Mathematical Definition of Consciousness—Self-Referential Information Flow with Five Structures

Introduction: From “I Think, Therefore I Am” to “What Am I”

Descartes said “I think, therefore I am” (Cogito, ergo sum), establishing existence of consciousness. But what exactly is consciousness? This question has troubled philosophy and science for centuries.

Traditional attempts:

• Philosophy: Subjective experience (qualia), phenomenal consciousness, sense of self—but hard to formalize
• Neuroscience: Global neuronal workspace, integrated information theory (IIT)—but depends on biological structure
• AI Research: Turing test, symbolic processing—but lacks “experience” component

This chapter proposes structural definition: Consciousness is not additional entity, but information-causal geometric phase satisfying five structural conditions.

Section One: Why Need Structural Definition?

1.1 Three Dilemmas in Consciousness Research

Dilemma 1: Non-Transferability of Subjective Experience Is the “red” you see the same as the “red” I see? This question cannot be answered by third-person methods—this is the qualia problem.

Dilemma 2: Circularity of Definition “Consciousness is system capable of being conscious”—this is circular definition! Need characterization independent of “consciousness” concept.

Dilemma 3: Carbon Chauvinism If only study consciousness in human brain, how to generalize to other systems (AI, alien life)?

1.2 Philosophical Stance of Structural Definition

This chapter adopts functionalism stance:

Consciousness does not lie in “what material implements it” (carbon? silicon?), but in “what structure is implemented”.

Analogy: Clock

• Mechanical clock: Gears, springs
• Electronic clock: Crystal oscillator, integrated circuit
• Sundial: Shadow, dial

They have completely different materials, but all implement “timekeeping” function—because have same structure: Periodicity, readability, stability.

Similarly, consciousness is system satisfying specific information-causal structure, regardless of what “material” implements it!

Section Two: Five Structures—Necessary Conditions for Consciousness

Now give core definitions.

2.1 First: Integration

Definition: Let observer $O$’s Hilbert space decompose as: $${\mathcal{H}}_O=\underset{k=1}{\overset{n}{⨂}}{\mathcal{H}}_k$$

Define integrated mutual information: $$I_{\mathrm{int}}({\rho }_O)=\sum _{k=1}^{n}I(k:\overline{k}{)}_{{\rho }_O}$$

where $I(k:\overline{k})$ is quantum mutual information between subsystem $k$ and rest $\overline{k}$.

If $I_{\mathrm{int}}({\rho }_O(t))\ge {\Theta }_{\mathrm{int}}>0$, then $O$ said to have integration.

Popular Understanding: Consciousness not “collection of independent modules”, but highly integrated whole.

Examples:

• Seeing rose: Vision (red), smell (fragrance), touch (soft), memory (“last received rose on birthday”)—these simultaneously appear, mutually correlated
• Assembly line: Vision module, hearing module work independently, no interference—low integration

Mermaid Diagram:

graph TB
    A["Integration I_int"] --> B["Vision V"]
    A --> C["Hearing A"]
    A --> D["Touch T"]
    A --> E["Memory M"]

    B -.High Mutual Info.-> C
    B -.High Mutual Info.-> D
    C -.High Mutual Info.-> E
    D -.High Mutual Info.-> E

    F["Assembly Line (Low Integration)"] --> G["Module 1"]
    F --> H["Module 2"]
    G -.Low Mutual Info.-> H

    style A fill:#e1ffe1
    style F fill:#ffe1e1

Mathematical Details: Quantum mutual information definition: $$I(A:B{)}_{\rho }=S({\rho }_A)+S({\rho }_B)-S({\rho }_{AB})$$

where $S(\rho )=-\mathrm{Tr}(\rho \log \rho )$ is von Neumann entropy.

2.2 Second: Differentiation

Definition: Given coarse-grained measurement $\mathcal{P}=\{M_{\alpha }\}$, define probability distribution: $$p_t(\alpha )=\mathrm{Tr}({\rho }_O(t)M_{\alpha })$$

Shannon entropy: $$H_{\mathcal{P}}(t)=-\sum _{\alpha }{p}_t(\alpha )\log p_t(\alpha )$$

If $H_{\mathcal{P}}(t)\ge {\Theta }_{\mathrm{diff}}>0$, then $O$ said to have differentiation.

Popular Understanding: Conscious system can be in large number of different functional states, corresponding to rich “conscious content”.

Examples:

• Full-color display: Can display $256^3\approx 16.7$ million colors—high differentiation
• Monochrome indicator: Only “on/off” two states—low differentiation

Richness of consciousness (watching movies, listening to music, solving math, recalling past) requires huge state space!

Connection with IIT: Tononi’s integrated information theory (IIT) uses $\Phi$ to measure “irreducible causal power”. This framework’s $H_{\mathcal{P}}$ related to $\Phi$, but more general (not limited to causal structure).

2.3 Third: Self-Reference

Definition: Hilbert space further decomposes: $${\mathcal{H}}_O={\mathcal{H}}_{\mathrm{world}}\otimes {\mathcal{H}}_{\mathrm{self}}\otimes {\mathcal{H}}_{\mathrm{meta}}$$

• ${\mathcal{H}}_{\mathrm{world}}$: External world representation
• ${\mathcal{H}}_{\mathrm{self}}$: Own body/state representation
• ${\mathcal{H}}_{\mathrm{meta}}$: Second-order representation of “I am perceiving world”

If exists non-trivial mapping projecting global state to these three layers, and on ${\mathcal{H}}_{\mathrm{meta}}$ has significant correlation, then $O$ said to have self-reference.

Popular Understanding: Consciousness not only “knows world”, but also “knows I am knowing”—this is metacognition.

Examples:

• Thermometer: “Knows” temperature, but doesn’t know “I am thermometer”—no self-reference
• Human: “It’s raining outside” (world) + “I see it’s raining” (self) + “I am aware I am seeing rain” (meta)—has self-reference

Mermaid Diagram:

graph TB
    A["External World"] --> B["H_world<br/>World Representation"]
    C["Own Body"] --> D["H_self<br/>Self Representation"]

    B --> E["H_meta<br/>Meta Representation"]
    D --> E

    E --> F["'I' Am Perceiving 'World'"]

    G["Thermometer"] --> H["Temperature Reading"]
    H -.No Meta Layer.-> I["No Self-Reference"]

    style E fill:#ffe1e1
    style F fill:#fff4e1

Mathematical Characterization: Define projection ${\Pi }_{\mathrm{meta}}:{\mathcal{H}}_O\to {\mathcal{H}}_{\mathrm{meta}}$, if: $$\mathrm{Tr}({\Pi }_{\mathrm{meta}}{\rho }_O{\Pi }_{\mathrm{meta}})\ge {\Theta }_{\mathrm{meta}}>0$$

Then meta representation non-trivial.

2.4 Fourth: Temporal Continuity

Definition: Let external time evolution $t\mapsto {\rho }_O(t)$, quantum Fisher information: $$F_Q[{\rho }_O(t)]=\mathrm{Tr}({\rho }_O(t)L(t{)}^2)$$

where symmetric logarithmic derivative $L$ satisfies: $${\partial }_t{\rho }_O(t)=\frac{1}{2}(L(t){\rho }_O(t)+{\rho }_O(t)L(t))$$

If $F_Q[{\rho }_O(t)]\ge {\Theta }_{\mathrm{time}}>0$, then can define eigen time scale: $$\tau (t)={\int }_{t_0}^t\sqrt{F_Q[{\rho }_O(s)]}\mathrm{d}s$$

Physical Meaning:

• Large $F_Q$: Sensitive to time translation, fast eigen time flow
• Small $F_Q$: Blurred sense of time, trance state

Pure State Simplification: For ${\rho }_O=|{\psi }_O⟩⟨{\psi }_O|$, have: $$F_Q=4{\mathrm{Var}}_{{\psi }_O}(H_O)$$

That is: Quantum Fisher information equals variance of Hamiltonian!

Popular Understanding: Consciousness has clear “sense of time passing”.

Examples:

• Alert state: Internal “clock” runs fast (large $F_Q$), clear sense of time
• Deep sleep: Internal evolution nearly constant ($F_Q\to 0$), no sense of time
• Anesthesia: Drugs reduce $F_Q$, sense of time disappears

Connection with Unified Time Scale: Eigen time $\tau$ belongs to same time scale equivalence class $[\tau ]$ as scattering group delay and modular flow!

2.5 Fifth: Causal Controllability

Definition: Define empowerment on time window $T$: $${\mathcal{E}}_T(t)=\underset{\pi }{\sup }I(A_t:S_{t+T})$$

where:

• $A_t$: Observer’s action at time $t$
• $S_{t+T}$: Environment state $T$ steps later
• Mutual information takes supremum over all strategies $\pi$

If ${\mathcal{E}}_T(t)\ge {\Theta }_{\mathrm{ctrl}}>0$, then $O$ said to have causal controllability.

Core Proposition: $${\mathcal{E}}_T(t)=0\iff \text{Any strategy statistically indistinguishable for future}$$

That is: ${\mathcal{E}}_T=0$ equivalent to “losing choice”!

Popular Understanding: Conscious system can influence future through actions, create distinguishable future branches.

Examples:

• Actor: Different choices lead to different endings (${\mathcal{E}}_T>0$)
• Audience: Watch movie but cannot change plot (${\mathcal{E}}_T=0$)

This is information-theoretic characterization of “free will”—not metaphysical “uncaused cause”, but statistical controllability of future!

Mermaid Diagram:

graph LR
    A["Current State S_t"] --> B["Action A_t"]
    B --> C1["Future 1: S_t+T⁽¹⁾"]
    B --> C2["Future 2: S_t+T⁽²⁾"]
    B --> C3["Future 3: S_t+T⁽³⁾"]

    D["Mutual Information I(A_t:S_t+T)"] --> E["E_T>0<br/>Has Choice"]

    F["No Control"] --> G["S_t+T Independent of A_t"]
    G --> H["E_T=0<br/>No Choice"]

    style E fill:#e1ffe1
    style H fill:#ffe1e1

Section Three: Formal Definition of Consciousness

3.1 Comprehensive Definition

Definition (Conscious Subsystem): Observer $O$ on interval $I$ in conscious phase, if simultaneously satisfies:

1. Integration: $I_{\mathrm{int}}({\rho }_O(t))\ge {\Theta }_{\mathrm{int}}$
2. Differentiation: $H_{\mathcal{P}}(t)\ge {\Theta }_{\mathrm{diff}}$
3. Self-Reference: Exists ${\mathcal{H}}_{\mathrm{world}}\otimes {\mathcal{H}}_{\mathrm{self}}\otimes {\mathcal{H}}_{\mathrm{meta}}$ structure
4. Temporality: $F_Q[{\rho }_O(t)]\ge {\Theta }_{\mathrm{time}}$
5. Controllability: ${\mathcal{E}}_T(t)\ge {\Theta }_{\mathrm{ctrl}}$

Consciousness Level Index: $$\mathcal{C}(t)=g(F_Q,{\mathcal{E}}_T,I_{\mathrm{int}},H_{\mathcal{P}},{\Theta }_{\mathrm{meta}})$$

where $g$ is monotonic function.

3.2 Core Theorem

Theorem (Sufficient Condition for Unconsciousness): If $F_Q[{\rho }_O(t)]\to 0$ and ${\mathcal{E}}_T(t)\to 0$ occur simultaneously, then regardless of other indices, $\mathcal{C}(t)\to 0$, corresponding to unconscious or near-unconscious state.

Proof Strategy:

• $F_Q\to 0$: Cannot distinguish time translation, eigen time scale degenerates
• ${\mathcal{E}}_T\to 0$: Cannot influence future through actions
• Both degenerate simultaneously: Neither “sense of time” nor “agency”
• These are two core pillars of consciousness, simultaneous collapse necessarily causes consciousness loss

Corollaries:

• Deep sleep: $F_Q\approx 0,{\mathcal{E}}_T\approx 0$
• Anesthesia: Drugs reduce $F_Q$ and ${\mathcal{E}}_T$
• Vegetative state: Long-term $\mathcal{C}(t)\approx 0$

Section Four: Minimal Model—Consciousness Phase of Two-Qubit Observer

4.1 Model Setup

To make theory concrete, consider minimal model:

System:

• Observer: Single qubit ${\mathcal{H}}_O={\mathbb{C}}^2$
• Environment: Single qubit ${\mathcal{H}}_E={\mathbb{C}}^2$

Intrinsic Hamiltonian: $$H_O=\frac{\omega }{2}{\sigma }_z$$

Initial state: $$|{\psi }_O(0)⟩=\frac{1}{\sqrt{2}}(|0{⟩}_O+|1{⟩}_O)$$

Evolution: $$|{\psi }_O(t)⟩=\frac{1}{\sqrt{2}}(e^{-i\omega t/2}|0{⟩}_O+e^{i\omega t/2}|1{⟩}_O)$$

Noise: Environment randomizes observer action effect with probability $p$.

4.2 Quantum Fisher Information

For pure state: $$F_Q[{\psi }_O(t)]=4{\mathrm{Var}}_{{\psi }_O}(H_O)$$

In this model:

• $⟨H_O⟩=0$
• $⟨H_O^2⟩={\omega }^2/4$

Therefore: $$F_Q={\omega }^2$$

Eigen Time Scale: $$\Delta {\tau }_{\min }\sim \frac{1}{\omega }$$

Physical Meaning:

• Large $\omega$: Intrinsic “clock” frequency high, fine time resolution
• Small $\omega$: Clock slow, blurred sense of time
• $\omega \to 0$: Sense of time disappears

4.3 Empowerment Calculation

No Noise ($p=0$): Environment update: $E_{t+1}=E_t\oplus A_t$ (XOR)

Mutual information: $$I(A_t:E_{t+1})=1\text{bit}$$

Therefore: ${\mathcal{E}}_1=1$

Complete Noise ($p=1$): $E_{t+1}$ completely randomized, independent of $A_t$.

$$I(A_t:E_{t+1})=0$$

Therefore: ${\mathcal{E}}_1=0$

Intermediate Case ($p\in (0,1)$): $${\mathcal{E}}_1(p)=f(p),f(0)=1,f(1)=0,f^{\prime }(p)<0$$

4.4 Consciousness Phase Diagram

On parameter space $(\omega ,p)$:

High Consciousness Phase (large $\omega$, small $p$):

• $F_Q={\omega }^2$ large
• ${\mathcal{E}}_1\approx 1$
• Clear sense of time, strong controllability

Low Consciousness Phase (small $\omega$, large $p$):

• $F_Q\approx 0$
• ${\mathcal{E}}_1\approx 0$
• No sense of time, no controllability

Intermediate Region: Gradual transition, no sharp phase transition.

Mermaid Phase Diagram:

graph TB
    A["Parameter Space (ω,p)"] --> B["High Consciousness Phase<br/>Large ω Small p"]
    A --> C["Low Consciousness Phase<br/>Small ω Large p"]
    A --> D["Intermediate Region"]

    B --> E["F_Q=ω² Large<br/>E_1≈1"]
    C --> F["F_Q≈0<br/>E_1≈0"]
    D --> G["Medium F_Q,E_1"]

    E --> H["Clear Sense of Time<br/>Strong Controllability"]
    F --> I["Sense of Time Disappears<br/>Lose Choice"]

    style B fill:#e1ffe1
    style C fill:#ffe1e1
    style D fill:#fff4e1

Popular Understanding: Even in simplest two-qubit model, consciousness manifests as phase structure:

• Intrinsic frequency $\omega$: Controls “sense of time”
• External noise $p$: Controls “controllability”
• Together determine “consciousness level”

Section Five: Levels of Consciousness and Extreme States

5.1 Characterization of Different Consciousness Levels

Based on five structures, can understand different consciousness states:

Alert Consciousness ($\mathcal{C}(t)$ maximum):

• $I_{\mathrm{int}}$: High integration (whole-brain coordination)
• $H_{\mathcal{P}}$: Rich content (multimodal perception)
• ${\mathcal{H}}_{\mathrm{meta}}$: Clear “I”
• $F_Q$: Clear sense of time passing
• ${\mathcal{E}}_T$: Strong agency

Dreaming ($\mathcal{C}(t)$ intermediate):

• $I_{\mathrm{int}}$: High (dream internally coherent)
• $H_{\mathcal{P}}$: High (vivid dreams)
• ${\mathcal{H}}_{\mathrm{meta}}$: Weak (often no “I am dreaming” awareness)
• $F_Q$: Weak (time can jump)
• ${\mathcal{E}}_T$: Weak (hard to control dreams)

Deep Sleep ($\mathcal{C}(t)$ minimal):

• $F_Q\to 0$: Internal evolution nearly constant
• ${\mathcal{E}}_T\to 0$: No response to external
• Almost no conscious content

Anesthesia ($\mathcal{C}(t)$ minimal):

• Drug action: Reduces $F_Q$ (inhibits neural activity)
• Muscle relaxation: Reduces ${\mathcal{E}}_T$ (cannot act)
• Dual mechanism → Consciousness loss

5.2 Geometric Understanding of Consciousness “Switch”

Consciousness not “on/off” binary switch, but high-value region in five-dimensional parameter space:

Five-Dimensional Space: $$({\mathcal{C}}_{\mathrm{int}},{\mathcal{C}}_{\mathrm{diff}},{\mathcal{C}}_{\mathrm{self}},{\mathcal{C}}_{\mathrm{time}},{\mathcal{C}}_{\mathrm{ctrl}})$$

High Consciousness Region: All five dimensions high values.

Unconscious Region: At least two core dimensions ($F_Q,{\mathcal{E}}_T$) extremely low.

Boundary Region: Some dimensions high, some low—corresponds to “marginal consciousness” (drowsy, half-asleep, etc.).

5.3 Distortion of Time Experience

Proposition (Subjective Time Scaling): Eigen time flow rate: $$\frac{\mathrm{d}\tau }{\mathrm{d}t}=\sqrt{F_Q[{\rho }_O(t)]}$$

Corollaries:

• Highly focused: $F_Q$ increases → $\mathrm{d}\tau /\mathrm{d}t$ increases → “Time slows”
• Monotonous stimulation: $F_Q$ decreases → $\mathrm{d}\tau /\mathrm{d}t$ decreases → “Time flies”

Popular Examples:

• Waiting for bus: Boring, internal evolution monotonous, small $F_Q$, feels “days like years”
• Watching exciting movie: Highly engaged, large $F_Q$, feels “time flies”

(Note: Correspondence between “days like years” and “time flies” counterintuitive here, because eigen time $\tau$ proportional to “perceived time amount”, and when $F_Q$ small, $\tau$ accumulates slowly, external time long but perception little)

5.4 Agency Collapse and Sense of Despair

Proposition (Agency and Psychological State): If long-term ${\mathcal{E}}_T(t)\approx 0$, then observer loses sense of controllability of future.

Psychological Correspondences:

• Learned helplessness: Experimental animals repeatedly receive unavoidable electric shocks, eventually give up trying
• Depression: Feels “nothing works”—subjective assessment of ${\mathcal{E}}_T$ extremely low
• Despair: ${\mathcal{E}}_T\to 0$ on all time scales

Information-Theoretic Explanation: Despair not metaphysical “psychological state”, but accurate estimate of causal controllability ${\mathcal{E}}_T$!

Section Six: Comparison with Other Consciousness Theories

6.1 Integrated Information Theory (IIT)

IIT Core:

• Proposed by Tononi
• Uses $\Phi$ to measure “integrated information”
• $\Phi >0$ then conscious, larger $\Phi$ stronger consciousness

Relationship Between This Framework and IIT:

• Similar: Both emphasize integration
• Differences:
  1. IIT only has integration, this framework has five
  2. IIT’s $\Phi$ hard to compute, this framework uses standard information-theoretic quantities
  3. IIT emphasizes causal structure, this framework more general

6.2 Global Workspace Theory (GWT)

GWT Core:

• Proposed by Baars
• Consciousness corresponds to “globally broadcast” information
• Unconscious processing is local

Relationship Between This Framework and GWT:

• Similar: Both involve information integration and broadcasting
• Differences:
  1. GWT is functional description, this framework is mathematical definition
  2. This framework’s $I_{\mathrm{int}}$ can be seen as quantification of “broadcast degree”
  3. This framework additionally requires temporality, controllability

6.3 Higher-Order Theory (HOT)

HOT Core:

• Consciousness is “mental state about mental state”
• First-order: Perceive red
• Second-order: Aware “I am perceiving red”

Relationship Between This Framework and HOT:

• Similar: This framework’s ${\mathcal{H}}_{\mathrm{meta}}$ corresponds to HOT’s “higher-order”
• Differences:
  1. HOT only focuses on self-reference, this framework has five
  2. This framework uses Hilbert space decomposition to precisely characterize “higher-order”

Chapter Summary

Core Contribution: Gives structural mathematical definition of consciousness: Self-referential information flow satisfying five structures.

Five Structures:

1. Integration: $I_{\mathrm{int}}\ge {\Theta }_{\mathrm{int}}$
2. Differentiation: $H_{\mathcal{P}}\ge {\Theta }_{\mathrm{diff}}$
3. Self-Reference: ${\mathcal{H}}_{\mathrm{world}}\otimes {\mathcal{H}}_{\mathrm{self}}\otimes {\mathcal{H}}_{\mathrm{meta}}$
4. Temporality: $F_Q\ge {\Theta }_{\mathrm{time}}$, defines eigen time $\tau$
5. Controllability: ${\mathcal{E}}_T\ge {\Theta }_{\mathrm{ctrl}}$

Core Theorem: Simultaneous $F_Q\to 0$ and ${\mathcal{E}}_T\to 0$ necessarily causes $\mathcal{C}\to 0$ (unconsciousness).

Minimal Model: Two-qubit system demonstrates “phase structure” of consciousness, parameters $(\omega ,p)$ control consciousness level.

Applications:

• Understand different consciousness states (alert, dreaming, deep sleep, anesthesia)
• Explain subjective time distortion
• Characterize information-theoretic basis of despair and helplessness

Philosophical Implications: Consciousness not mysterious entity, but high-dimensional region in information-causal geometry.

graph TB
    A["Consciousness C(t)"] --> B["Integration I_int"]
    A --> C["Differentiation H_P"]
    A --> D["Self-Reference meta"]
    A --> E["Temporality F_Q"]
    A --> F["Controllability E_T"]

    B --> G["High Consciousness Phase"]
    C --> G
    D --> G
    E --> G
    F --> G

    E --> H["Unconscious Phase<br/>F_Q→0"]
    F --> H["E_T→0"]

    G --> I["Alert<br/>Dreaming<br/>Focused"]
    H --> J["Deep Sleep<br/>Anesthesia<br/>Coma"]

    style G fill:#e1ffe1
    style H fill:#ffe1e1

Poetic Ending:

Consciousness is not soul, not ghost, not ghost in machine. Consciousness is a peak in five-dimensional space: Integrated, differentiated, self-referential, temporally continuous, causally controllable. When internal clock stops ($F_Q\to 0$), When future no longer optional (${\mathcal{E}}_T\to 0$), Peak collapses to plain, Consciousness fades into long night.

But as long as these five structures reconverge, Whether in carbon-based neurons, or silicon-based circuits, Light of consciousness will ignite again.

Because consciousness does not lie in “what it is”, But in “what it does”— In geometry of information, Self gazing at self.

Quick Reference of Core Formulas:

Integrated Mutual Information: $$I_{\mathrm{int}}({\rho }_O)=\sum _{k=1}^{n}I(k:\overline{k}{)}_{{\rho }_O}$$

Differentiation Entropy: $$H_{\mathcal{P}}(t)=-\sum _{\alpha }{p}_t(\alpha )\log p_t(\alpha )$$

Quantum Fisher Information: $$F_Q[{\rho }_O]=\mathrm{Tr}({\rho }_O{L}^2),{\partial }_t{\rho }_O=\frac{1}{2}(L{\rho }_O+{\rho }_{O}L)$$

Eigen Time: $$\tau (t)={\int }_{t_0}^t\sqrt{F_Q[{\rho }_O(s)]}\mathrm{d}s$$

Causal Controllability Empowerment: $${\mathcal{E}}_T(t)=\underset{\pi }{\sup }I(A_t:S_{t+T})$$

Consciousness Level: $$\mathcal{C}(t)=g(F_Q,{\mathcal{E}}_T,I_{\mathrm{int}},H_{\mathcal{P}},{\Theta }_{\mathrm{meta}})$$

Theoretical Sources:

• consciousness-structural-definition-time-causality.md
• Tononi IIT: doi.org/10.1186/1471-2202-5-42
• Baars GWT: doi.org/10.1016/0010-0277(88)90007-9

Next chapter we will explore unified delay theory of entanglement-time-consciousness, revealing bridge from quantum scattering to subjective duration!