Meta Theory of the Zeckendorf-Hilbert Universe

19.3 Free Energy Principle: Prediction Error Minimization as Physical Hamiltonian Minimization

In Sections 19.1 and 19.2, we described observers as dynamical systems that correct deviations between internal models and external environment through self-referential update operator $U_{\text{self}}$. This “correction” mechanism was summarized by Karl Friston in biology and neuroscience as Free Energy Principle (FEP): any self-organizing system capable of resisting environmental entropy increase must have dynamics dedicated to minimizing variational free energy.

This section elevates this biological principle to a fundamental physics theorem. We will prove that in QCA discrete ontology, so-called “variational free energy” is not an abstract statistical quantity, but the expectation value of observer’s effective Hamiltonian (Physical Hamiltonian). Observer’s process of minimizing free energy is physically equivalent to system relaxation toward ground state. This not only unifies physics (principle of least action) with biology (adaptation principle), but also provides an analytical solution to “why life exists”: life is the dynamical configuration matter must adopt to maintain low-entropy states in a universe full of uncertainty.

19.3.1 Algebraic Definition of Variational Free Energy

In statistical inference, free energy $F$ is an upper bound of “surprise” ($-\ln P(o)$). In QCA framework, we need to rewrite this quantity using operator algebra language.

Let observer’s boundary (sensory) state be $o$ (corresponding to projection or state ${\rho }_{\partial }$ on boundary algebra ${\mathcal{A}}_{\partial }$).

Let external environment’s “hidden variables” or true state be $s$ (corresponding to environmental algebra ${\mathcal{A}}_{\text{ext}}$).

Observer’s internal state (belief/recognition density) is $q(s)$ (corresponding to state ${\rho }_{\text{int}}$ on internal algebra ${\mathcal{A}}_{\text{int}}$).

Observer possesses a Generative Model $P(s,o)$ about the world, encoded in observer’s internal structural Hamiltonian $H_{\text{model}}$:

$$P(s,o)\propto \exp \left(-\beta ⟨s,o|H_{\text{model}}|s,o⟩\right)$$

Definition 19.3.1 (Quantum Variational Free Energy)

For given sensory input $o$, observer’s variational free energy functional $\mathcal{F}$ is defined as relative entropy between internal state ${\rho }_{\text{int}}$ and generative model posterior distribution (plus logarithm of sensory evidence):

$$\mathcal{F}({\rho }_{\text{int}},o)\equiv D_{\text{KL}}({\rho }_{\text{int}}\Vert P(\cdot |o))-\ln P(o)$$

Expanded, it consists of two parts:

$$\mathcal{F}=\underset{\text{Energy: Prediction Error}}{\underbrace{⟨E(s,o){⟩}_{{\rho }_{\text{int}}}}}-\underset{\text{Entropy: Uncertainty}}{\underbrace{S({\rho }_{\text{int}})}}$$

where $⟨E⟩$ is expected energy (cost from prediction inaccuracy), $S$ is internal Shannon entropy (ability to maintain multiple hypotheses).

19.3.2 Physical Equivalence: Free Energy as Hamiltonian

In classical FEP, free energy is a pure information quantity. But in QCA universes, information is physical (Landauer’s principle). We will prove that $\mathcal{F}$ is actually the expectation value of observer’s effective Hamiltonian.

Theorem 19.3.2 (Free Energy-Hamiltonian Duality Theorem)

Let observer’s self-referential dynamics be driven by $U_{\text{self}}$ defined in Section 19.2, and system be in quasi-steady state. Then observer subsystem’s physical free energy (thermodynamic free energy) and variational free energy (information-theoretic free energy) are proportional in value:

$$F_{\text{phys}}\equiv ⟨H_{\text{eff}}⟩-T{S}_{\text{vn}}\cong k_{B}T\cdot {\mathcal{F}}_{\text{info}}$$

where $H_{\text{eff}}=H_{\text{model}}+H_{\text{coupling}}$ is observer’s effective Hamiltonian.

Proof Outline:

1. Energy Term: $H_{\text{coupling}}\propto ({\hat{M}}_{\text{ext}}-{\hat{M}}_{\text{int}}{)}^2$ (Section 19.2). Its expectation value is precisely squared prediction error, corresponding to energy term (Accuracy term) in $\mathcal{F}$.

2. Entropy Term: $S_{\text{vn}}$ is von Neumann entropy, corresponding to entropy term (Complexity term) in $\mathcal{F}$.

3. Minimization Dynamics: Physical systems tend to evolve to thermodynamic equilibrium states where $F_{\text{phys}}$ is minimal. Therefore, observer’s physical relaxation process (cooling) naturally achieves minimization of variational free energy ${\mathcal{F}}_{\text{info}}$.

Conclusion: “Understanding the world” is physically equivalent to “lowering one’s own energy level”. Observer adjusting internal state ${\rho }_{\text{int}}$ to match external world is not to pursue truth, but to minimize interaction potential energy with environment (i.e., eliminate excitation from prediction errors).

19.3.3 Perception and Action: Bidirectional Minimization Paths

Minimizing $\mathcal{F}({\rho }_{\text{int}},s,o)$ can be achieved by changing two variables, corresponding to two basic modes of living systems.

1. Perceptual Inference

• Operation: Change ${\rho }_{\text{int}}$ (internal beliefs), keep $o$ (sensory input) unchanged.

• Physical Process: Fix boundary conditions, system internal degrees of freedom relax to ground state.

• Meaning: Update beliefs to explain data. For example, seeing a blurry shadow, brain interprets it as “cat” rather than “ghost,” because “cat” hypothesis better eliminates prediction error (lower free energy).

2. Active Inference

• Operation: Change $o$ (sensory input), keep ${\rho }_{\text{int}}$ (prior goal) unchanged.

• Physical Process: Exert force on environment through actuators, changing environmental state $s$, thereby changing feedback $o$.

• Meaning: Act to change the world to conform to expectations. For example, feeling cold (prediction error: body temperature below target), not only modify belief “I am now cold,” but put on clothes (action), making sensory input conform to prior model “body temperature 37°C” again.

Theorem 19.3.3 (Physical Necessity of Active Inference)

If observer’s internal model ${\rho }_{\text{prior}}$ is rigid (i.e., locked by extremely stable memory subsystem ${\mathcal{M}}_{\text{mem}}$, such as gene-encoded survival instincts), then when prediction error $\hat{E}$ increases, the system cannot completely eliminate free energy through pure perception (modifying ${\rho }_{\text{int}}$) (because that would mean admitting one “should” die).

To survive (maintain ${\rho }_{\text{int}}\approx {\rho }_{\text{prior}}$), the system must activate active inference channel, doing work on the environment.

This means: stubborn priors (beliefs) are the physical power source of life’s actions.

19.3.4 Strange Attractors and Life Characteristics

Free energy principle explains life’s stability from the perspective of phase space geometry.

Definition 19.3.4 (Non-Equilibrium Steady State / NESS)

Living systems are not in thermal equilibrium (maximum entropy), but in Non-Equilibrium Steady State (NESS).

In QCA phase space, this corresponds to a Strange Attractor.

• Attractor region is the set of low free energy states (survival domain).

• Free energy $\mathcal{F}$ acts as Lyapunov Function in phase space.

• All life activities (metabolism, predation, escape) are processes of phase space flow converging toward attractor.

Corollary 19.3.5 (Ergodicity Breaking of Self-referential Systems)

An observer following FEP has evolution trajectory confined to extremely tiny phase space volume near the attractor. It does not traverse all possible microscopic states like gas molecules (ergodicity breaking). This “picky” behavior toward phase space is what we call “order” or “life characteristics.”

19.3.5 Summary

This section completed the leap from physical dynamics to cognitive dynamics through free energy principle.

1. Unity: Physical potential energy minimization $=$ information surprise minimization.

2. Duality: Minimization can be achieved by changing thoughts (perception) or changing the world (action).

3. Teleology: Life’s “purpose” is to maintain itself in free energy basin (strange attractor), resisting thermodynamic dissipation.

At this point, we have constructed a physical agent capable of perceiving, predicting, and acting. But does such a system possess consciousness? Or is it merely a complex automaton?

In the next section 19.4, we will explore deep connections between strange attractors and life characteristics, providing final preparation for Part XI on topological physics of consciousness.