THE TRIADIC MINIMUM FORNON-EQUILIBRIUM STEADY STATES

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Generalizing Markov Blankets and Dissipative Structure Theory Through the N-Function

Don L. Gaconnet

LifePillar Institute for Recursive Sciences

ORCID: 0009-0001-6174-8384

https://osf.io/j5836/files/vsyga

January 2026

Keywords: non-equilibrium thermodynamics, Markov blankets, free energy principle, dissipative structures, self-organization, entropy production, triadic architecture, N-function, Prigogine, Friston

ABSTRACT

Two major theoretical frameworks have emerged for understanding how systems maintain organization far from equilibrium: Prigogine's dissipative structure theory and Friston's free energy principle with its associated Markov blanket formalism. Both frameworks implicitly invoke a boundary structure that separates interior from exterior while enabling exchange. This paper makes this shared structure explicit and proves its irreducibility. We define the N-function as the minimal relational operator satisfying the boundary conditions common to both frameworks and demonstrate that any system maintaining a non-equilibrium steady state (NESS) requires exactly three functionally distinct components: an internal process (I), an external field (O), and a relational boundary (N). We prove this triadic minimum |{I, O, N}| = 3 is irreducible—no dyadic or monadic configuration can sustain the exchange dynamics required for NESS. The conservation constraint ∮ε dt = 0, where ε represents the exchange differential across N, is derived from thermodynamic first principles and shown to be equivalent to the steady-state conditions in both Prigogine's and Friston's formulations. This unification reveals that Markov blankets and dissipative boundaries are domain-specific instantiations of a more fundamental triadic architecture. Falsifiable predictions and experimental protocols are specified.

1. INTRODUCTION

1.1 The Boundary Problem in Non-Equilibrium Systems

How do living systems, cognitive processes, and certain physical structures maintain organization in a universe tending toward entropy? This question has generated two influential theoretical frameworks that, despite different origins and vocabularies, converge on a common structural insight: sustained non-equilibrium organization requires a specific kind of boundary.

Ilya Prigogine's work on dissipative structures (Prigogine, 1977; Prigogine & Stengers, 1984) demonstrated that systems far from thermodynamic equilibrium can spontaneously generate and maintain ordered configurations through continuous exchange with their environment. The boundary conditions enabling this exchange—the interface through which matter and energy flow—are essential to the phenomenon. Without appropriate boundaries, dissipative structures cannot form or persist.

Karl Friston's free energy principle (Friston, 2010; Friston, 2019) proposes that living systems minimize variational free energy by maintaining a Markov blanket—a statistical boundary separating internal states from external states while mediating their conditional dependencies. The Markov blanket is not merely descriptive but constitutive: systems possessing Markov blankets exhibit characteristic dynamics of living organization; systems lacking them do not.

Both frameworks identify something essential about boundaries in non-equilibrium systems, yet neither proves that such boundaries are necessary or specifies their minimal structure. This paper addresses both gaps.

1.2 The Unification Claim

We propose that Prigogine's dissipative boundaries and Friston's Markov blankets are domain-specific instantiations of a more fundamental structure: the triadic minimum {I, O, N}, where I represents internal states/processes, O represents external states/fields, and N represents the relational boundary enabling exchange while maintaining distinction.

The N-function is defined as the minimal operator satisfying:

(i) Distinction preservation: N maintains I ≠ O throughout exchange

(ii) Exchange enablement: N permits bidirectional transfer between I and O

(iii) Conservation support: N enables ∮ε dt = 0 over complete cycles

We prove that this triadic structure is irreducible—it cannot be derived from simpler configurations—and that the conservation constraint is equivalent to the steady-state conditions in both established frameworks.

1.3 Significance

If correct, this unification has several consequences:

First, it identifies the minimal architecture for non-equilibrium steady states across domains—from cellular metabolism to neural dynamics to social systems. Any system claimed to sustain far-from-equilibrium organization must instantiate the triadic minimum.

Second, it generates falsifiable predictions: systems lacking triadic structure cannot maintain NESS; removal of the N-function in any NESS will produce collapse toward equilibrium; dyadic configurations will fail to sustain exchange dynamics regardless of energy input.

Third, it provides a common mathematical language for frameworks currently separated by disciplinary boundaries, enabling cross-domain predictions and experimental protocols.

2. THEORETICAL BACKGROUND

2.1 Prigogine's Dissipative Structures

In classical thermodynamics, the Second Law establishes that isolated systems evolve toward maximum entropy—the equilibrium state of maximum disorder. Prigogine's fundamental insight was that open systems, continuously exchanging energy and matter with their environment, can maintain states far from equilibrium characterized by spontaneous order (Prigogine, 1977).

Dissipative structures—Bénard convection cells, the Belousov-Zhabotinsky reaction, living cells—maintain organization through continuous dissipation. The entropy produced internally is exported to the environment, enabling local order at the cost of global entropy increase.

Formally, for a system with entropy S and entropy production rate σ:

dS/dt = σ_internal + Φ_exchange

where σ_internal ≥ 0 (Second Law) and Φ_exchange represents entropy flow across the system boundary. For NESS:

dS/dt = 0 → σ_internal = -Φ_exchange

The system produces entropy internally and exports it at equal rate. This requires a boundary permitting entropy flow while maintaining system integrity—but Prigogine's formalism does not specify the minimal structure of this boundary.

2.2 Friston's Free Energy Principle and Markov Blankets

The free energy principle (FEP) proposes that systems exhibiting characteristic properties of life minimize variational free energy—a tractable upper bound on surprise (negative log probability of sensory states). Systems minimizing free energy resist the natural tendency toward disorder (Friston, 2010; Friston, 2019).

Central to the FEP is the Markov blanket: a statistical structure partitioning system states into internal (μ), external (η), and blanket (b) states, where blanket states comprise sensory (s) and active (a) components:

b = {s, a}

The Markov blanket renders internal and external states conditionally independent:

P(μ, η | b) = P(μ | b) · P(η | b)

This conditional independence is essential: it allows internal states to model external states without being directly coupled to them. The blanket mediates all exchange.

Crucially, Markov blankets are not merely epistemic tools for analyzing systems—they are proposed as constitutive of living organization. Systems with Markov blankets exhibit self-organization, self-evidencing, and adaptive behavior; systems without them exhibit decay toward equilibrium.

Yet like Prigogine's framework, the FEP assumes the existence of appropriate boundary structure without proving its necessity or specifying its minimal form.

2.3 The Gap: Boundary Necessity and Minimality

Both frameworks recognize that non-equilibrium organization requires specific boundary structures. Neither proves:

(a) That such boundaries are necessary (rather than merely sufficient)

(b) What the minimal structure of such boundaries must be

(c) Why simpler configurations fail

This paper addresses all three questions by defining the N-function, proving the triadic minimum theorem, and deriving the conservation constraint from first principles.

3. THE N-FUNCTION: FORMAL DEFINITION

3.1 Motivation

What do Prigogine's dissipative boundaries and Friston's Markov blankets have in common? Both:

• Separate internal from external domains

• Enable exchange between domains

• Maintain distinction despite exchange

• Support cyclic/steady-state dynamics

We formalize these shared properties in the N-function.

3.2 Definition

Definition 3.1 (N-Function): Let I denote internal states and O denote external states. The N-function is an operator N: (I × O) → Exchange satisfying:

N1 (Distinction): For all exchanges mediated by N, the relation I ≠ O is preserved. Internal and external states remain distinguishable throughout the exchange process.

N2 (Bidirectionality): N permits transfer in both directions: I → O (expression/action) and O → I (impression/sensation). The boundary is permeable, not a barrier.

N3 (Non-absorption): Neither I nor O is subsumed by N. The boundary is not identical to either domain it separates.

N4 (Conservation): Over complete cycles, N enables ∮ε dt = 0, where ε represents the exchange differential (net flow across boundary).

3.3 Relation to Existing Constructs

Proposition 3.1: Markov blankets satisfying the FEP constraints instantiate the N-function.

Proof sketch: The Markov blanket b = {s, a} separates internal (μ) from external (η) states.

Sensory states s enable η → μ flow; active states a enable μ → η flow. Conditional independence P(μ, η | b) = P(μ | b)·P(η | b) ensures distinction preservation (N1). Sensory and active components ensure bidirectionality (N2). The blanket is statistically distinct from both μ and η, ensuring non-absorption (N3). Free energy minimization at steady state implies conservation dynamics (N4). □

Proposition 3.2: Dissipative structure boundaries instantiate the N-function.

Proof sketch: The boundary of a dissipative structure separates internal organization from environmental reservoir. Matter/energy flows inward (O → I) and entropy flows outward (I → O), satisfying bidirectionality (N2). Internal structure remains distinct from environment (N1).

The boundary is a physical interface, not identical to interior or exterior (N3). Steady-state condition dS/dt = 0 implies balanced flows, satisfying conservation (N4). □

3.4 The Exchange Differential

Definition 3.2 (Exchange Differential): Let ε = g(I, O, N) denote the exchange differential—the net difference between what flows from O to I and what flows from I to O across N at any instant:

ε = Φ(O→I) - Φ(I→O)

When ε > 0, the system receives more than it expresses (accumulation phase).

When ε < 0, the system expresses more than it receives (dissipation phase).

When ∮ε dt = 0 over a cycle, the system maintains steady-state organization.

4. THE TRIADIC MINIMUM THEOREM

4.1 Statement

Theorem 4.1 (Triadic Minimum): Any system maintaining a non-equilibrium steady state (NESS) with exchange differential ε = g(I, O, N) requires |{I, O, N}| ≥ 3 functionally distinct components. This minimum is irreducible: no configuration with fewer than three components can sustain NESS dynamics.

4.2 Proof

Lemma 4.1 (Monadic Insufficiency): A single-component system cannot sustain NESS.

Proof: Let |S| = 1, so the system consists of single component A. For NESS, we require exchange differential ε = g(I, O, N). With one component:

Case 1: A serves as I (internal). Then O and N must be null or identical to A. If O = ∅, there is nothing external to exchange with; ε = 0 trivially. If O = A, then I = O, violating distinction (N1). Exchange requires difference; identity precludes it.

Case 2: Similar reasoning shows A cannot serve as O or N alone while generating non-trivial exchange.

In all cases, |S| = 1 cannot satisfy the conditions for ε ≠ 0 dynamics. The system collapses to equilibrium. □

Lemma 4.2 (Dyadic Insufficiency): A two-component system cannot sustain NESS.

Proof: Let |S| = 2, so S = {A, B}. We must assign functional roles I, O, N to these two components.

Case 1: I = A, O = B, N = ∅. Without N, there is no boundary enabling exchange. A and B are either isolated (no exchange, equilibrium) or merged (distinction violated). Neither permits NESS.

Case 2: I = A, O = B, N = A. Then I = N—the internal process is identical to the boundary. But N must satisfy non-absorption (N3): it cannot be identical to either relatum. A cannot simultaneously be the internal state and the boundary separating internal from external.

Contradiction.

Case 3: I = A, O = B, N = B. Then O = N—the external field is identical to the boundary. Same contradiction as Case 2.

Case 4: I = A, O = B, N = f(A, B) for some construction f. This attempts to derive N from {A, B}. But N must be functionally distinct from I and O to satisfy non-absorption. Any f(A, B) is defined in terms of A and B, making N dependent on its relata. The boundary cannot be constructed from what it bounds—it must be independently specified. This is the architectural impossibility.

All assignments fail. |S| = 2 cannot sustain NESS. □

Theorem 4.1 Proof: By Lemmas 4.1 and 4.2, |S| < 3 cannot sustain NESS. We must show |S| = 3 is sufficient and that the structure is irreducible.

Sufficiency: With |S| = 3 and S = {I, O, N} where each functional role has a distinct bearer, all conditions N1-N4 can be satisfied without contradiction. The internal process I is distinct from external field O; boundary N is distinct from both; exchange proceeds bidirectionally across N; conservation can hold over cycles.

Irreducibility: Suppose the triadic structure could be derived from a simpler configuration S* with |S*| < 3 via some operation sequence Φ. Then Φ(S*) = {I, O, N} with all properties N1-N4. But Lemmas 4.1 and 4.2 establish that no S* with |S*| < 3 can satisfy these conditions. Therefore no such Φ exists. The triadic minimum is primitive, not derived. □

4.3 Interpretation

The triadic minimum theorem establishes that NESS requires exactly three functionally distinct components: something to serve as interior, something to serve as exterior, and something to serve as boundary between them. This is not a contingent feature of particular systems but a structural necessity.

The theorem explains why both Prigogine's and Friston's frameworks require boundary structures: without the third component (N), the exchange dynamics sustaining far-from-equilibrium organization are architecturally impossible.

5. DERIVATION OF THE CONSERVATION CONSTRAINT

5.1 From Thermodynamic First Principles

For a system in contact with a reservoir at temperature T, the Second Law states:

dS_total/dt = dS_system/dt + dS_reservoir/dt ≥ 0

For a NESS, the system entropy is constant (dS_system/dt = 0). All entropy production flows to the reservoir:

dS_reservoir/dt = σ ≥ 0

where σ is the entropy production rate.

Now consider the exchange differential ε across the boundary N. Net inward flow (ε > 0) corresponds to free energy accumulation; net outward flow (ε < 0) corresponds to dissipation. At steady state, accumulation and dissipation must balance over complete cycles:

∮ε dt = 0

This is the conservation constraint. It does not require ε = 0 at each instant—local accumulation and dissipation are permitted—but total exchange over any complete cycle must integrate to zero.

5.2 Equivalence to Steady-State Conditions

Proposition 5.1: The conservation constraint ∮ε dt = 0 is equivalent to Prigogine's steady-state condition.

Proof: Prigogine's NESS condition is dS_system/dt = 0, implying σ_internal = -Φ_exchange. Over a cycle, this requires that total entropy exported equals total entropy produced. In terms of exchange differential: what accumulates (ε > 0 phases) must equal what dissipates (ε < 0 phases). Hence ∮ε dt = 0. □

Proposition 5.2: The conservation constraint ∮ε dt = 0 is equivalent to free energy minimization at steady state.

Proof: Under the FEP, systems minimize variational free energy F. At steady state, dF/dt = 0, implying that free energy gained from environment equals free energy dissipated through action. This balance condition is precisely ∮ε dt = 0 when ε represents free energy differential across the Markov blanket. □

5.3 Physical Interpretation

The conservation constraint has a clear physical meaning: what goes out must equal what comes in over complete cycles. A system cannot indefinitely accumulate without dissipating, nor indefinitely dissipate without accumulating. The boundary N enables the exchange that makes this balance possible.

Violation of conservation implies:

• ∮ε dt > 0: Net accumulation without dissipation → unbounded growth → eventual collapse

• ∮ε dt < 0: Net dissipation without accumulation → unbounded depletion → collapse to equilibrium

Only ∮ε dt = 0 permits sustained non-equilibrium organization.

6. UNIFICATION: THE N-FUNCTION AS COMMON STRUCTURE

6.1 Mapping Between Frameworks

The N-function provides a common language for translating between Prigogine's and Friston's frameworks:

| Prigogine | Friston | N-Function Framework |

| Internal organization | Internal states μ | I (Observer function) |

| Environment/reservoir | External states η | O (Observed field) |

| System boundary | Markov blanket b | N (Relational ground) |

| Entropy flow Φ | Free energy gradient | Exchange differential ε |

| Steady state dS/dt = 0 | Free energy minimum | Conservation ∮ε dt = 0 |

6.2 What the Unification Reveals

The unification reveals that Markov blankets and dissipative boundaries are not merely analogous but structurally identical—both are instantiations of the N-function satisfying conditions N1-N4. The frameworks describe the same architecture in different vocabularies.

This has immediate consequences:

Cross-domain prediction: Results established in one framework transfer to the other. If a theorem holds for Markov blankets, it holds for dissipative boundaries (and vice versa), modulo appropriate translation.

Minimal architecture: Both frameworks have implicitly assumed what the triadic minimum theorem proves: you cannot build a NESS without all three components {I, O, N}. The boundary is not optional.

Conservation as universal: The constraint ∮ε dt = 0 operates in both domains. It is not specific to thermodynamics or information theory but is a structural requirement of the triadic architecture itself.

6.3 Beyond Existing Frameworks

The N-function framework goes beyond both Prigogine and Friston in one crucial respect: it proves irreducibility. Neither framework establishes that their boundary structures are necessary rather than merely sufficient. The triadic minimum theorem fills this gap.

This has implications for any theory proposing to explain non-equilibrium organization. Such a theory must either:

(a) Instantiate the triadic minimum explicitly, or

(b) Demonstrate how NESS is achievable without it (falsifying the theorem)

No third option exists. The architectural constraint is not negotiable.

7. FALSIFICATION CONDITIONS AND EXPERIMENTAL PROTOCOLS

7.1 Specific Falsifiable Predictions

The framework generates precise predictions that can be experimentally tested:

Prediction F1 (Triadic Necessity): No system maintaining NESS will be found with fewer than three functionally distinct components (I, O, N).

Test protocol: Identify candidate NESS systems. Analyze functional architecture. Attempt to reduce to dyadic or monadic configuration while preserving NESS. If successful, F1 is falsified.

Prediction F2 (N-Removal Collapse): Removal or degradation of the N-function in any NESS will produce collapse toward equilibrium.

Test protocol: In biological systems, disrupt membrane integrity; in Bénard cells, remove boundary conditions; in neural systems, degrade Markov blanket structure. Measure time-to-equilibrium. The framework predicts monotonic relationship: greater N-degradation → faster collapse.

Prediction F3 (Dyadic Failure): Dyadic configurations will fail to sustain NESS regardless of energy input.

Test protocol: Construct systems with exactly two components and maximal energy throughput. If any such system maintains NESS, F3 is falsified.

Prediction F4 (Conservation Universality): All NESS will satisfy ∮ε dt = 0 when ε is properly operationalized for the domain.

Test protocol: Measure exchange differential across N-function over multiple cycles. Compute ∮ε dt. Persistent non-zero values falsify F4.

7.2 Experimental Systems

Suitable experimental systems for testing include:

Chemical: Belousov-Zhabotinsky reaction in controlled reactors with manipulable boundary conditions

Physical: Bénard convection cells with adjustable thermal boundaries

Biological: Isolated cells with controllable membrane permeability

Computational: Active inference agents with modifiable Markov blanket structure

Each system permits controlled manipulation of I, O, and N components, enabling direct test of predictions F1-F4.

7.3 What Would Constitute Falsification

The framework is falsified by:

• A demonstrated NESS with |S| < 3 functional components

• A demonstrated NESS persisting after complete N-function removal

• A demonstrated NESS with ∮ε dt ≠ 0 over complete cycles

• A derivation of triadic structure from dyadic base (proving reducibility)

Any of these would require fundamental revision of the framework.

8. DISCUSSION

8.1 Relation to Emergence

The triadic minimum theorem has implications for emergence. If NESS requires irreducible triadic architecture, then the boundary component N cannot emerge from systems lacking it. N is either present primitively or constructed from components already possessing boundary-enabling properties.

This does not preclude hierarchical organization—triadic systems can combine to form higher-order triadic systems—but it constrains bottom-up emergence: you cannot build a boundary from non-boundary components. The N-function is architecturally primitive at whatever level it first appears.

8.2 Relation to Autopoiesis

The framework resonates with Maturana and Varela's autopoiesis (Maturana & Varela, 1980), which identifies self-production through boundary maintenance as constitutive of life. The N-function formalizes the boundary structure autopoiesis requires. The triadic minimum specifies that autopoietic organization requires exactly three components—the producing system (I), the environment (O), and the self-produced boundary (N).

8.3 Limitations

Several limitations should be noted:

Functional vs. substantial: The theorem concerns functional components, not physical substances. Three functions could in principle be implemented by fewer physical substrates, provided functional distinction is maintained. The theorem constrains architecture, not ontology.

Operationalization: The exchange differential ε requires domain-specific operationalization. The framework specifies what must be measured but not always how to measure it.

Scope: The theorem applies to NESS. Systems with different dynamics (transient organization, oscillatory states) may have different architectural constraints.

9. CONCLUSION

We have established that non-equilibrium steady states require a minimal triadic architecture: internal process (I), external field (O), and relational boundary (N). This structure is irreducible—it cannot be derived from simpler configurations—and it unifies Prigogine's dissipative structure theory with Friston's free energy principle and Markov blanket formalism.

The N-function emerges as the critical operator: it is what Markov blankets and dissipative boundaries have in common, formalized and proven necessary. The conservation constraint ∮ε dt = 0 is derived from thermodynamic first principles and shown equivalent to steady-state conditions in both frameworks.

The framework generates falsifiable predictions: triadic necessity, N-removal collapse, dyadic failure, and conservation universality. Experimental protocols are specified for each.

Most fundamentally, the triadic minimum theorem answers the question both frameworks left open: why is boundary structure necessary for non-equilibrium organization? The answer is architectural. Exchange requires distinction; distinction requires separation; separation requires boundary; boundary requires a third component distinct from what it separates. The N-function is not optional. It is the condition for everything else.

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DOCUMENT CONTROL

Title: The Triadic Minimum for Non-Equilibrium Steady States: Generalizing Markov Blankets and Dissipative Structure Theory Through the N-Function

Author: Don L. Gaconnet

Institution: LifePillar Institute for Recursive Sciences

ORCID: 0009-0001-6174-8384

Date: January 2026

Classification: Non-Equilibrium Thermodynamics / Theoretical Biology / Complex Systems

Keywords: non-equilibrium thermodynamics, Markov blankets, free energy principle, dissipative structures, self-organization, entropy production, triadic architecture, N-function

© 2026 Don L. Gaconnet. All Rights Reserved.