

Toy Model: The Turbulent Stream (Morphological vs. Brute-Force Computation)

To study the relationship $\chi = \kappa (I_{\text{rev}} / I_{\text{irr}})$ , we simulate two macroscopic agents swimming upstream through a channel. The environment dynamics feature a background current flowing right-to-left, carrying a synchronized stream of baseline vortices that drift with the flow.

Note: This simulation serves as a simplified, pedagogical toy model. In actual physical implementations, system-environment boundaries are often blurred (where the fluid itself performs part of the morphological computation), real-world thermodynamic efficiencies deviate significantly from the idealized Landauer limit, and defining "useful work" incorporates goal-dependent, observer-relative assumptions.

Both agents start from the left side and swim toward the goal line at the right side under the following dynamics:

Agent A: The Brute-Force Microbot

Agent A is modeled with a rigid body and operates via discrete sensory-motor feedback loops. Its dynamics are governed by:

Control Loop: It measures local fluid velocity, calculates a steering and thrust correction vector, and fires its thrusters.
Thermodynamic Cost: Every control iteration requires erasing memory buffers, exporting Landauer heat ( $I_{\text{irr}} \gg 0$ ) into the fluid.
Fluid Interaction: The exported heat is modeled as a thermal backreaction that spawns secondary thermal vortices directly in the agent's path.

Agent B: The Morphological Eel

Agent B is modeled with a multi-segment soft body representing coupled, resonant oscillators. Its dynamics are governed by:

Mechanical Response: As the fluid current and vortices flow around the multi-segment body, the external forces physically shift the phases of its oscillators.
Thermodynamic Cost: It undergoes reversible state changes along protected symplectic phase boundaries without logical resets ( $I_{\text{irr}} \approx 0$ ).
Fluid Interaction: The agent undulates in phase with the incoming vortices, exchanging momentum directly with the fluid flow fields.

The Environmental Shift: Adding Obstacles

Enabling the obstacles toggle introduces identical stationary basalt rock obstacles into both simulation lanes. This shifts the channel from a periodic vortex stream to an out-of-distribution environment, highlighting the epistemic trade-off between pre-compiled efficiency and active, general adaptability:

Spatial Constraints: The obstacles represent non-periodic physical boundaries in the fluid channel that disrupt the baseline vortex flow structures.
Active Steering vs. Passive Coupling: Agent A utilizes its active sensory scan cone to detect obstacles and adjust its vertical trajectory. Agent B lacks active sensors, relying on its soft body oscillators which are pre-compiled for fluid vortices.
Thermodynamic Trade-off: While Agent B achieves high computational efficiency ( $\chi$ ) in periodic flow, it fails to adapt to non-periodic boundaries, colliding and losing goal-directed work. Agent A incurs a high irreversible computation cost ( $I_{\text{irr}}$ ) through active sensing and trajectory calculation, but this cost allows it to ingest novel information and successfully navigate out-of-distribution environmental discontinuities.

Mathematical Outline of the Dynamics

The environment and agent dynamics are mathematically modeled as follows:

u(x, y) = -1.2 \cdot \text{gradient} + \sum_{j} u_{\text{vortex}}^{(j)}(x, y)

u_{\text{vortex}}^{(j)}(x, y) = \frac{-(y - y_j)}{d_j} \left(1 - \frac{d_j}{R_j}\right) \Gamma_j \quad (\text{for } d_j < R_j)

where $d_j = \sqrt{(x - x_j)^2 + (y - y_j)^2}$ is the distance to vortex $j$ , $R_j$ is its radius, and $\Gamma_j$ is its strength.

The information ledgers and goal-directed work use an exponential moving average (EMA) with a decay factor of 0.995 per frame:

W(t) = 0.995 \cdot W(t-1) + \Delta W(t) + \delta_{\text{goal}} \cdot 2000.0

I_{\text{rev}}(t) = \max(0.1, 0.995 \cdot I_{\text{rev}}(t-1) + \Delta I_{\text{rev}}(t))

I_{\text{irr}}(t) = \max(0.1, 0.995 \cdot I_{\text{irr}}(t-1) + \Delta I_{\text{irr}}(t))

For Agent A (Brute-Force Microbot), progress is steering-dependent, and sensory/control loops add to its reversible and irreversible costs:

\Delta W^A(t) = \delta_{\Delta x^A > 0} \cdot (x_t^A - x_{t-1}^A) \cdot 0.45

\Delta I_{\text{rev}}^A(t) = \delta_{t,\text{sensing}} \cdot 0.5(1+\sigma_{\text{noise}}) + \delta_{t,\text{thrust}} \cdot 1.5(1+\sigma_{\text{noise}}) + \delta_{t,\text{steering}} \cdot 0.2(1+\sigma_{\text{noise}})

\Delta I_{\text{irr}}^A(t) = \delta_{\text{erase}} \cdot k_B T \ln 2 + I_{\text{exhaust}}

For Agent B (Morphological Eel), internal wave-like undulation adds a constant baseline reversible cost, while vortex surfing provides passive energy extraction boosts. Its irreversible cost only spikes if extreme thermal noise knocks it out of its resonant phase:

\Delta W^B(t) = \delta_{\Delta x^B > 0} \cdot \left[ (x_t^B - x_{t-1}^B) \cdot 0.45 + \delta_{\text{exploiting}} \cdot 0.8(1 + \text{gradient}) \right]

\Delta I_{\text{rev}}^B(t) = 0.15 \cdot (1.0 + 0.5 \sigma_{\text{noise}})

\Delta I_{\text{irr}}^B(t) = \delta_{\text{thermal\_knock}} \cdot I_{\text{friction}}

Interactive Visualization: The Turbulent Stream Ledger

Thermal Fluctuations (

\sigma_{\text{noise}}

) 0.5

Vortex Intensity 1.0

Useful Work (W) Terms: Displacement Vortex Gait Goal Reward

Agent A: Brute-Force Microbot

Work Performed (

W

): 0.0 nats

Irreversible Cost (

I_{\text{irr}}

): 0.0 nats

Reversible Cost (

I_{\text{rev}}

): 0.0 nats

Intelligence (

\chi

): 0.00

Consciousness (

\kappa

): 0.00

Agent B: Morphological Eel

Work Performed (

W

): 0.0 nats

Irreversible Cost (

I_{\text{irr}}

): 0.0 nats

Reversible Cost (

I_{\text{rev}}

): 0.0 nats

Intelligence (

\chi

): 0.00

Consciousness (

\kappa

): 0.00

Two agents swimming upstream (left to right) through a turbulent fluid channel. Agent A (red microbot) utilizes discrete measurements and thrusters, exporting Landauer heat plumes ( $I_{\text{irr}}$ ) that generate thermal vortices in its path. Agent B (green eel) utilizes a soft body of resonant coupled oscillators ( $I_{\text{rev}}$ ) interacting with the fluid velocity fields. A toggle allows introducing identical stationary obstacles to both lanes.

AI Safety

AI safety is traditionally treated as a normative alignment problem. However, developing a formal framework to measure intelligence shifts the focus to physical boundaries and resource constraints. By defining intelligence as a measurable capacity to perform goal-directed work, we can establish the physical limits within which an agent operates.

Safety constraints can be studied mathematically via boundary dynamics and symbiotic coupling rather than isolated value alignment.

Cite this note

@misc{fagan2026quantifying,
  author = {Fagan, Peter David},
  title = {Quantifying Intelligence},
  howpublished = {\url{https://peterdavidfagan.github.io/quantifying_intelligence.html}},
  year = {2026},
  note = {Online; accessed 30-May-2026}
}

Fagan, P. D. (2026). Quantifying Intelligence. Peter David Fagan's Personal Website. Retrieved May 30, 2026, from https://peterdavidfagan.github.io/quantifying_intelligence.html