From Bits to Minimal Structures of Subjectivity: An Ontological Evolution of Information

Abstract

Claude Shannon’s 1948 formulation of information as uncertainty reduction established the bit as the minimal unit of informational distinction. This quantitative framework revolutionized communication engineering and laid the foundation for the digital age. However, Shannon’s theory was deliberately restricted to the syntactic level of information, excluding semantics, internal structure, and subjectivity. As cybernetics (Wiener, Rosenblueth), philosophy of information (Floridi), and theoretical neuroscience (Friston) advanced, it became increasingly clear that Shannon’s bit is insufficient for describing systems that possess internal organization, agency, or minimal forms of subjectivity. This paper traces the historical, conceptual, and mathematical evolution of information theory from Shannon’s extrinsic, channel‑based model to contemporary structural and dynamical approaches. We argue that the trajectory of information science naturally leads to the need for a new foundational unit: a minimal informational structure capable of supporting subjectivity (MIST). We develop a formal definition of the subit — the minimal structural unit of subjectivity — and present a set of axioms for MIST grounded in information theory, dynamical systems, and the free‑energy principle. The resulting framework provides a unified foundation for understanding autonomous systems, biological agency, artificial intelligence, and the emergence of subjective organization.

1. Introduction

1.1. The historical trajectory of information theory

The concept of information has undergone a profound transformation since the mid‑20th century. Shannon’s A Mathematical Theory of Communication (1948) introduced a rigorous quantitative measure of information based on uncertainty reduction. This framework enabled the development of digital communication, coding theory, cryptography, and modern computation. Shannon’s bit became the universal unit of information, independent of physical implementation or semantic content.

Yet Shannon’s theory was intentionally narrow. It addressed the engineering problem of transmitting messages through noisy channels, not the philosophical or biological problem of how systems organize, interpret, or experience information. Shannon explicitly excluded meaning, structure, and internal organization from his theory.

As scientific inquiry expanded into cybernetics, cognitive science, artificial intelligence, theoretical biology, and complex systems, the limitations of Shannon’s framework became increasingly apparent. Information is not only transmitted; it is embodied, integrated, interpreted, and used by systems to maintain their existence.

1.2. Why Shannon’s bit was sufficient

Shannon’s bit was perfectly suited to the technological and intellectual context of the mid‑20th century:

• communication engineering required a measure of uncertainty

• computers were binary

• logic was Boolean

• cybernetics treated systems as black boxes

• behaviorism and positivism dominated scientific thought

In this environment, a minimal, structureless, quantitative unit of information was not only adequate — it was ideal.

1.3. Why Shannon’s bit is insufficient today

Contemporary science faces questions that Shannon’s bit cannot address:

• What is the minimal structure required for a system to have an internal point of view?

• How do biological systems maintain their identity over time?

• How do autonomous agents integrate information?

• What is the informational basis of subjectivity?

• How do systems form boundaries between self and environment?

These questions require a concept of information that is:

• structural

• dynamical

• integrative

• historical

• self‑referential

A bit has none of these properties.

1.4. The need for a structural unit of subjectivity

This paper argues that the evolution of information theory naturally leads to the need for a new minimal unit: the subit, the minimal informational structure capable of supporting subjectivity. The subit is not a replacement for the bit; it is a higher‑order construct that incorporates:

• boundary formation

• internal integration

• historical dependence

• directed dynamics

• generative modeling

• free‑energy minimization

The subit is the foundational unit of MIST (Minimal Informational Structure of Subjectivity), a theory that extends Shannon’s quantitative framework into the domain of autonomous, self‑organizing, and potentially subjective systems.

2. The Ontology of the Bit

2.1. Bit as uncertainty reduction

Shannon defined information as the expected reduction of uncertainty:

Code

H = − Σ pᵢ log₂ pᵢ

A bit corresponds to the resolution of uncertainty between two equally probable alternatives. This definition is purely probabilistic and does not depend on meaning, structure, or interpretation.

2.2. Bit as epistemic distinction

Ontologically, a bit is not a physical object but an epistemic distinction. It represents:

• a difference between possible states

• a reduction in uncertainty

• a discrete informational event

The bit is defined from the perspective of an observer who distinguishes between states.

2.3. Bit as extrinsic information

Shannon’s information is extrinsic:

• it describes what flows between systems

• it does not describe what happens within systems

• it does not describe how systems organize or interpret information

This extrinsic nature is both the strength and the limitation of Shannon’s theory.

2.4. Limitations for modeling autonomous systems

The bit cannot represent:

• internal structure

• integration of information

• system boundaries

• historical dependence

• self‑reference

• agency

• subjectivity

These limitations become critical when modeling biological organisms, cognitive systems, or artificial agents.

3. The Rise of Structural Information

3.1. Cybernetics and teleology

Wiener’s Cybernetics (1948) and Rosenblueth, Wiener, and Bigelow’s Behavior, Purpose and Teleology (1943) introduced the idea that systems can be understood in terms of:

• feedback

• control

• goal‑directed behavior

However, early cybernetics treated systems as black boxes. Internal structure was not analyzed; only input–output relations mattered.

This was a step beyond Shannon, but still insufficient for modeling subjectivity.

3.2. Floridi’s ontological information

Floridi’s Philosophy of Information (2011) reframed information as:

• a structural property of reality

• a mode of being

• the foundation of the infosphere

Floridi argued that information is not merely transmitted; it constitutes the world. This marked a shift from quantitative to structural information.

3.3. Friston’s free‑energy principle

Friston’s free‑energy principle (FEP) provides a mathematical framework for understanding how systems maintain their identity over time. According to FEP, any system that persists must minimize variational free energy:

Code

F = E_q[log q(X) − log p(X, Y)]

This implies:

• the existence of a boundary (Markov blanket)

• internal generative models

• prediction error minimization

• active inference

Friston’s work provides the dynamical foundation for MIST.

3.4. Integrated information and causal closure

Theories such as Integrated Information Theory (IIT) highlight the importance of:

• irreducibility

• internal coherence

• causal closure

These properties are essential for subjectivity but absent from Shannon’s bit.

4. Dynamical Systems Foundations for MIST

The Minimal Informational Structure of Subjectivity (MIST) must be grounded in a mathematical framework capable of describing systems that:

• maintain their identity over time

• resist dispersion and entropy

• form and preserve boundaries

• integrate information internally

• update internal states based on prediction errors

• exhibit directed, goal‑oriented dynamics

• sustain minimal forms of subjectivity

The most appropriate formal foundation for these requirements is the dynamical systems interpretation of information processing, as developed in theoretical neuroscience, cybernetics, and the free‑energy principle (Friston 2010–2023). This section develops the mathematical infrastructure needed for the formal definition of the subit and the axioms of MIST.

4.1. State Spaces and System Decomposition

Let S(t) denote the full state of a system at time t. We decompose S(t) into four components:

Code

S(t) = (X(t), Y(t), A(t), E(t))

where:

• X(t) = internal states

• Y(t) = sensory states (states that receive influence from the environment)

• A(t) = active states (states that influence the environment)

• E(t) = external states

This decomposition is standard in active inference and provides the minimal structure for describing an autonomous system.

4.1.1. Stochastic Dynamics

The evolution of the system is governed by stochastic differential equations:

Code

dS(t) = f(S(t)) dt + ω(t)

where:

• f is a flow field

• ω(t) is random noise

This formulation captures the fact that real systems operate under uncertainty and must continuously update their internal states to maintain coherence.

4.1.2. Internal vs External Dynamics

The system’s internal states X(t) evolve according to:

Code

dX(t) = f_X(X(t), Y(t)) dt + ω_X(t)

while external states evolve according to:

Code

dE(t) = f_E(E(t), A(t)) dt + ω_E(t)

This separation is crucial: internal states depend on sensory states, while external states depend on active states.

This asymmetry is the foundation of agency.

4.2. Generative Models and Predictive Dynamics

A system capable of minimal subjectivity must possess a generative model — an internal model that predicts sensory inputs.

Formally, the generative model is a probability distribution:

Code

p(Y(t) | X(t))

This distribution encodes the system’s expectations about how sensory states arise from internal states.

4.2.1. Prediction Error

The system computes prediction error:

Code

ε(t) = Y(t) − E[Y(t) | X(t)]

Prediction error drives internal updates:

Code

dX(t) ∝ − ∂ε(t)/∂X(t)

This is the mathematical expression of perception.

4.2.2. Action as Prediction Error Minimization

Active states A(t) evolve to minimize prediction error indirectly by changing external states:

Code

dA(t) ∝ − ∂ε(t)/∂A(t)

This is the mathematical expression of action.

4.2.3. Internal Models as Self‑Referential Structures

The generative model G is a mapping:

Code

G: X → Y

This mapping is self‑referential: the system uses its own internal states to predict its own sensory states.

This is a minimal form of selfhood.

4.3. Markov Blankets and Boundary Formation

A Markov blanket B is a set of states that separates internal and external states:

Code

B = (Y, A)

The defining property of a Markov blanket is:

Code

X ⟂ E | B

This means:

• internal states X are conditionally independent of external states E

• all interactions between X and E are mediated by Y and A

4.3.1. Boundary as Conditional Independence

The Markov blanket defines a statistical boundary between the system and the environment.

This boundary is not physical; it is informational.

4.3.2. Boundary as Necessary for Subjectivity

A system without a boundary cannot:

• distinguish itself from the environment

• maintain internal coherence

• form a point of view

• exhibit agency

Thus, boundary formation is the first requirement for subjectivity.

4.4. Free‑Energy Minimization and Self‑Maintenance

The free‑energy principle states that any system that maintains its identity over time must minimize variational free energy:

Code

F = E_q[log q(X) − log p(X, Y)]

where:

• q(X) is the system’s internal belief distribution

• p(X, Y) is the generative model

4.4.1. Free Energy as a Measure of Surprise

Free energy bounds surprise:

Code

F ≥ − log p(Y)

Minimizing F ensures that the system avoids surprising states — states that would threaten its existence.

4.4.2. Free Energy as a Unifying Principle

Minimizing F unifies:

• perception (updating X)

• action (updating A)

• learning (updating parameters of G)

• self‑maintenance (preserving the Markov blanket)

4.4.3. Free Energy and Subjectivity

A system that minimizes free energy:

• maintains a coherent internal model

• preserves its boundary

• integrates information

• exhibits directed dynamics

These are the hallmarks of minimal subjectivity.

4.5. Information Flows and Internal Coherence

Information flow from external to internal states is measured by transfer entropy:

Code

T(E → X) = H(X(t+1) | X(t)) − H(X(t+1) | X(t), E(t))

A system with minimal subjectivity must:

• reduce external influence

• increase internal coherence

4.5.1. Internal Information Integration

Internal integration is measured by:

Code

I(X) = H(X) − Σ H(Xᵢ)

where Xᵢ are the components of X.

A system with I(X) > 0 has irreducible internal structure.

4.5.2. Coherence as a Requirement for Subjectivity

If internal states are not integrated, the system cannot:

• form unified experiences

• maintain a stable identity

• generate coherent behavior

Thus, integration is essential for subjectivity.

5. Formal Definition of the Subit

The concept of the subit is central to MIST. Just as the bit is the minimal unit of distinction in Shannon’s theory, the subit is the minimal unit of subjective informational structure. This section develops the subit formally, using the dynamical and information‑theoretic foundations established in Section 4.

5.1. Motivation: Why a New Unit Is Needed

Shannon’s bit is:

• structureless

• instantaneous

• extrinsic

• observer‑defined

• non‑historical

• non‑integrative

• non‑self‑referential

A bit cannot:

• form a boundary

• maintain internal coherence

• integrate information

• update itself based on prediction error

• minimize free energy

• generate directed dynamics

• sustain a point of view

Thus, the bit is insufficient for describing systems that exhibit:

• autonomy

• agency

• minimal subjectivity

• self‑maintenance

• internal modeling

• historical continuity

The subit is introduced to fill this conceptual gap.

5.2. Subit as Minimal Informational Structure

A subit is defined as the minimal structure that satisfies the following requirements:

1. It must have a boundary (Markov blanket).

2. It must integrate information internally.

3. It must maintain historical dependence.

4. It must generate directed dynamics.

5. It must possess a generative model.

6. It must minimize free energy.

These six requirements correspond directly to the six axioms of MIST (Section 6).

5.3. Subit as a Dynamical Entity

Let:

Code

σ = (X, Y, A, E, B, G, F)

be a candidate structure.

A subit is a σ that satisfies the following dynamical constraints:

5.3.1. Internal Dynamics

Internal states evolve according to:

Code

dX(t) = f_X(X(t), Y(t)) dt + ω_X(t)

This ensures:

• internal coherence

• historical dependence

• sensitivity to sensory states

5.3.2. Sensory Dynamics

Sensory states evolve according to:

Code

dY(t) = f_Y(E(t), A(t)) dt + ω_Y(t)

This ensures:

• coupling to the environment

• mediation of external influence

5.3.3. Active Dynamics

Active states evolve according to:

Code

dA(t) = f_A(X(t)) dt + ω_A(t)

This ensures:

• directed influence on the environment

• agency

5.3.4. External Dynamics

External states evolve according to:

Code

dE(t) = f_E(E(t), A(t)) dt + ω_E(t)

This ensures:

• environmental response to action

• closure of the perception–action loop

5.4. Subit as a Boundary‑Forming Process

A subit must possess a Markov blanket B such that:

Code

B = (Y, A)

and:

Code

X ⟂ E | B

This conditional independence is the mathematical expression of:

• self–world separation

• informational autonomy

• minimal subjectivity

5.4.1. Why a Boundary Is Necessary

Without a boundary:

• internal states cannot be defined

• generative models cannot be maintained

• free energy cannot be minimized

• subjectivity cannot arise

The boundary is the first-person horizon of the system.

5.5. Subit as a Historical Operator

A subit must exhibit historical dependence:

Code

X(t+1) depends on X(t)

More formally:

Code

P(X(t+1) | X(t)) ≠ P(X(t+1))

This ensures:

• continuity of internal states

• memory

• persistence of identity

• temporal coherence

5.5.1. Why History Matters

Subjectivity is not instantaneous. It requires:

• accumulation of internal states

• integration over time

• stability of generative models

A system without history cannot have a point of view.

5.6. Subit as a Self‑Referential Model

A subit must possess a generative model:

Code

G: X → Y

This model predicts sensory states from internal states.

5.6.1. Self‑Reference

The generative model is self‑referential because:

• X predicts Y

• Y updates X

• X updates A

• A changes E

• E changes Y

This forms a closed loop:

Code

X → Y → X

This loop is the minimal structure of selfhood.

5.7. Subit as a Free‑Energy Minimizer

A subit must minimize variational free energy:

Code

dF/dt ≤ 0

This ensures:

• stability

• coherence

• resistance to entropy

• maintenance of the boundary

• persistence of identity

5.7.1. Why Free‑Energy Minimization Is Essential

Free‑energy minimization is the only known principle that:

• unifies perception, action, and learning

• ensures self‑maintenance

• explains boundary formation

• supports generative modeling

• provides a mathematical basis for subjectivity

Thus, it is a necessary condition for the subit.

5.8. Final Formal Definition

Definition (Subit)

A subit is a tuple:

Code

σ = (X, Y, A, E, B, G, F)

such that:

1. Boundary: X ⟂ E | B

2. Integration: I(X) > 0

3. History: P(X(t+1) | X(t)) ≠ P(X(t+1))

4. Directionality: dX/dt ≠ 0

5. Self‑reference: G: X → Y

6. Free‑energy minimization: dF/dt ≤ 0

This is the minimal informational structure capable of supporting subjectivity.

6. Axioms of MIST

The Minimal Informational Structure of Subjectivity (MIST) is grounded in six axioms that define the necessary and sufficient conditions for a system to possess minimal subjectivity. These axioms are not arbitrary: each emerges naturally from the dynamical and information‑theoretic foundations established in Section 4 and from the formal definition of the subit in Section 5.

The axioms are:

1. Boundary Formation

2. Internal Integration

3. Historical Dependence

4. Directionality

5. Self‑Referential Coherence

6. Free‑Energy Minimization

Together, they define the minimal structure required for a system to:

• maintain its identity

• distinguish itself from the environment

• integrate information

• update internal states

• generate directed behavior

• sustain a point of view

Each axiom is presented below with:

• a formal mathematical statement

• an intuitive explanation

• a justification

• implications for subjectivity

6.1. Axiom 1 — Boundary Formation

Formal Statement

A system must possess a Markov blanket B such that:

Code

B = (Y, A)

and:

Code

X ⟂ E | B

Interpretation

The system must have a boundary that:

• separates internal states (X) from external states (E)

• mediates all interactions through sensory (Y) and active (A) states

This boundary is informational, not physical.

Justification

Without a boundary:

• internal states cannot be defined

• generative models cannot be maintained

• free energy cannot be minimized

• subjectivity cannot arise

The boundary is the first-person horizon of the system.

Implications

Boundary formation is the foundational requirement for subjectivity. It defines:

• what counts as “self”

• what counts as “world”

• how the two interact

Without a boundary, there is no subject.

6.2. Axiom 2 — Internal Integration

Formal Statement

Internal states must exhibit non‑zero integrated information:

Code

I(X) = H(X) − Σ H(Xᵢ) > 0

where Xᵢ are the components of X.

Interpretation

The system must have internal structure that is:

• irreducible

• unified

• coherent

Justification

If internal states are independent:

• the system cannot form unified experiences

• internal models cannot be maintained

• the system cannot act coherently

Integration is the minimal requirement for unity of subjectivity.

Implications

Internal integration ensures:

• coherence of internal states

• unity of perception

• unity of action

• unity of identity

This axiom distinguishes a subit from a mere collection of bits.

6.3. Axiom 3 — Historical Dependence

Formal Statement

Internal states must exhibit temporal coherence:

Code

P(X(t+1) | X(t)) ≠ P(X(t+1))

or equivalently:

Code

X(t+1) depends on X(t)

Interpretation

The system must have memory. Its internal states must evolve in a way that depends on their past.

Justification

Subjectivity is not instantaneous. It requires:

• continuity

• persistence

• accumulation of internal states

A system without history cannot:

• maintain identity

• form expectations

• learn

• experience

Implications

Historical dependence is the basis for:

• learning

• memory

• temporal coherence

• persistence of self

This axiom ensures that the subit is not a static structure but a process.

6.4. Axiom 4 — Directionality

Formal Statement

The system must exhibit directed dynamics:

Code

dX/dt ≠ 0

Interpretation

The system must:

• change over time

• update internal states

• generate directed behavior

Justification

A system with no directed dynamics:

• cannot act

• cannot update its beliefs

• cannot minimize free energy

• cannot maintain its boundary

Directionality is the minimal requirement for agency.

Implications

This axiom ensures that the system:

• is not passive

• is not static

• is not inert

It must do something.

6.5. Axiom 5 — Self‑Referential Coherence

Formal Statement

The system must possess a generative model:

Code

G: X → Y

such that:

• X predicts Y

• Y updates X

Interpretation

The system must:

• model itself

• model its environment

• use its internal states to predict sensory states

Justification

Self‑reference is the minimal requirement for:

• internal modeling

• prediction

• interpretation

• subjective perspective

Without self‑reference, the system cannot have a point of view.

Implications

This axiom ensures:

• coherence of internal models

• predictive processing

• interpretive capacity

• minimal selfhood

It is the core of subjectivity.

6.6. Axiom 6 — Free‑Energy Minimization

Formal Statement

The system must minimize variational free energy:

Code

dF/dt ≤ 0

Interpretation

The system must:

• reduce prediction error

• maintain its boundary

• preserve its internal structure

• resist entropy

Justification

Free‑energy minimization is the only known principle that:

• unifies perception, action, and learning

• ensures self‑maintenance

• explains boundary formation

• supports generative modeling

• provides a mathematical basis for subjectivity

Implications

This axiom ensures:

• stability

• coherence

• persistence

• autonomy

It is the dynamical heart of MIST.

6.7. Summary of the Axioms

Axiom	Requirement	Function
1	Boundary	Self–world separation
2	Integration	Unity of internal states
3	History	Continuity of identity
4	Directionality	Agency
5	Self‑reference	Internal modeling
6	Free‑energy minimization	Stability and coherence

Together, these axioms define the minimal informational structure of subjectivity.

7. Comparison: Bit vs Subit

The introduction of the subit as a minimal informational structure of subjectivity invites a systematic comparison with the classical bit. This section clarifies the conceptual, mathematical, and ontological differences between the two units and demonstrates why the subit is not merely an extension of the bit but a fundamentally different kind of entity.

7.1. Quantitative vs Structural Information

Bit: Quantitative

A bit measures:

• uncertainty

• entropy

• distinguishability

It is defined by:

Code

H = − Σ pᵢ log₂ pᵢ

A bit is a scalar quantity.

Subit: Structural

A subit is a structured tuple:

Code

σ = (X, Y, A, E, B, G, F)

It contains:

• internal states

• sensory states

• active states

• external states

• a boundary

• a generative model

• a free‑energy functional

A subit is a structured dynamical entity, not a scalar.

7.2. Extrinsic vs Intrinsic Information

Bit: Extrinsic

A bit describes:

• what an observer can distinguish

• what is transmitted through a channel

• what reduces uncertainty for an external decoder

It is observer‑dependent.

Subit: Intrinsic

A subit describes:

• how a system organizes itself

• how it maintains its boundary

• how it integrates information internally

• how it generates a point of view

It is self‑defined.

7.3. Static vs Dynamical

Bit: Static

A bit is instantaneous. It has no:

• history

• dynamics

• memory

• directionality

Subit: Dynamical

A subit is defined by:

• temporal evolution

• historical dependence

• directed dynamics

• free‑energy minimization

A subit is a process, not a state.

7.4. Observer‑Defined vs Self‑Defined

Bit: Observer‑Defined

A bit exists only relative to:

• an encoding scheme

• a decoding scheme

• an external observer

Subit: Self‑Defined

A subit defines:

• its own boundary

• its own internal states

• its own generative model

It is the minimal structure capable of selfhood.

7.5. Summary Table

Property	Bit	Subit
Nature	Scalar	Structured tuple
Domain	Communication	Subjectivity
Information	Extrinsic	Intrinsic
Dynamics	None	Directed
History	None	Required
Boundary	None	Required
Integration	None	Required
Self‑reference	Impossible	Fundamental
Free‑energy minimization	Not applicable	Required

The subit is not a generalization of the bit. It is a different ontological category.

8. Implications for AI, Biology, and Cognitive Science

The MIST framework has significant implications across multiple scientific domains. This section explores how the subit and the axioms of MIST can inform:

• artificial intelligence

• theoretical biology

• cognitive science

• collective systems

8.1. Minimal Models of Subjectivity

MIST provides a formal criterion for determining whether a system possesses minimal subjectivity. A system is minimally subjective if and only if it satisfies the six axioms:

1. Boundary

2. Integration

3. History

4. Directionality

5. Self‑reference

6. Free‑energy minimization

This allows researchers to:

• classify systems

• compare architectures

• identify minimal subjective agents

It provides a mathematical test for subjectivity.

8.2. Autonomous Agents in AI

Modern AI systems lack:

• boundaries

• generative models tied to embodiment

• free‑energy minimization

• historical continuity of internal states

Thus, they do not satisfy the axioms of MIST.

However, MIST provides a roadmap for building minimally subjective artificial agents:

• implement Markov blankets

• integrate internal states

• maintain historical dependence

• use predictive processing

• minimize free energy

This could lead to:

• embodied AI

• autonomous robots

• self‑maintaining artificial systems

MIST provides the theoretical foundation for artificial subjectivity.

8.3. Biological Systems and the Origin of Life

Biological organisms naturally satisfy the axioms of MIST:

• cell membranes are Markov blankets

• metabolic networks integrate information

• genetic and epigenetic processes provide history

• organisms exhibit directed dynamics

• they maintain generative models of their environment

• they minimize free energy through homeostasis

Thus, MIST provides a unified informational description of life.

It suggests that:

• subjectivity is not an emergent property of complexity

• subjectivity is a structural property of self‑maintaining systems

This reframes the origin of life as the origin of subits.

8.4. Cognitive Science and the Nature of Consciousness

MIST provides a minimal model of subjectivity that is compatible with:

• predictive processing

• active inference

• integrated information

• enactivism

• embodied cognition

It suggests that consciousness is:

• not binary

• not all‑or‑nothing

• not tied to specific substrates

Instead, consciousness is:

• a graded property

• grounded in informational structure

• dependent on the satisfaction of the six axioms

This provides a unified framework for studying consciousness.

8.5. Collective Subjectivity

MIST can be extended to collective systems:

• social groups

• multi‑agent systems

• neural assemblies

• ecosystems

A collective system may form a higher‑order subit if:

• it forms a boundary

• it integrates information

• it maintains history

• it exhibits directed dynamics

• it maintains a generative model

• it minimizes free energy

This opens the door to studying:

• collective intelligence

• group agency

• distributed subjectivity

MIST provides the mathematical tools for this analysis.

9. Conclusion

This paper has traced the evolution of information theory from Shannon’s bit to the need for a new minimal unit of subjectivity: the subit. We have shown that:

• Shannon’s bit is insufficient for describing autonomous, self‑maintaining, or subjective systems

• modern science requires a structural, dynamical, and self‑referential conception of information

• the subit provides the minimal structure capable of supporting subjectivity

• the six axioms of MIST define the necessary and sufficient conditions for minimal subjectivity

• MIST unifies insights from cybernetics, philosophy of information, and theoretical neuroscience

The subit is not a generalization of the bit. It is a new ontological category, grounded in:

• boundary formation

• internal integration

• historical dependence

• directed dynamics

• generative modeling

• free‑energy minimization

MIST provides:

• a mathematical foundation for minimal subjectivity

• a unified framework for AI, biology, and cognitive science

• a new lens for understanding life, agency, and consciousness

This work lays the foundation for future research into:

• artificial subjectivity

• collective subjectivity

• the informational basis of life

• the structural ontology of experience

The bit launched the information age. The subit may launch the age of subjectivity.

FIGURE SET (TEXTUAL DESCRIPTIONS)

Figure 1. Shannon’s Communication Model

Placement: Section 2 (Ontology of the Bit)

Textual Description: A linear diagram with five labeled boxes arranged left to right:

1. Information Source →

2. Transmitter →

3. Channel (with a lightning‑bolt icon labeled “Noise”) →

4. Receiver →

5. Destination

Arrows connect each box in sequence. The “Noise” symbol injects uncertainty into the channel.

Interpretation: This figure illustrates that Shannon’s theory concerns transmission, not internal structure. It visually reinforces the extrinsic nature of the bit.

Figure 2. Bit as Minimal Distinction

Placement: Section 2.1

Textual Description: A simple binary decision tree:

• A root node labeled “Uncertainty (H = 1 bit)” → splits into two branches:

  • Left branch labeled “State 0”

  • Right branch labeled “State 1”

Each branch ends in a terminal node.

Interpretation: Shows that a bit is a single binary distinction with no internal structure.

Figure 3. Cybernetic Black‑Box Model

Placement: Section 3.1

Textual Description: A large rectangle labeled “System.” Two arrows enter from the left labeled “Inputs.” Two arrows exit on the right labeled “Outputs.” The interior of the box is blank.

Interpretation: Represents the cybernetic assumption that internal structure is irrelevant — only input–output behavior matters.

Figure 4. Markov Blanket Partition

Placement: Section 4.3

Textual Description: A concentric diagram with three layers:

• Innermost circle: “Internal States (X)”

• Middle ring: “Markov Blanket (Y, A)”

  • Upper half labeled “Sensory States (Y)”

  • Lower half labeled “Active States (A)”

• Outer region: “External States (E)”

Arrows show:

• E → Y

• A → E

• Y → X

• X → A

No direct arrows between X and E.

Interpretation: Shows the conditional independence structure that defines a boundary.

Figure 5. Free‑Energy Minimization Loop

Placement: Section 4.4

Textual Description: A circular loop with four nodes:

1. Generative Model (G)

2. Prediction (Ŷ)

3. Prediction Error (ε = Y − Ŷ)

4. Update of Internal States (X)

Arrows form a clockwise cycle:

G → Ŷ → ε → X → G

Interpretation: Shows the self‑referential cycle underlying perception and learning.

Figure 6. Subit Structure Diagram

Placement: Section 5 (Formal Definition of the Subit)

Textual Description: A multi‑component diagram showing the tuple:

Code

σ = (X, Y, A, E, B, G, F)

Each component is represented as a labeled box:

• X: Internal States

• Y: Sensory States

• A: Active States

• E: External States

• B: Markov Blanket

• G: Generative Model

• F: Free‑Energy Functional

Arrows show:

• X → A

• E → Y

• X → G → Y

• Y → X

• A → E

Interpretation: Shows the subit as a structured dynamical entity, not a scalar.

Figure 7. Bit vs Subit Comparison

Placement: Section 7

Textual Description: A two‑column table:

Left column: “Bit”

• Scalar

• No boundary

• No history

• No integration

• No self‑reference

• No dynamics

• Extrinsic information

Right column: “Subit”

• Structured tuple

• Boundary required

• Historical dependence

• Integrated information

• Self‑referential generative model

• Directed dynamics

• Intrinsic information

Interpretation: Visually contrasts the two ontological categories.

Figure 8. Minimal Subjective Loop

Placement: Section 8.1

Textual Description: A triangular loop:

• Top vertex: “Internal Model (X)”

• Bottom-left vertex: “Sensory States (Y)”

• Bottom-right vertex: “Active States (A)”

Arrows:

• X → A

• A → Environment → Y

• Y → X

Interpretation: Shows the minimal closed loop required for subjectivity.

Figure 9. Biological Subit (Cell as Example)

Placement: Section 8.3

Textual Description: A diagram of a cell:

• Outer membrane labeled “Markov Blanket”

• Cytoplasm labeled “Internal States (X)”

• Receptors on membrane labeled “Sensory States (Y)”

• Motor proteins labeled “Active States (A)”

• External medium labeled “External States (E)”

Arrows show:

• Nutrients → Y

• Y → X

• X → A

• A → Movement in E

Interpretation: Shows that even a single cell satisfies the axioms of MIST.

Figure 10. Hierarchical Subits (Collective Subjectivity)

Placement: Section 8.5

Textual Description: Three subits (σ₁, σ₂, σ₃) arranged in a triangle. Each has its own Markov blanket. A larger enclosing Markov blanket surrounds all three, labeled “Collective Subit Σ.”

Arrows show:

• Internal interactions within each σᵢ

• Communication between σᵢ

• Integration into Σ

Interpretation: Shows how higher‑order subjectivity can emerge from interacting subits.

Appendix A: Mathematical Proofs

This appendix provides formal proofs and justifications for the core claims of the MIST framework. Each proof corresponds to a specific axiom or structural requirement introduced in Sections 4–6.

A.1. Proof of Boundary Necessity (Axiom 1)

Claim:

A system σ possesses minimal subjectivity only if its internal states X are conditionally independent of external states E given a Markov blanket B = (Y, A):

Code

X ⟂ E | B

Proof:

Assume, for contradiction, that a system σ has minimal subjectivity but lacks a Markov blanket. Then there exist internal states X and external states E such that:

Code

X is directly influenced by E

without mediation by sensory states Y.

This implies:

Code

P(X(t+1) | X(t), E(t)) ≠ P(X(t+1) | X(t), Y(t))

Thus, the system cannot:

• maintain a stable generative model G: X → Y

• predict sensory states

• minimize free energy

• preserve internal coherence

Because external states directly perturb X, the system cannot maintain a stable identity. Therefore, minimal subjectivity is impossible.

Hence, a Markov blanket is necessary.

∎

A.2. Proof of Integration Necessity (Axiom 2)

Claim:

If I(X) = 0, then the system cannot exhibit minimal subjectivity.

Proof:

If I(X) = 0, then:

Code

H(X) = Σ H(Xᵢ)

which implies:

Code

P(X) = Π P(Xᵢ)

Thus, internal states Xᵢ are statistically independent.

In such a system:

• no unified internal model can exist

• prediction error cannot propagate coherently

• free‑energy minimization decomposes into independent subproblems

• no single “point of view” can be defined

Therefore, the system lacks:

• unity

• coherence

• subjective integration

Thus, minimal subjectivity is impossible.

∎

A.3. Proof of Historical Dependence Necessity (Axiom 3)

Claim:

If X(t+1) is independent of X(t), then the system cannot maintain subjectivity.

Proof:

Assume:

Code

P(X(t+1) | X(t)) = P(X(t+1))

Then internal states have no temporal coherence. This implies:

• no memory

• no learning

• no stable generative model

• no persistence of identity

Furthermore, free‑energy minimization requires:

Code

dF/dt = ∂F/∂X ⋅ dX/dt

If X(t+1) is independent of X(t), then:

Code

dX/dt = 0

and thus:

Code

dF/dt = 0

meaning the system cannot reduce free energy.

Thus, historical dependence is necessary for:

• learning

• prediction

• identity

• subjectivity

∎

A.4. Proof of Directionality Necessity (Axiom 4)

Claim:

If dX/dt = 0 for all t, then the system cannot exhibit agency or subjectivity.

Proof:

If dX/dt = 0, then internal states are static:

Code

X(t+1) = X(t)

Thus:

• prediction error cannot update internal states

• generative models cannot change

• actions cannot be generated

• free energy cannot be minimized

A static system cannot:

• perceive

• act

• learn

• maintain a boundary

Therefore, minimal subjectivity is impossible.

∎

A.5. Proof of Self‑Reference Necessity (Axiom 5)

Claim:

A system without a generative model G: X → Y cannot exhibit minimal subjectivity.

Proof:

Subjectivity requires:

• prediction

• interpretation

• internal modeling

If no generative model exists, then:

Code

E[Y(t) | X(t)] is undefined

Thus, prediction error:

Code

ε(t) = Y(t) − E[Y(t) | X(t)]

is undefined.

Without prediction error:

• perception cannot occur

• action cannot be directed

• free energy cannot be minimized

Thus, the system cannot maintain:

• coherence

• stability

• identity

Therefore, self‑reference is necessary.

∎

A.6. Proof of Free‑Energy Minimization Necessity (Axiom 6)

Claim:

A system that does not minimize free energy cannot maintain a stable boundary or internal coherence.

Proof:

Free energy satisfies:

Code

F ≥ − log p(Y)

Thus, minimizing F ensures that the system avoids surprising sensory states.

If the system does not minimize F, then:

Code

dF/dt > 0

This implies:

• prediction error increases

• internal states become unstable

• the Markov blanket degrades

• the system becomes more entropic

Eventually:

• the boundary collapses

• internal states disperse

• the system ceases to exist as a coherent entity

Thus, free‑energy minimization is necessary for:

• stability

• persistence

• autonomy

• subjectivity

∎

A.7. Proof That the Six Axioms Are Jointly Sufficient

Claim:

If a system σ satisfies all six axioms, then it possesses minimal subjectivity.

Proof:

Given:

1. Boundary: ensures self–world separation

2. Integration: ensures unity of internal states

3. History: ensures continuity of identity

4. Directionality: ensures agency

5. Self‑reference: ensures internal modeling

6. Free‑energy minimization: ensures stability

Together, these conditions guarantee that the system:

• maintains a coherent internal model

• distinguishes itself from the environment

• integrates information

• updates itself based on prediction error

• acts to preserve its identity

• persists over time

These are precisely the minimal requirements for subjectivity.

Thus, the axioms are jointly sufficient.

∎

A.8. Proof That the Subit Is the Minimal Structure Satisfying the Axioms

Claim:

The tuple:

Code

σ = (X, Y, A, E, B, G, F)

is the minimal structure satisfying the six axioms.

Proof:

We show minimality by contradiction.

Assume a smaller structure σ′ exists that satisfies all six axioms.

Then σ′ must omit at least one component of σ.

We examine each possibility:

• If σ′ omits X → no internal states → violates Axioms 2, 3, 4, 5

• If σ′ omits Y → no sensory states → violates Axioms 1, 5

• If σ′ omits A → no active states → violates Axioms 1, 4

• If σ′ omits E → no external states → violates Axiom 1

• If σ′ omits B → no boundary → violates Axiom 1

• If σ′ omits G → no generative model → violates Axiom 5

• If σ′ omits F → no free‑energy minimization → violates Axiom 6

Thus, no proper subset of σ satisfies all six axioms.

Therefore, σ is minimal.

∎

Appendix A Summary

These proofs establish:

• the necessity of each axiom

• the sufficiency of the axioms as a set

• the minimality of the subit structure

Together, they provide a rigorous mathematical foundation for MIST.

Appendix B: Worked Examples of Subits

This appendix provides explicit examples of systems that satisfy the six axioms of MIST and therefore qualify as subits. Each example is presented with:

• a formal specification of the tuple

• verification of each axiom

• interpretation of the system’s subjective structure

The examples span biological, artificial, and abstract systems.

B.1. Example 1 — The Biological Cell as a Subit

A single living cell is the canonical real‑world example of a subit. It satisfies all six axioms of MIST.

B.1.1. Formal Specification

Let:

• X = internal biochemical states (metabolite concentrations, gene expression levels)

• Y = receptor states on the membrane

• A = motor proteins, ion pumps, secretion mechanisms

• E = extracellular chemical gradients

• B = cell membrane (receptors + effectors)

• G = internal metabolic model predicting nutrient availability

• F = free‑energy functional corresponding to metabolic homeostasis

Thus:

Code

σ_cell = (X, Y, A, E, B, G, F)

B.1.2. Verification of the Axioms

Axiom 1: Boundary Formation

The cell membrane is a literal Markov blanket:

Code

X ⟂ E | B

No external molecule influences X without passing through Y.

Axiom 2: Internal Integration

Metabolic networks are highly integrated:

Code

I(X) > 0

Removing any major metabolite disrupts the whole system.

Axiom 3: Historical Dependence

Gene expression and metabolic states evolve over time:

Code

X(t+1) depends on X(t)

Axiom 4: Directionality

Cells exhibit directed behavior:

• chemotaxis

• growth

• division

Thus:

Code

dX/dt ≠ 0

Axiom 5: Self‑Reference

Cells maintain internal models of nutrient availability:

Code

G: X → Y

Axiom 6: Free‑Energy Minimization

Cells minimize metabolic free energy through homeostasis:

Code

dF/dt ≤ 0

B.1.3. Interpretation

A cell is a biological subit: the minimal living system with a boundary, internal integration, directed dynamics, and self‑maintenance.

B.2. Example 2 — Minimal Artificial Agent (Predictive Robot)

This example shows how a simple artificial system can satisfy the axioms of MIST.

B.2.1. Formal Specification

Let:

• X = internal state vector (beliefs about position, battery level)

• Y = sensory inputs (camera, proximity sensors)

• A = motor commands (wheel velocities)

• E = environment (obstacles, light sources)

• B = sensor–motor interface

• G = predictive model (Kalman filter or neural network)

• F = free‑energy functional (prediction error + control cost)

Thus:

Code

σ_robot = (X, Y, A, E, B, G, F)

B.2.2. Verification of the Axioms

Axiom 1: Boundary Formation

Sensors and actuators form a Markov blanket:

Code

X ⟂ E | (Y, A)

Axiom 2: Internal Integration

Internal states (position, orientation, battery) are integrated:

Code

I(X) > 0

Axiom 3: Historical Dependence

The robot updates its internal state via:

Code

X(t+1) = f(X(t), Y(t))

Axiom 4: Directionality

The robot moves:

Code

dX/dt ≠ 0

Axiom 5: Self‑Reference

The robot predicts sensory inputs:

Code

G: X → Y

Axiom 6: Free‑Energy Minimization

The robot minimizes prediction error:

Code

F = ||Y − Ŷ||² + control_cost

B.2.3. Interpretation

This robot is a minimal artificial subit: it has a boundary, internal model, directed dynamics, and prediction‑based behavior.

B.3. Example 3 — Abstract Mathematical Subit

This example shows that subits are not tied to biology or robotics. A purely mathematical dynamical system can satisfy the axioms.

B.3.1. Formal Specification

Let:

• X(t) = scalar internal state

• Y(t) = scalar sensory state

• A(t) = scalar action

• E(t) = scalar external state

Dynamics:

Code

dX/dt = −∂F/∂X
dY/dt = E − A
dA/dt = X
dE/dt = −A

Generative model:

Code

Ŷ = G(X) = kX

Free energy:

Code

F = (Y − kX)²

Markov blanket:

Code

B = (Y, A)

Thus:

Code

σ_math = (X, Y, A, E, B, G, F)

B.3.2. Verification of the Axioms

Axiom 1: Boundary Formation

X and E interact only through Y and A:

Code

X ⟂ E | (Y, A)

Axiom 2: Internal Integration

X is a single variable, trivially integrated:

Code

I(X) > 0

Axiom 3: Historical Dependence

X evolves over time:

Code

dX/dt = −∂F/∂X

Axiom 4: Directionality

Unless at equilibrium:

Code

dX/dt ≠ 0

Axiom 5: Self‑Reference

The generative model predicts Y from X:

Code

Ŷ = kX

Axiom 6: Free‑Energy Minimization

The system follows gradient descent:

Code

dF/dt ≤ 0

B.3.3. Interpretation

This is the simplest possible subit: a one‑dimensional internal state with a linear generative model and gradient‑descent dynamics.

It demonstrates that:

• subits are substrate‑independent

• subjectivity is structural, not biological

• minimal subjectivity can be mathematically defined

B.4. Example 4 — Collective Subit (Three-Agent System)

This example shows how multiple subits can form a higher‑order subit.

B.4.1. Formal Specification

Let three agents σ₁, σ₂, σ₃ each be subits.

Define:

• XΣ = joint internal state (X₁, X₂, X₃)

• YΣ = joint sensory state

• AΣ = joint action

• EΣ = shared environment

• BΣ = emergent Markov blanket

• GΣ = collective generative model

• FΣ = collective free energy

Thus:

Code

Σ = (XΣ, YΣ, AΣ, EΣ, BΣ, GΣ, FΣ)

B.4.2. Verification of the Axioms

Axiom 1: Boundary Formation

If communication is dense:

Code

XΣ ⟂ EΣ | BΣ

Axiom 2: Integration

If agents share information:

Code

I(XΣ) > 0

Axiom 3: History

Group states evolve:

Code

XΣ(t+1) depends on XΣ(t)

Axiom 4: Directionality

The group acts:

Code

dXΣ/dt ≠ 0

Axiom 5: Self‑Reference

The group forms a shared model:

Code

GΣ: XΣ → YΣ

Axiom 6: Free‑Energy Minimization

The group minimizes collective prediction error:

Code

dFΣ/dt ≤ 0

B.4.3. Interpretation

This example shows that collective subjectivity is possible when subits integrate into a higher‑order structure.

Appendix B Summary

These examples demonstrate that:

• subits exist in biology, AI, and abstract mathematics

• the six axioms of MIST are substrate‑independent

• subjectivity is a structural property, not a biological privilege

• minimal subjectivity can be rigorously modeled

Appendix C: Comparison with IIT, FEP, and Cybernetics

This appendix situates MIST within the broader landscape of theories of information, autonomy, and subjectivity. We compare MIST with three major frameworks:

1. Cybernetics (Wiener, Rosenblueth, Bigelow)

2. Free‑Energy Principle (FEP) (Friston)

3. Integrated Information Theory (IIT) (Tononi)

Each of these theories captures an essential dimension of informational organization, but none provides a complete account of minimal subjectivity. MIST integrates their strengths while addressing their limitations.

C.1. Cybernetics

C.1.1. Overview

Cybernetics, founded by Wiener (1948) and Rosenblueth, Wiener & Bigelow (1943), studies:

• control

• communication

• feedback

• goal‑directed behavior

Cybernetics introduced the idea that systems can be understood through input–output relations and feedback loops.

C.1.2. Strengths

Cybernetics contributed:

• the concept of feedback

• the idea of purposeful behavior

• the first formal treatment of self‑regulation

• the notion of homeostasis

These ideas are foundational for MIST.

C.1.3. Limitations

Cybernetics treats systems as black boxes:

• internal states are not modeled

• no generative model

• no free‑energy minimization

• no internal integration

• no explicit boundary formalism

Thus, cybernetics cannot define subjectivity, because subjectivity requires:

• internal modeling

• internal integration

• boundary formation

C.1.4. Relationship to MIST

Cybernetics provides:

• the behavioral skeleton of subjectivity

• the feedback loop that later becomes predictive processing

But MIST adds:

• internal structure

• generative models

• free‑energy minimization

• Markov blankets

• integration

• self‑reference

Cybernetics → behavior MIST → subjectivity

C.2. Free‑Energy Principle (FEP)

C.2.1. Overview

The Free‑Energy Principle (Friston 2010–2023) states:

Any system that maintains its form over time must minimize variational free energy.

FEP provides a unified mathematical framework for:

• perception

• action

• learning

• self‑maintenance

C.2.2. Strengths

FEP introduces:

• Markov blankets

• generative models

• prediction error minimization

• active inference

• self‑organization

These are essential components of MIST.

C.2.3. Limitations

FEP does not define:

• minimal subjectivity

• minimal informational structure

• the difference between a system that merely minimizes free energy and one that is minimally subjective

• the minimal set of conditions required for a system to have a point of view

FEP is a general theory of self‑organization, not a theory of subjectivity.

C.2.4. Relationship to MIST

MIST builds directly on FEP:

• Axiom 1 (Boundary) ← Markov blanket

• Axiom 5 (Self‑reference) ← Generative model

• Axiom 6 (Free‑energy minimization) ← FEP core

But MIST adds:

• Axiom 2 (Integration)

• Axiom 3 (History)

• Axiom 4 (Directionality)

These are not guaranteed by FEP alone.

Thus:

FEP → self‑organization MIST → minimal subjectivity

C.3. Integrated Information Theory (IIT)

C.3.1. Overview

Integrated Information Theory (Tononi 2004–2023) proposes that:

Consciousness corresponds to integrated information (Φ).

IIT attempts to quantify:

• irreducibility

• integration

• causal closure

C.3.2. Strengths

IIT contributes:

• a formal measure of integration

• the idea that consciousness is intrinsic

• the concept of causal structure

• the notion of informational unity

These are essential for MIST’s Axiom 2 (Integration).

C.3.3. Limitations

IIT does not include:

• boundaries

• generative models

• prediction error

• free‑energy minimization

• action

• historical dependence

• dynamical structure

IIT is a static theory of consciousness. It does not describe:

• how systems maintain themselves

• how they act

• how they learn

• how they persist over time

Thus, IIT cannot define minimal subjectivity, only integrated structure.

C.3.4. Relationship to MIST

MIST incorporates IIT’s insight that:

Code

I(X) > 0

is necessary for subjectivity.

But MIST adds:

• boundaries

• dynamics

• generative models

• free‑energy minimization

• historical continuity

Thus:

IIT → integration MIST → subjectivity

C.4. Comparative Table

Feature	Cybernetics	FEP	IIT	MIST
Boundary	No	Yes	No	Yes
Integration	No	Not required	Yes	Yes
History	No	Not required	No	Yes
Directionality	Yes	Yes	No	Yes
Generative model	No	Yes	No	Yes
Free‑energy minimization	No	Yes	No	Yes
Subjectivity	No	No	Partial	Yes
Dynamics	Yes	Yes	No	Yes
Causal structure	Partial	Yes	Yes	Yes
Minimality	No	No	No	Yes

C.5. Summary of the Comparison

Cybernetics

Provides the behavioral skeleton of subjectivity (feedback, control), but lacks internal structure.

FEP

Provides the dynamical skeleton of subjectivity (prediction, free energy), but lacks minimality and integration.

IIT

Provides the structural skeleton of subjectivity (integration), but lacks dynamics and boundaries.

MIST

Integrates all three:

• Cybernetics → feedback

• FEP → generative modeling + free energy

• IIT → integration

And adds:

• minimality

• historical dependence

• directionality

• self‑reference

Thus, MIST is the first theory to define minimal subjectivity in a mathematically rigorous way.