Chapter 5: Geometric Characterization of Free Will—Information-Theoretic Foundation of Empowerment and Causal Control

Introduction: Millennia-Long Confusion of Free Will

“Do I really have free will?”

This question has troubled philosophers for millennia. Determinists argue: Universe is causal chain, each “choice” is inevitable result of past states—free will is illusion. Non-determinists counter: Quantum uncertainty, chaotic dynamics prove future undetermined—free will has space.

But this debate deadlocked, because lacks operational definition: What is “freedom”? How to measure? How to test?

This chapter will start from information geometry and causal theory, give operational geometric characterization of free will:

$$\boxed{\text{Free Will}\equiv \text{Causal Control}\equiv \text{Empowerment }{\mathcal{E}}_T}$$

where ${\mathcal{E}}_T$ defined as:

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

That is “maximum mutual information of causal influence on future state $S_{t+T}$ through action $A_t$”.

This definition transforms abstract “freedom” concept into measurable, optimizable, physically constrained geometric quantity.

graph TB
    subgraph "Hierarchical Structure of Free Will"
        A["Physical Layer<br/>Non-Equilibrium Supply Δ F≠0"]
        B["Causal Layer<br/>Intervention Channel P(S'|do(A))≠P(S')"]
        C["Information Layer<br/>Empowerment E<sub>T</sub>>0"]
        D["Geometric Layer<br/>Causal Control Manifold (M,G)"]
        E["Experience Layer<br/>Subjective Sense of Freedom"]
    end

    A --> B
    B --> C
    C --> D
    D --> E

    style C fill:#e1f5ff
    style D fill:#fff4e1
    style E fill:#ffe1f5

Core Insight: Triple Constraints of Free Will

Physical Foundation Theorem: Observer has operational freedom, if and only if simultaneously satisfies:

1. Controllability: Exists non-degenerate intervention channel $P(S^{\prime }|do(A))$
2. Non-Equilibrium Supply: Exists steady-state free energy flow $⟨W⟩\ge \Delta F-k_{B}T⟨I⟩$
3. Barrier Separation: Exists Markov blanket $(M,A,S_{\mathrm{int}},S_{\mathrm{ext}})$, making influence of internal states on outcomes distinguishable

Meaning: Free will not “uncaused cause”, but causal intervention ability under physical constraints—it needs energy, needs channel, needs information, but within these constraints, it is real, measurable.

Part One: Empowerment—Information Measure of Causal Control

1.1 From Mutual Information to Directed Information

Classic Mutual Information:

$$I(X:Y)=H(Y)-H(Y|X)=D_{\mathrm{KL}}(p(x,y)\Vert p(x)p(y))$$

Measures “correlation” of two random variables, but doesn’t distinguish causal direction: $I(X:Y)=I(Y:X)$.

Directed Information (Massey, 1990):

For sequences $M^n=(M_1,\dots ,M_n)$ and $Y^n=(Y_1,\dots ,Y_n)$, define:

$$I(M^n\to Y^n)=\sum _{t=1}^{n}I(M^t:Y_t\mid Y^{t-1})$$

where $M^t=(M_1,\dots ,M_t)$, $Y^{t-1}=(Y_1,\dots ,Y_{t-1})$.

Meaning: $I(M^n\to Y^n)$ measures “cumulative causal influence of $M$ on $Y$”—it considers temporal order, distinguishes causal direction.

Properties:

• $I(M^n\to Y^n)\ne I(Y^n\to M^n)$ (asymmetric)
• $I(M^n\to Y^n)=0$ if and only if $Y_t\perp M^t\mid Y^{t-1}$ holds for all $t$ (no causal influence)

graph LR
    subgraph "No Causal Influence"
        A1["M<sub>1</sub>"] --> A2["M<sub>2</sub>"] --> A3["M<sub>3</sub>"]
        B1["Y<sub>1</sub>"] --> B2["Y<sub>2</sub>"] --> B3["Y<sub>3</sub>"]
    end

    subgraph "With Causal Influence"
        C1["M<sub>1</sub>"] --> C2["M<sub>2</sub>"] --> C3["M<sub>3</sub>"]
        D1["Y<sub>1</sub>"] --> D2["Y<sub>2</sub>"] --> D3["Y<sub>3</sub>"]
        C1 -.->|causal| D2
        C2 -.->|causal| D3
    end

    E["I(M<sup>n</sup>→Y<sup>n</sup>)=0"] -.corresponds to.- A3
    F["I(M<sup>n</sup>→Y<sup>n</sup>)>0"] -.corresponds to.- C3

    style A3 fill:#fff4e1
    style C3 fill:#ffe1f5

1.2 Definition and Physical Meaning of Empowerment

Definition 1.1 ($k$-Step Empowerment) (Klyubin, Polani, Nehaniv, 2005)

At time $t$, observer’s $k$-step Empowerment defined as:

$${\mathcal{E}}_k(t)=\underset{\pi }{\sup }I(A_{t:t+k-1}:S_{t+k}\mid S_t)$$

where:

• $A_{t:t+k-1}=(A_t,A_{t+1},\dots ,A_{t+k-1})$ is action sequence
• $S_t,S_{t+k}$ are initial and terminal perceptual states
• ${\sup }_{\pi }$ means taking supremum over all possible strategies $\pi$

Physical Meaning:

• ${\mathcal{E}}_k$ measures “maximum ability of observer to exert distinguishable influence on future state through $k$-step actions”
• Equivalent to supremum of “action–sensor channel capacity”
• When ${\mathcal{E}}_k=0$, observer’s actions have no detectable influence on future state—completely no freedom

Analogy: Imagine person locked in room. If pushing wall, shouting, knocking door cannot change any observable result (wall doesn’t move, no one hears), then ${\mathcal{E}}_k=0$—he “has no freedom”. Conversely, if pushing door can open, shouting can attract rescue, then ${\mathcal{E}}_k>0$—he “has freedom”.

1.3 Empowerment and Causal Intervention

In Pearl causal framework, intervention operator $do(A)$ means “externally force $A$ to take certain value”, different from conditioning $p(Y|A)$.

Proposition 1.1 (Empowerment and Intervention Distinguishability)

If exists $a,a^{\prime }\in \mathcal{A}$ such that:

$$p(S_{t+k}\mid do(A_t=a),S_t)\ne p(S_{t+k}\mid do(A_t=a^{\prime }),S_t)$$

then ${\mathcal{E}}_k(t)>0$. Conversely, if for all $a,a^{\prime }$ above equality holds, then ${\mathcal{E}}_k(t)=0$.

Proof Sketch: Intervention distinguishable $\Rightarrow$ actions carry causal information $\Rightarrow$ $I(A:S_{t+k}|S_t)>0$. $\square$

Meaning: Positive Empowerment is necessary and sufficient condition for causal intervention detectable. This bridges “free will” with Pearl causal theory.

1.4 Geometric Characterization of Empowerment

On control manifold $(\mathcal{M},G)$, Empowerment can be represented as curvature quantity on information geometry:

Proposition 1.2 (Fisher Metric Representation of Empowerment)

Under local linearization approximation, $k$-step Empowerment can be written as:

$${\mathcal{E}}_k(t)\approx \frac{1}{2}\log \det \left(I+{\Sigma }_A^{-1}{J}_T^{\top }{J}_T\right)$$

where:

• $J_T=\frac{\partial S_{t+T}}{\partial A_t}$ is Jacobian matrix (causal sensitivity)
• ${\Sigma }_A$ is covariance of action space
• $I$ is identity matrix

Meaning: Empowerment proportional to log determinant of “causal sensitivity matrix”—geometrically, it measures “local volume amplification rate from action space to state space”.

Analogy: Imagine joystick controlling robot. If slight movement of joystick causes large change in robot position, then singular values of Jacobian $J_T$ large, Empowerment ${\mathcal{E}}_k$ high—operator “has power”. Conversely, if pushing joystick hard, robot doesn’t move, then $J_T\approx 0$, ${\mathcal{E}}_k\approx 0$—operator “powerless”.

Part Two: Physical Foundation Theorem—Thermodynamic Constraints of Free Will

2.1 Feedback Thermodynamics and Information–Work Duality

Classic Second Law of Thermodynamics:

$$⟨W⟩\ge \Delta F$$

That is average work at least equals free energy change.

Generalized Second Law with Measurement–Feedback (Sagawa–Ueda, 2008):

$$⟨W⟩\ge \Delta F-k_{B}T⟨I⟩$$

where $I$ is mutual information acquired by measurement.

Meaning: Information can be converted to work—Maxwell’s demon doesn’t violate second law, but needs pay entropy cost of “erasing information” (Landauer principle).

Feedback Jarzynski Equality:

$$⟨\exp \left(-\beta W+\beta \Delta F-I\right)⟩=1$$

This is probability-1 identity, stronger than inequality.

graph LR
    A["Measurement M<br/>Acquire Mutual Information I"] --> B["Feedback Control<br/>Choose Strategy π"]
    B --> C["Apply Work W<br/>Change System"]
    C --> D["Free Energy Change Δ F"]

    E["Thermodynamic Constraint<br/>⟨W⟩≥Δ F-k<sub>B</sub>T⟨I⟩"]

    A -.information.-> E
    D -.energy.-> E

    style A fill:#e1f5ff
    style C fill:#ffe1f5
    style E fill:#fff4e1

2.2 Markov Blanket and Barrier Separation

Definition 2.1 (Markov Blanket) (Pearl, 2000; Friston, 2010)

For random variable set $(S_{\mathrm{int}},S_{\mathrm{ext}},A,M)$, say $(A,M)$ is Markov blanket of $S_{\mathrm{int}}$ relative to $S_{\mathrm{ext}}$, if:

$$S_{\mathrm{int}}\perp S_{\mathrm{ext}}\mid (A,M)$$

That is internal states and external states conditionally independent on blanket.

Physical Meaning:

• $S_{\mathrm{int}}$: Observer internal states (like neural activity, memory)
• $S_{\mathrm{ext}}$: External environment states
• $A$: Actions (output)
• $M$: Observations (input)

Markov blanket $(A,M)$ is “system boundary”—internal–external interaction must pass through this boundary.

Proposition 2.1 (Barrier Separation Condition)

If exists Markov blanket, and:

$$I(S_{\mathrm{int}}:S_{\mathrm{ext}}\mid A)>0$$

then action $A$ carries distinguishable causal information of internal states on external states.

Meaning: Markov blanket cannot “completely block” causal flow—otherwise changes in internal states have no influence on external states, Empowerment zero.

2.3 Physical Foundation Theorem

Theorem 2.1 (Physical Foundation of Free Will)

Observer has operational freedom on time domain $[0,T]$, if and only if simultaneously satisfies:

(i) Controllability: Exists non-degenerate intervention channel $P(S_{t+1}\mid A_t,S_t)$, i.e.:

$$\exists a,a^{\prime }\in \mathcal{A},P(S_{t+1}\mid do(A_t=a),S_t)\ne P(S_{t+1}\mid do(A_t=a^{\prime }),S_t)$$

(ii) Non-Equilibrium Supply: Exists sustained free energy flow or entropy flow, satisfying:

$$⟨W⟩\ge \Delta F-k_{B}T⟨I⟩>0$$

That is system not in thermal equilibrium.

(iii) Barrier Separation: Exists Markov blanket $(A,M,S_{\mathrm{int}},S_{\mathrm{ext}})$, making causal influence of internal states on external states detectable through action $A$:

$$I(S_{\mathrm{int}}:S_{\mathrm{ext}}\mid A,M)>0$$

Then exists strategy $\pi$ such that:

$$I(M^T\to S^T)>0,{\mathcal{E}}_k>0$$

And constrained by thermodynamic inequality $⟨W⟩\ge \Delta F-k_{B}T⟨I⟩$.

Proof Idea:

1. Controllability $\Rightarrow$ Empowerment Positive: Non-degenerate channel guarantees ${\sup }_{\pi }I(A:S_{t+k}|S_t)>0$ (channel capacity non-zero)

2. Barrier Separation $\Rightarrow$ Directed Information Positive: Markov blanket allows $P(S_{\mathrm{ext}}|do(A))\ne P(S_{\mathrm{ext}})$, therefore $I(M^T\to S^T)>0$ (Massey directed information)

3. Non-Equilibrium Supply $\Rightarrow$ Thermodynamically Feasible: Sagawa–Ueda feedback Jarzynski equality guarantees work–information duality consistency

$\square$

Meaning: This theorem transforms “free will” from philosophical concept into conjunction of three operational physical conditions. Each condition can be experimentally tested.

2.4 Thermodynamic Cost of Free Will

Corollary 2.2 (Energy Lower Bound of Freedom)

If observer maintains Empowerment ${\mathcal{E}}_k>ϵ>0$ within time $T$, must consume minimum average work:

$$⟨W{⟩}_{\min }\ge k_{B}Tϵ$$

Meaning: Maintaining free will requires sustained energy supply. When energy exhausted (like fatigue, sleep, coma), Empowerment tends to zero, free will lost.

Biological Evidence:

• Brain accounts for 2% of body weight, but consumes 20% of basal metabolism—maintaining neural plasticity and action control
• Glucose supply interruption (hypoglycemia) causes confusion, decision ability decline—${\mathcal{E}}_k\downarrow$
• Sleep deprivation reduces prefrontal activity, impulse control weakened—causal control impaired

Part Three: Geometric Structure of Free Will—Control Manifold and Variational Principle

3.1 Definition of Control Manifold

Recall Chapter 0, observer moves on joint manifold ${\mathcal{E}}_Q=\mathcal{M}\times {\mathcal{S}}_Q$. Now, we view Empowerment ${\mathcal{E}}_k$ as scalar field on control manifold $\mathcal{M}$:

$${\mathcal{E}}_k:\mathcal{M}\to {\mathbb{R}}_{\ge 0}$$

Definition 3.1 (Causal Control Manifold)

Control manifold $(\mathcal{M},G,{\mathcal{E}}_k)$ is triple:

• $\mathcal{M}$: Parameter space (like strategy parameters, control parameters)
• $G$: Complexity metric (like Fisher metric)
• ${\mathcal{E}}_k$: Empowerment scalar field

On $\mathcal{M}$, define Empowerment gradient flow:

$$\dot{\theta }(t)=\eta {\nabla }_G{\mathcal{E}}_k(\theta (t))$$

where ${\nabla }_G$ is gradient operator with respect to metric $G$, $\eta >0$ is learning rate.

Physical Meaning: Observer through gradient ascent, maximizes causal control ${\mathcal{E}}_k$—this is instinctive drive: Biological systems tend to enhance control ability over environment.

graph TB
    subgraph "Empowerment Landscape"
        A["Low Empowerment<br/>E<sub>k</sub>≈0<br/>No Control"]
        B["Medium Empowerment<br/>E<sub>k</sub>>0<br/>Partial Control"]
        C["High Empowerment<br/>E<sub>k</sub>≫0<br/>Strong Control"]
    end

    A -->|gradient flow| B
    B -->|gradient flow| C

    D["Constraint<br/>Thermodynamic Cost ⟨W⟩↑"] -.limits.- C

    style A fill:#fff4e1
    style C fill:#ffe1f5
    style D fill:#e1f5ff

3.2 Maximum Empowerment Principle

Hypothesis 3.1 (Maximum Empowerment Hypothesis)

Biological system’s strategy $\pi$ tends to maximize long-term average Empowerment:

$${\pi }^{\ast }=\arg \underset{\pi }{\max }{\mathbb{E}}_{s\sim p_{\pi }}\left[{\mathcal{E}}_k(s)\right]$$

Under constraints:

• Resource constraint: ${\mathbb{E}}_{\pi }[c(a)]\le C_{\max }$
• Thermodynamic constraint: $⟨W⟩\ge \Delta F-k_{B}T⟨I⟩$

Theoretical Basis:

• Empowerment maximization can be derived from “maximum entropy reinforcement learning” (Salge et al., 2014)
• Evolutionary pressure favors high-Empowerment strategies (stronger survival ability)
• Neuroscience evidence: Dopamine system encodes “expected control” (Sharot & Sunstein, 2020)

Engineering Applications:

• Robot navigation: Maximize entropy of future reachable positions—Empowerment guides exploration
• Game AI: Learn strategies “controlling key resources”—enhance strategic choice space
• Neural prosthetics: Design interfaces “giving patients maximum control freedom”

3.3 Duality of Variational Free Energy and Empowerment

In free energy principle (Friston, 2010), observer minimizes variational free energy:

$$\mathcal{F}(\varphi ,\theta )={\mathbb{E}}_{q_{\varphi }}[\log q_{\varphi }(s|o)-\log p_{\theta }(s,o)]$$

where $q_{\varphi }$ is internal generative model, $p_{\theta }$ is true distribution.

Proposition 3.1 (Duality of Free Energy and Empowerment)

Under Markov blanket setting, minimizing variational free energy $\mathcal{F}$ equivalent to maximizing expected Empowerment:

$$\underset{\varphi ,\pi }{\min }\mathcal{F}(\varphi ,\theta )\iff \underset{\pi }{\max }{\mathbb{E}}_{q_{\varphi }}\left[{\mathcal{E}}_k(s)\right]$$

Under appropriate regularization and time discounting.

Meaning: Free energy principle and Empowerment maximization are different formulations of same optimization objective—former from “prediction error minimization” perspective, latter from “causal control maximization” perspective.

Part Four: Hierarchical Structure of Free Will—From Determinism to Creativity

4.1 Level Zero: No Freedom (${\mathcal{E}}_k=0$)

Characteristics:

• Observer’s actions have no detectable influence on future state
• Examples: Completely passive observer, patients in coma or deep anesthesia

Extreme Cases:

• Physical death: ${\mathcal{E}}_k=0$, $I(M^T\to S^T)=0$
• Completely deterministic system: Although has “actions”, results completely determined by initial conditions, no real choice

4.2 Level One: Passive Choice (${\mathcal{E}}_k>0$, Single Peak)

Characteristics:

• Empowerment positive but small, choice space limited
• Empowerment landscape has only one global optimum—“only correct choice”

Examples:

• Emergency escape: Only safe exit in fire—has “freedom” (can choose not to run), but rational choice unique
• Chess endgame: Master facing “forced win”—has “freedom” to choose other moves, but only one optimal solution

Geometric Picture: Empowerment landscape as below, only one sharp peak.

4.3 Level Two: Active Choice (${\mathcal{E}}_k\gg 0$, Multiple Peaks)

Characteristics:

• Empowerment positive and large, choice space rich
• Empowerment landscape has multiple local optima—“multiple paths all viable”

Examples:

• Career choice: Multiple career paths can achieve “controlling life” goal
• Artistic creation: Multiple styles can express theme—creative choice

Geometric Picture: Empowerment landscape has multiple peaks, observer can “explore” between different peaks.

graph TB
    subgraph "Level One: Single Peak Landscape"
        A1["Unique Optimum"]
        A2["Suboptimum"] --> A1
        A3["Suboptimum"] --> A1
    end

    subgraph "Level Two: Multiple Peak Landscape"
        B1["Local Optimum 1"]
        B2["Local Optimum 2"]
        B3["Local Optimum 3"]
        B4["Valley"] --> B1
        B4 --> B2
        B4 --> B3
    end

    style A1 fill:#fff4e1
    style B1 fill:#ffe1f5
    style B2 fill:#ffe1f5
    style B3 fill:#ffe1f5

4.4 Level Three: Meta-Freedom ($\partial {\mathcal{E}}_k/\partial \theta \ne 0$)

Characteristics:

• Not only choose action $a$, but also choose “how to modify own control manifold $\mathcal{M}$”
• Observer changes own strategy space, value function, goals—“reshaping self”

Examples:

• Learning new skills: Expand action space $\mathcal{A}$, thus enhance Empowerment
• Value reassessment: Change objective function $u(s,a)$, thus change Empowerment landscape
• Self-transformation: Through training, education, therapy change neural circuits—modify parameter space $\Theta$ of ${\pi }_{\theta }$

Philosophical Meaning: Meta-freedom is “freedom of freedom”—choosing what kind of chooser to become. This is highest form of human free will.

Part Five: Experimental Testing of Free Will—Operational Protocols

5.1 Behavioral Estimation of Empowerment

Protocol E1 (Two-Alternative Forced Choice Task)

1. Present two options $a_1,a_2\in \mathcal{A}$
2. Observer chooses one
3. Record subsequent state $S_{t+k}$
4. Repeat $N$ trials, estimate conditional distribution $p(S_{t+k}|a_i,S_t)$
5. Calculate Empowerment:

$${\hat{\mathcal{E}}}_k=I(A:S_{t+k}|S_t)=D_{\mathrm{KL}}(p(a,s^{\prime }|s)\Vert p(a|s)p(s^{\prime }|s))$$

Expected Results:

• If ${\hat{\mathcal{E}}}_k\approx 0$: Actions have no influence on results—no freedom
• If ${\hat{\mathcal{E}}}_k>0$: Actions have distinguishable influence on results—has freedom

5.2 Detection of Causal Intervention

Protocol E2 (do-Calculus Experiment)

1. Randomly assign action $a\sim p(a)$ (external intervention, not subject choice)
2. Observe result $S_{t+k}$
3. Estimate intervention distribution $p(S_{t+k}|do(a),S_t)$
4. Compare with observational distribution $p(S_{t+k}|a,S_t)$

Criterion:

• If $p(S^{\prime }|do(a))\ne p(S^{\prime }|a)$: Exists confounders, non-causal effect
• If $p(S^{\prime }|do(a))=p(S^{\prime }|a)$: Observational distribution already captures causal effect

Experimental Evidence:

• Drug clinical trials: Randomized controlled trials (RCT) vs observational studies
• Neural modulation: Transcranial magnetic stimulation (TMS) vs natural movement

5.3 Measurement of Thermodynamic Cost

Protocol E3 (Work–Information Martingale Test)

Under continuous monitoring, construct martingale:

$${\Gamma }_t=\exp \left(-\beta W_t+\beta \Delta F_t-I_t\right)$$

where:

• $W_t$: Cumulative work (can estimate through energy monitoring or metabolic rate)
• $\Delta F_t$: Free energy change
• $I_t$: Cumulative mutual information

Criterion: Feedback Jarzynski equality requires $\mathbb{E}[{\Gamma }_t]=1$.

Verification:

• Test if ${\Gamma }_t$ is martingale (conditional expectation $\mathbb{E}[{\Gamma }_{t+1}|{\mathcal{F}}_t]={\Gamma }_t$)
• Estimate Jensen lower bound $⟨W⟩\ge \Delta F-k_{B}T⟨I⟩$ whether holds

Biological Applications:

• Measure ATP consumption of neural activity (like fMRI BOLD signal)
• Estimate energy cost of decision process
• Verify hypothesis “high-Empowerment strategies need higher energy”

5.4 Identification of Markov Blanket

Protocol E4 (Conditional Independence Test)

1. Define candidate blanket $(A,M)$ (like: sensory organ input + motor organ output)
2. Test conditional independence: $S_{\mathrm{int}}\perp S_{\mathrm{ext}}\mid (A,M)$
3. Use kernel independence test (like HSIC) or Bayesian network inference

Expectation:

• If independence holds: $(A,M)$ is valid Markov blanket
• If independence doesn’t hold: Need expand blanket (add more mediator variables)

Neuroscience Applications:

• Identify neural representation of “self–world boundary”
• Verify Markov blanket hypothesis of free energy principle

Part Six: Philosophical Postscript of Free Will—Reconciliation of Determinism and Freedom

6.1 Geometric Reconstruction of Compatibilism

Classic Compatibilism (Hume, Ayer): Free will and determinism can be compatible, because “freedom” means “acting according to own will”, not “uncaused cause”.

Extension of This Theory: Free will $\equiv$ Empowerment ${\mathcal{E}}_k>0$, this completely compatible with determinism, because:

1. Causal Chain Unbroken: Empowerment still in causal network—action $A$ determined by internal state $S_{\mathrm{int}}$
2. Freedom is Relative: ${\mathcal{E}}_k$ measures “choice space relative to constraints”, not “absolute unconstrained”
3. Operational Testable: Doesn’t depend on metaphysical “freedom” concept, only depends on measurable information quantity

Geometric Picture: Deterministic system is geodesic flow on $\mathcal{M}$, but “multi-peak structure” of Empowerment landscape allows “branching paths”—within deterministic framework, still has “freedom of choice”.

6.2 From Free Will to Moral Responsibility

Traditional Argument: If no free will, then no moral responsibility—because “cannot choose” means “no need to be responsible”.

Response of This Theory: Moral responsibility doesn’t need “absolute freedom”, only needs:

1. Causal Intervention Ability: ${\mathcal{E}}_k>0$, actor has distinguishable influence on results
2. Predictability: Actor’s internal model $q_{\varphi }$ can predict consequences of actions
3. Adjustability: Actor can modify ${\pi }_{\theta }$ through learning (meta-freedom)

Corollary:

• Infants or severe mental disorder patients: ${\mathcal{E}}_k\approx 0$ or $q_{\varphi }$ missing—exempt from responsibility
• Normal adults: ${\mathcal{E}}_k>0$ and $q_{\varphi }$ mature—bear responsibility
• Borderline cases (like mild cognitive impairment): ${\mathcal{E}}_k$ proportional to degree of responsibility—partial responsibility

6.3 Emergence of Free Will

Question: Empowerment ${\mathcal{E}}_k$ defined at macroscopic level (actions $A$, states $S$ are coarse-grained variables). Does microscopic particle level have “freedom”?

Answer: Free will is emergent phenomenon:

1. Macroscopic Causal Effectiveness: Coarse-grained variables $(A,S)$ can have higher causal effective information (Tononi, Hoel, 2013)
2. Scale-Dependent: Empowerment ${\mathcal{E}}_k$ maximum at appropriate coarse-graining scale—this is natural scale of “free will”
3. Downward Causation: Macroscopic intentions constrain microscopic dynamics through neural implementation—although microscopic follows physical laws, macroscopic still has “control”

Philosophical Meaning: Free will not at fundamental particle level, but emerges at appropriate organizational level—like “liquid” not at single water molecule level, but emerges in macroscopic collective behavior.

6.4 Cost and Finitude of Freedom

Existentialist Insight (Sartre): “Man is condemned to freedom”—freedom is burden, because choice means responsibility.

Quantification of This Theory: Cost of freedom is thermodynamic cost $⟨W⟩$ and cognitive load:

$$\text{Cost of Freedom}=k_{B}T{\mathcal{E}}_k+\text{Decision Complexity}$$

Corollary:

• When energy exhausted (fatigue) or cognitive overload (decision fatigue), observer tends to “autopilot”—${\mathcal{E}}_k\downarrow$
• High freedom (multiple options, high uncertainty) causes anxiety, paralysis—“paradox of choice”
• Moderate constraints (like habits, norms) can reduce cognitive cost, while preserving core freedom

Conclusion: Geometric Truth of Free Will

This chapter starts from information geometry and causal theory, gives operational definition of free will:

$$\boxed{{\mathcal{E}}_k(t)=\underset{\pi }{\sup }I(A_{t:t+k-1}:S_{t+k}\mid S_t)}$$

Core Theorems Review:

1. Physical Foundation Theorem (Theorem 2.1): Free will needs triple conditions—controllability, non-equilibrium supply, barrier separation

2. Thermodynamic Cost (Corollary 2.2): $⟨W⟩\ge k_{B}T{\mathcal{E}}_k$—freedom needs energy

3. Compatibilism: Empowerment compatible with determinism—freedom is “choice space within constraints”

4. Hierarchical Structure: No freedom (${\mathcal{E}}_k=0$) $\to$ passive choice $\to$ active choice $\to$ meta-freedom

Experimental Path:

• Behavioral estimation: Two-alternative task $\to$ ${\hat{\mathcal{E}}}_k$
• Causal detection: Random intervention $\to$ $p(S^{\prime }|do(a))$
• Thermodynamic verification: Work–information martingale $\to$ Jarzynski equality
• Neural identification: Markov blanket $\to$ conditional independence

Philosophical Significance:

• Free will not “uncaused cause”, but causal control under physical constraints
• It is measurable, optimizable, resource-limited—but within these constraints, it is real
• It emerges at appropriate organizational level, not at fundamental particle level

Next chapter (Chapter 6) will explore multi-observer consensus geometry, revealing how individual free will couples through social networks, forming collective decisions and consensus emergence.

References

Empowerment Theory

• Klyubin, A. S., Polani, D., & Nehaniv, C. L. (2005). Empowerment: A universal agent-centric measure of control. IEEE Congress on Evolutionary Computation.
• Salge, C., Glackin, C., & Polani, D. (2014). Empowerment–an introduction. In Guided Self-Organization: Inception (pp. 67-114).

Causal Theory

• Pearl, J. (2000). Causality: Models, Reasoning, and Inference. Cambridge University Press.
• Massey, J. L. (1990). Causality, feedback and directed information. Proc. Int. Symp. Inf. Theory Applic.(ISITA-90), 303-305.

Thermodynamics and Information

• Sagawa, T., & Ueda, M. (2008). Second law of thermodynamics with discrete quantum feedback control. Physical Review Letters, 100(8), 080403.
• Jarzynski, C. (1997). Nonequilibrium equality for free energy differences. Physical Review Letters, 78(14), 2690.

Free Energy Principle

• Friston, K. (2010). The free-energy principle: a unified brain theory? Nature Reviews Neuroscience, 11(2), 127-138.
• Parr, T., Pezzulo, G., & Friston, K. J. (2022). Active Inference: The Free Energy Principle in Mind, Brain, and Behavior. MIT Press.

Philosophy

• Hume, D. (1748). An Enquiry Concerning Human Understanding.
• Sartre, J.-P. (1943). L’Être et le néant (Being and Nothingness).
• Dennett, D. C. (1984). Elbow Room: The Varieties of Free Will Worth Wanting. MIT Press.

This Collection

• This collection: Observer–World Section Structure (Chapter 1)
• This collection: Structural Definition of Consciousness (Chapter 2)
• This collection: Attention–Time–Knowledge Graph (Chapter 4)
• This collection: Value–Meaning Unification: Optimal Geometry of Ethical Values and Physical Foundation of Free Will (Source theory document)