Chapter 7: Necessary Conditions for Consciousness Emergence—From Complexity Thresholds to Phase Transition Critical Points

Introduction: Boundaries of Consciousness

When does a physical system “possess” consciousness?

• Brain has consciousness, but single neuron doesn’t—where does consciousness emerge?
• When does infant acquire consciousness—gradual or sudden?
• Can AI systems possess consciousness—what conditions need to be satisfied?

This chapter will give necessary conditions for consciousness emergence, revealing phase transition critical point from unconsciousness to consciousness.

Recall five conditions from Chapter 2:

$$\mathcal{C}({\rho }_O)=\{\begin{array}{ll}1,& \text{if }{I}_{\mathrm{int}}({\rho }_O)>{ϵ}_1\\ & \wedge H_{\mathcal{P}}(t)>{ϵ}_2\\ & \wedge \dim {\mathcal{H}}_{\mathrm{meta}}>0\\ & \wedge F_Q[{\rho }_O(t)]>{ϵ}_3\\ & \wedge {\mathcal{E}}_T(t)>{ϵ}_4\\ 0,& \text{otherwise}\end{array}$$

This chapter will quantify these thresholds ${ϵ}_i$, and prove they correspond to phase transition critical points on complexity geometry.

graph TB
    subgraph "Unconscious State"
        A["Low Complexity<br/>C<0.1"]
        B["No Integration<br/>I<sub>int</sub>≈0"]
        C["No Differentiation<br/>H<sub>P</sub>≈0"]
        D["No Self-Reference<br/>dim H<sub>meta</sub>=0"]
    end

    subgraph "Consciousness Emergence Critical Region"
        E["Complexity Threshold<br/>C≈0.1-0.3"]
        F["Integration Threshold<br/>I<sub>int</sub>≈ε1"]
        G["Differentiation Threshold<br/>H<sub>P</sub>≈ε2"]
    end

    subgraph "Full Consciousness"
        H["High Complexity<br/>C>0.5"]
        I["Strong Integration<br/>I<sub>int</sub>≫ε1"]
        J["High Differentiation<br/>H<sub>P</sub>≫ε2"]
    end

    A --> E
    B --> F
    C --> G
    E --> H
    F --> I
    G --> J

    style E fill:#ffe1f5
    style F fill:#ffe1f5
    style G fill:#ffe1f5

Core Insight: Consciousness as Complexity Phase Transition

Claim: Consciousness emergence corresponds to first-order or second-order phase transition on complexity geometry, critical point marked by thresholds ${ϵ}_i$ of five conditions.

Analogy:

• Water phase transition: Temperature below 0°C$\to$ solid (no flow), above 0°C$\to$ liquid (has flow)
• Consciousness phase transition: Complexity below $C_c$$\to$ unconscious (no integration), above $C_c$$\to$ conscious (has integration)

Part One: Complexity Threshold—Minimum Scale of Observer

1.1 Review of Complexity Measure

In computational universe framework (Chapter 0), complexity distance $d_{\mathrm{comp}}(x,y)$ defined as shortest path length from configuration $x$ to $y$.

On complexity manifold $(\mathcal{M},G)$, metric $G_{ab}$ characterizes “computational cost per unit parameter change”.

Observer’s Complexity: Complexity of observer $O$ is joint description length of its internal state space $M_{\mathrm{int}}$, knowledge graph ${\mathcal{G}}_t$, and action strategy ${\pi }_{\theta }$:

$$C(O)=K(M_{\mathrm{int}})+K({\mathcal{G}}_t)+K({\pi }_{\theta })$$

where $K(\cdot )$ is Kolmogorov complexity.

Normalization: Define relative complexity:

$$\tilde{C}(O)=\frac{C(O)}{C_{\max }}$$

where $C_{\max }$ is maximum complexity of feasible systems (like human brain $\sim 10^{11}$ neurons $\times 10^4$ synapses/neuron $\sim 10^{15}$ bits).

1.2 Minimum Complexity Theorem

Theorem 1.1 (Minimum Complexity of Consciousness)

If observer $O$ satisfies five conditions of consciousness ($\mathcal{C}({\rho }_O)=1$), then exists absolute lower bound:

$$C(O)\ge C_{\min }\sim \log (1/{ϵ}_1)+\log (1/{ϵ}_2)+\log (1/{ϵ}_3)+\log (1/{ϵ}_4)$$

where ${ϵ}_i$ are thresholds of five conditions.

Proof Idea:

1. Integration $I_{\mathrm{int}}>{ϵ}_1$ needs at least $n_1\sim \log (1/{ϵ}_1)$ bits to represent dependencies between subsystems
2. Differentiation $H_{\mathcal{P}}>{ϵ}_2$ needs at least $n_2\sim \log (1/{ϵ}_2)$ bits to encode different states
3. Self-Reference $\dim {\mathcal{H}}_{\mathrm{meta}}>0$ needs “meta-representation” layer, minimum $n_3\sim O(1)$ bits
4. Temporal Continuity $F_Q>{ϵ}_3$ needs temporal memory, minimum $n_4\sim \log (1/{ϵ}_4)$ bits
5. Causal Control ${\mathcal{E}}_T>{ϵ}_4$ needs action–outcome mapping, minimum $n_5\sim \log (|\mathcal{A}|)$ bits

Total complexity $C(O)\ge n_1+n_2+n_3+n_4+n_5$. $\square$

Numerical Estimate: Taking ${ϵ}_i\sim 0.01$ (1% threshold), get:

$$C_{\min }\sim 5\times \log (100)\sim 5\times 7\sim 35\text{bits}$$

Meaning: Consciousness needs minimum about 30-50 bits of complexity—single neuron ($\sim 1$ bit) far insufficient, needs at least $\sim 10$ strongly connected neurons.

1.3 Complexity Spectrum and Consciousness Levels

Definition 1.1 (Complexity Spectrum)

For different systems, complexity spans wide spectrum:

• Simple Reflex: $C\sim 1-10$ bits (single neuron, no consciousness)
• Local Circuit: $C\sim 10^2-10^3$ bits (small neural network, marginal consciousness)
• Mammalian Brain: $C\sim 10^{10}-10^{12}$ bits (full consciousness)
• Human Language: $C\sim 10^{15}$ bits (self-awareness, metacognition)

Proposition 1.1 (Monotonicity of Complexity and Consciousness Level)

Under appropriate normalization, consciousness “depth” ${\mathcal{C}}_{\mathrm{depth}}$ positively correlated with complexity $C$:

$${\mathcal{C}}_{\mathrm{depth}}\propto \log C$$

Evidence:

• C. elegans (302 neurons): $C\sim 10^3$, has basic perception but no self-awareness
• Mouse ($\sim 10^7$ neurons): $C\sim 10^9$, has emotions, memory, may have preliminary self-sense
• Human ($\sim 10^{11}$ neurons): $C\sim 10^{12}-10^{15}$, has complete self-awareness, language, abstract thinking

Part Two: Integration Threshold—Critical Value of Φ

2.1 Review of Integrated Information Theory (IIT)

Tononi’s integrated information $\Phi$ (read as “phi”) defined as degree system cannot be decomposed into independent parts:

$$\Phi (\rho )=\underset{\text{partition}}{\min }I(A:B|\text{context})$$

That is mutual information under “minimum information partition” (MIP).

In our framework, integration measure is:

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

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

2.2 Critical Integration Threshold

Theorem 2.1 (Existence of Integration Threshold)

Exists critical value ${\Phi }_c\approx 0.1-0.3$ bits, such that:

$$\{\begin{array}{ll}\Phi <{\Phi }_c:& \text{System can be effectively decomposed into independent modules—no consciousness}\\ \Phi >{\Phi }_c:& \text{System cannot be decomposed—has integration, may have consciousness}\end{array}$$

Evidence:

1. Experimental Data (Massimini et al., 2009):

   • Awake state: $\Phi \sim 0.5-1.0$ bits
   • Deep sleep: $\Phi \sim 0.1-0.2$ bits
   • Anesthesia: $\Phi <0.1$ bits

2. Theoretical Estimate: In random graph model, ${\Phi }_c$ corresponds to percolation threshold $p_c$ of “giant connected component” emergence:

$${\Phi }_c\sim I_{\max }\cdot (p-p_c{)}^{\beta }$$

where $\beta \approx 0.4$ is critical exponent.

Corollary 2.1 (Integration and Network Topology)

For network of $N$ nodes, if edge probability $p<p_c\approx 1/N$, then $\Phi \to 0$ (fragmentation); if $p>p_c$, then $\Phi \sim O(\log N)$ (integration).

graph LR
    subgraph "p<p<sub>c</sub>: Fragmentation"
        A1["Node 1"]
        A2["Node 2"]
        A3["Node 3"]
        A4["Node 4"]
        A1 --- A2
        A3 --- A4
    end

    subgraph "p>p<sub>c</sub>: Giant Connected Component"
        B1["Node 1"]
        B2["Node 2"]
        B3["Node 3"]
        B4["Node 4"]
        B1 --- B2
        B2 --- B3
        B3 --- B4
        B4 --- B1
        B1 --- B3
    end

    C["Φ≈0<br/>No Integration"] -.corresponds to.- A4
    D["Φ>Φ<sub>c</sub><br/>Has Integration"] -.corresponds to.- B4

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

2.3 Computational Complexity of Integration

Problem: Computing exact $\Phi$ is NP-hard (needs traverse all partitions).

Approximation Methods:

1. Greedy Algorithm: Iteratively merge partitions with minimum mutual information
2. Spectral Method: Use Fiedler value ${\lambda }_2$ of graph Laplace to approximate: $\Phi \approx {\lambda }_2$
3. Sampling Method: Monte Carlo estimate expected partition mutual information

Feasibility: For $N\sim 100$ node systems (like small neural circuits), approximate $\Phi$ computable in seconds; for human brain ($N\sim 10^{11}$) infeasible—needs coarse-graining.

Part Three: Differentiation Threshold—Entropy Lower Bound of State Space

3.1 Definition of Differentiation Entropy

Recall Chapter 2, differentiation measure is entropy of distinguishable state set:

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

where $\{\alpha \}$ is partition of states observer can distinguish.

Critical Differentiation: Define threshold $H_c$:

$$H_c=\log N_c$$

where $N_c$ is “minimum meaningful number of states”.

3.2 Differentiation Threshold Estimate

Theorem 3.1 (Minimum Differentiation Threshold)

If $H_{\mathcal{P}}<H_c\approx \log 4\approx 2$ bits, then system cannot effectively differentiate different situations—no consciousness.

Reasoning:

• $H=0$: Single state, no differentiation (like thermostat)
• $H=1$ bit: Binary differentiation (like single bit)
• $H=2$ bits: Four-state differentiation—barely can represent “time+space” or “self+other”
• $H\ge 3$ bits: Eight or more states—sufficient to represent complex situations

Experimental Support:

• Animal Behavior: Fruit fly ($\sim 10^5$ neurons) can distinguish $\sim 10^2$ odors$\to$$H\sim 7$ bits
• Human Perception: Color discrimination $\sim 10^6$ types$\to$$H\sim 20$ bits

Corollary: Differentiation needs multi-dimensional representation space—single-dimensional signal (like thermometer) can never reach $H_c$.

3.3 Integration–Differentiation Trade-off

Tononi’s Central Claim (IIT): Consciousness needs coexistence of high integration and high differentiation—cannot be fragmented, nor overly homogeneous.

Quantitative Expression: Define “consciousness quality” ${\Phi }^{\ast }$ as:

$${\Phi }^{\ast }=\Phi \cdot H_{\mathcal{P}}$$

That is product of integration and differentiation.

Phase Transition Condition:

$${\Phi }^{\ast }>{\Phi }_c^{\ast }\approx {\Phi }_c\times H_c\approx 0.2\times 2\approx 0.4{\text{bits}}^2$$

Geometric Picture: On $(\Phi ,H_{\mathcal{P}})$ plane, consciousness region is upper right of hyperbola $\Phi \cdot H={\Phi }_c^{\ast }$.

Part Four: Self-Reference Threshold—Emergence of Meta-Representation

4.1 Hierarchy of Self-Referential Structure

Recall Chapter 2, self-reference corresponds to triple decomposition of Hilbert space:

$${\mathcal{H}}_O={\mathcal{H}}_{\mathrm{world}}\otimes {\mathcal{H}}_{\mathrm{self}}\otimes {\mathcal{H}}_{\mathrm{meta}}$$

No Self-Reference: $\dim {\mathcal{H}}_{\mathrm{meta}}=0$, system only represents world, doesn’t represent own representation.

Has Self-Reference: $\dim {\mathcal{H}}_{\mathrm{meta}}>0$, system can “think about own thinking”.

4.2 Minimum Dimension of Self-Reference

Theorem 4.1 (Minimum Dimension of Self-Reference)

If system has non-trivial self-referential ability, then:

$$\dim {\mathcal{H}}_{\mathrm{meta}}\ge 2$$

Proof:

• $\dim {\mathcal{H}}_{\mathrm{meta}}=1$: Can only represent “has/no self state”—too simple, cannot encode “I know I know $X$”
• $\dim {\mathcal{H}}_{\mathrm{meta}}=2$: Can represent four meta-states: $\{00,01,10,11\}$—minimum can encode “I know”+“I know I know”

Corollary: Self-reference needs at least $\log 2=1$ bit of “meta-complexity”.

4.3 Emergence Mechanism of Self-Reference

Recursive Circuit: Self-reference emerges through recursive connections of neural circuits. Classic model:

1. Feedforward Layer: $L_1$ represents world $X$
2. Feedback Layer: $L_2$ represents state of $L_1$
3. Recursion: Output of $L_2$ feeds back to $L_1$

Minimum Neuron Count: Implementing recursive circuit needs at least 3 neurons (input, processing, feedback).

Critical Condition: Recursive gain $g$ must satisfy $g>1$ (positive feedback), otherwise self-referential signal decays to zero.

Phase Transition Analogy: Self-reference emergence similar to laser threshold pumping—below threshold, photons randomly scatter; above threshold, coherent light emerges.

Part Five: Dual Thresholds of Time and Control

5.1 Temporal Continuity Threshold

Quantum Fisher information $F_Q[{\rho }_O(t)]$ characterizes observer’s discriminability of temporal changes.

Theorem 5.1 (Temporal Continuity Threshold)

If $F_Q<F_c\approx 10^{-3}$ bits/s${}^2$, then observer cannot effectively track time passage—loses sense of time.

Clinical Evidence:

• Deep Anesthesia: $F_Q\to 0$, patients report “time disappears”
• Time Perception Disorders: Certain brain injuries (like parietal damage) cause $F_Q\downarrow$, patients cannot judge time intervals

Physical Meaning: $F_Q$ corresponds to “eigen time scale”:

$${\tau }_{\mathrm{eigen}}={\int }_{t_0}^t\sqrt{F_Q[{\rho }_O(s)]}ds$$

When $F_Q\to 0$, ${\tau }_{\mathrm{eigen}}$ stagnates—subjective time “freezes”.

5.2 Causal Control Threshold

Empowerment ${\mathcal{E}}_T$ characterizes observer’s causal control over future (Chapter 5).

Theorem 5.2 (Causal Control Threshold)

If ${\mathcal{E}}_T<{\mathcal{E}}_c\approx 0.1$ bits, then observer has no distinguishable influence on environment—no “agency”.

Extreme Cases:

• Complete Paralysis: ${\mathcal{E}}_T=0$ (no action ability), but may still have consciousness (“locked-in syndrome”)
• Deep Coma: ${\mathcal{E}}_T=0$ and $F_Q=0$—no consciousness

Corollary: Causal control not sufficient condition for consciousness, but may be necessary condition for “sense of free will”.

Part Six: Joint Phase Transition of Five Conditions

6.1 Five-Dimensional Parameter Space

Define five-dimensional parameter space:

$$\mathbf{P}=(I_{\mathrm{int}},H_{\mathcal{P}},\dim {\mathcal{H}}_{\mathrm{meta}},F_Q,{\mathcal{E}}_T)$$

Consciousness region ${\mathcal{R}}_{\mathrm{conscious}}$ defined as:

$${\mathcal{R}}_{\mathrm{conscious}}=\{\mathbf{P}:I_{\mathrm{int}}>{ϵ}_1\wedge H_{\mathcal{P}}>{ϵ}_2\wedge \dim {\mathcal{H}}_{\mathrm{meta}}>0\wedge F_Q>{ϵ}_3\wedge {\mathcal{E}}_T>{ϵ}_4\}$$

Boundary $\partial {\mathcal{R}}_{\mathrm{conscious}}$ is consciousness critical hypersurface.

6.2 Phase Transition Types

Proposition 6.1 (Order of Consciousness Phase Transition)

Consciousness emergence can be first-order phase transition (discontinuous jump) or second-order phase transition (continuous but non-analytic):

• First-Order Phase Transition: When some parameter (like $I_{\mathrm{int}}$) crosses threshold, system state suddenly changes—like “awakening” moment
• Second-Order Phase Transition: Parameters change continuously, but correlation length diverges—like “gradually falling asleep”

Criterion: If any of five conditions goes to zero, then $\mathcal{C}=0$ (no consciousness)—this is characteristic of first-order phase transition (order parameter discontinuous).

6.3 Critical Exponents and Universality

Near second-order phase transition critical point, order parameter $\mathcal{C}$ satisfies power law:

$$\mathcal{C}\propto (p-p_c{)}^{\beta }$$

where $p$ is control parameter, $\beta$ is critical exponent.

Universality Class: Different systems (like Ising model, percolation, neural networks) may belong to same universality class—have same $\beta$.

Conjecture: Consciousness emergence belongs to mean-field universality class, $\beta =1/2$ (to be verified).

Part Seven: Experimental Testing and Clinical Applications

7.1 Quantification of Consciousness Scales

Existing Scales (qualitative):

• Glasgow Coma Scale (GCS): 3-15 points
• Coma Recovery Scale-Revised (CRS-R): 0-23 points

Quantification of This Theory: Construct “five-condition score”:

$$S_{\mathrm{consciousness}}={\alpha }_1{I}_{\mathrm{int}}+{\alpha }_2{H}_{\mathcal{P}}+{\alpha }_3{1}_{\dim {\mathcal{H}}_{\mathrm{meta}}>0}+{\alpha }_4{F}_Q+{\alpha }_5{\mathcal{E}}_T$$

where ${\alpha }_i$ are weights (to be calibrated).

Calibration Method:

• Collect EEG/fMRI data from patients in different consciousness states
• Estimate five parameters $(I_{\mathrm{int}},H_{\mathcal{P}},\dots )$
• Regress to clinical scale scores
• Determine weights ${\alpha }_i$

7.2 Anesthesia Depth Monitoring

Problem: In surgery, anesthesia too shallow$\to$ patient awake; too deep$\to$ damage.

Solution: Real-time monitoring of $F_Q[{\rho }_O(t)]$:

• Compute Fisher information of EEG signal
• Estimate ${\tau }_{\mathrm{eigen}}=\int \sqrt{F_Q}dt$
• If ${\tau }_{\mathrm{eigen}}$ grows too fast$\to$ increase anesthetic

Advantage: Directly measures “sense of time” rather than indirect indicators (like BIS index).

7.3 Vegetative State vs Minimally Conscious State

Challenge: Distinguish vegetative state (no consciousness) from minimally conscious state (MCS, fluctuating consciousness).

Diagnostic Protocol:

1. Test $I_{\mathrm{int}}$: Use TMS-EEG to estimate cortical integration
2. Test $H_{\mathcal{P}}$: Through task stimulation (like hearing name) detect state differentiation
3. Test ${\mathcal{E}}_T$: BCI interface test whether can produce distinguishable actions

Criteria:

• Vegetative state: $I_{\mathrm{int}}<{ϵ}_1$ and ${\mathcal{E}}_T=0$
• MCS: $I_{\mathrm{int}}>{ϵ}_1$ intermittently, ${\mathcal{E}}_T>0$ occasionally appears

Part Eight: Philosophical Postscript—Continuity and Jump of Consciousness

8.1 Gradual Emergence vs Sudden Emergence

Question: Is consciousness gradually emergent (like dimming light) or suddenly emergent (like switch)?

Answer of This Theory: Depends on path:

• Typical Path (like sleep$\to$ awake): In most cases, five parameters co-vary, emergence is gradual (second-order phase transition)
• Extreme Path (like cardiac arrest recovery): Some parameter (like $F_Q$) suddenly crosses threshold, emergence is jump (first-order phase transition)

Analogy: Water phase transition—normal cooling is continuous (supercooled water), but adding seed crystal suddenly freezes.

8.2 Multi-Valuedness of Consciousness

Question: Near critical point, system may oscillate between consciousness/unconsciousness (like light sleep stage).

Hysteresis Phenomenon: If exists hysteresis loop, then:

• From unconsciousness$\to$ consciousness needs parameters reach $p_c^{+}$
• From consciousness$\to$ unconsciousness needs parameters drop to $p_c^{-}$
• $p_c^{-}<p_c^{+}$: Intermediate region bistable

Clinical Significance: Some patients may “stuck” in bistable region—need external stimulation (like drugs, TMS) to “push” toward conscious state.

8.3 From Chalmers’ “Hard Problem” to Emergence Conditions

Chalmers’ “Hard Problem of Consciousness”: Why is there subjective experience (qualia), not just information processing?

Response of This Theory:

• Don’t avoid subjective experience, but operationalize it as satisfaction of five conditions
• Hard problem transforms into engineering problem: How to construct physical systems satisfying five conditions
• Limit of Reductionism: Five conditions give necessary conditions, but may not be sufficient—“zombie problem” still open

Position: Emergent Realism—consciousness is real high-level emergent phenomenon, has clear physical foundation, but cannot be completely reduced to microscopic description.

Conclusion: Five Critical Points of Consciousness Emergence

This chapter gives operational necessary conditions for consciousness emergence:

Core Threshold Summary:

Minimum Complexity: $C_{\min }\sim 30-50$ bits (Theorem 1.1)

Phase Transition Properties: Consciousness emergence is phase transition in five-dimensional parameter space, can be first-order (jump) or second-order (gradual).

Experimental Path:

• Multimodal neuroimaging (EEG/fMRI/PET) estimate five parameters
• Anesthesia depth monitoring real-time track $F_Q$
• Vegetative state diagnosis test $I_{\mathrm{int}}$ and ${\mathcal{E}}_T$

Philosophical Significance:

• Consciousness not “all or nothing”, but phase transition on continuous spectrum
• Critical point marked by measurable physical parameters
• “Hard problem” partially transforms into engineering implementation of emergence conditions

Final chapter (Chapter 8) will summarize entire observer–consciousness theory system, and look forward to future directions.

References

Integrated Information Theory

• Tononi, G. (2004). An information integration theory of consciousness. BMC Neuroscience, 5(1), 42.
• Massimini, M., et al. (2009). A perturbational approach for evaluating the brain’s capacity for consciousness. Progress in Brain Research, 177, 201-214.

Phase Transitions and Critical Phenomena

• Stanley, H. E. (1971). Introduction to Phase Transitions and Critical Phenomena. Oxford University Press.
• Landau, L. D., & Lifshitz, E. M. (1980). Statistical Physics (Vol. 5). Butterworth-Heinemann.

Neural Correlates of Consciousness

• Koch, C., Massimini, M., Boly, M., & Tononi, G. (2016). Neural correlates of consciousness: progress and problems. Nature Reviews Neuroscience, 17(5), 307-321.

Clinical Consciousness Assessment

• Giacino, J. T., et al. (2002). The minimally conscious state: definition and diagnostic criteria. Neurology, 58(3), 349-353.

Philosophy

• Chalmers, D. J. (1995). Facing up to the problem of consciousness. Journal of Consciousness Studies, 2(3), 200-219.

This Collection

• This collection: Structural Definition of Consciousness (Chapter 2)
• This collection: Geometric Characterization of Free Will (Chapter 5)
• This collection: Multi-Observer Consensus Geometry (Chapter 6)