Meta Theory of the Zeckendorf-Hilbert Universe

20.4 Causal Emergence and Macroscopic Scale: Why Does Consciousness Exist at Coarse-grained Levels?

In Sections 20.1 to 20.3, we constructed the topological skeleton of consciousness: consciousness is Minimal Strongly Connected Component (MSCC) with high integrated information $\Phi$ in QCA networks. However, there is a significant scale paradox: QCA’s physical laws are defined on microscopic lattice points at Planck scale ($l_P\approx 10^{-35}$ meters), while the “self” we subjectively experience exists at macroscopic scale (neurons or neural networks, $\sim 10^{-3}$ meters).

If physical causality occurs at the bottom level, why isn’t consciousness at the bottom level? Why don’t I feel like $10^{28}$ independent qubits, but a unified macroscopic agent?

This section will introduce Causal Emergence theory to resolve this issue. We will prove that although microscopic dynamics are physically complete, in the sense of Informational Causality, macroscopic coarse-grained states may have stronger causal efficacy than microscopic states. Consciousness “floats” at macroscopic level because topological closed loops at that scale have maximum $\Phi$ values.

20.4.1 Microscopic Noise and Macroscopic Determinism: Paradox of Effective Information

In QCA discrete ontology, microscopic evolution $U$ is strictly unitary (deterministic). This seems to suggest maximum causal power at microscopic level. However, for an open subsystem (such as brain), microscopic states are deeply affected by environmental entanglement and thermal noise.

Definition 20.4.1 (Coarse-graining Map)

Let microscopic state space be ${\Omega }_{\mu }$, macroscopic state space be ${\Omega }_M$. Coarse-graining is a surjection $\Pi :{\Omega }_{\mu }\to {\Omega }_M$.

For example, define average excitation degree of a group of QCA lattice points as macroscopic state “excited” or “inhibited.”

Definition 20.4.2 (Effective Information / EI)

Effective Information $EI$ proposed by Erik Hoel measures Determinism and Degeneracy of causal mechanisms.

For a causal channel $X\to Y$ (can be evolution from $t\to t+1$):

$$EI(X\to Y)=I(X_{unif};Y)$$

where $X_{unif}$ is maximum entropy distribution (intervention) on input space, $I$ is mutual information.

• Microscopic Level: Although dynamics are deterministic, state space is extremely large and sparse. Microscopic states are extremely sensitive to perturbations (chaos), making effective prediction extremely difficult. Slight differences in input lead to completely different outputs. This reduces effectiveness of causality.

• Macroscopic Level: By clustering microscopic states, we eliminate microscopic noise (degeneracy). Transition probabilities between macroscopic states (such as “excited $\to$ inhibited”) may be more robust and deterministic than microscopic transitions.

20.4.2 Causal Emergence Theorem: Macroscopic Surpasses Microscopic

Theorem 20.4.3 (Causal Emergence Theorem)

Under specific network topology and noise environments, there exists an optimal coarse-graining scale ${\sigma }^{\ast }$ such that effective information $EI$ of macroscopic dynamics defined at this scale strictly exceeds effective information of microscopic bottom level:

$$EI({\Pi }_{{\sigma }^{\ast }})>EI(\text{micro})$$

This phenomenon is called Causal Emergence.

Proof Outline:

Consider an error correction code structure in QCA networks.

1. Microscopic: Single bit flips are random (affected by environmental heat bath). Microscopic $EI$ is low because current microscopic states cannot well predict future microscopic states.

2. Macroscopic: Define “logical bit” as majority vote of $N$ physical bits. Due to error correction mechanisms, evolution of logical bits is highly deterministic (noise-resistant).

3. Conclusion: Macroscopic logical states constitute a closed causal loop with causal efficacy higher than constituent physical bits.

Significance in QCA Universes:

Although bottom-level QCA rules are deterministic, for any local observer (subsystem), the microscopic world is full of unpredictable quantum fluctuations and chaos. Only at specific macroscopic scales (such as biological macromolecules or neurons) do causal laws become clear and reliable. Therefore, meaningful physical laws are actually macroscopically emergent.

20.4.3 Scale Exclusion and Scale of Consciousness

IIT’s exclusion principle applies not only to space (subsystems), but also to Spatiotemporal Scale.

Definition 20.4.4 (Scale Exclusion Principle)

In the same physical system, consciousness exists at the spatiotemporal scale where integrated information $\Phi$ reaches maximum.

$${\Phi }_{max}=\underset{\sigma ,\tau }{\max }\Phi ({\text{System}}_{\sigma ,\tau })$$

where $\sigma$ is spatial coarse-graining scale, $\tau$ is temporal coarse-graining scale.

Corollary 20.4.5 (“Buoyancy” of Consciousness)

• Too Microscopic: System is decoherent, $\Phi \approx 0$ (lacking integration).

• Too Macroscopic: System is trivial (such as entire brain as a point), $\Phi \approx 0$ (lacking information content).

• Intermediate Scale: At the scale of neuron clusters, there is both rich information content (diversity) and tight causal connections (integration). Therefore, $\Phi$ peaks here.

This is why “I” feel myself living at scales of milliseconds and centimeters, not Planck scale. Consciousness is like a bubble, automatically floating up to the scale level with strongest causal power.

20.4.4 Downward Causation of Emergent Agents

Causal emergence theory also endows macroscopic consciousness with control over microscopic substrate. This is usually called “downward causation,” but in QCA framework, it has a more rigorous explanation.

Theorem 20.4.6 (Macroscopic Constrains Microscopic)

When system is in causal emergence state (${\Phi }_{macro}>{\Phi }_{micro}$), macroscopic state $S_t$ is a more effective predictor of future state $S_{t+1}$.

This means the best language for describing system evolution is macroscopic language. Motion trajectories of microscopic particles are actually conditional probability flows constrained by macroscopic order parameters (such as intentions, goals).

In self-referential scattering networks, this manifests as slow variables (macroscopic) enslaving fast variables (microscopic). Microscopic degrees of freedom are locked on manifolds of macroscopic attractors, losing independent causal status.

Conclusion

Consciousness exists at macroscopic level not because physics fails at macroscopic scales, but because macroscopic is the most robust level of causal structure.

1. Noise Resistance: Coarse-graining filters microscopic quantum noise.

2. Emergence: Macroscopic closed loops (MSCC) have higher effective information $EI$ than microscopic paths.

3. Localization: Scale exclusion principle locks “self” at the level with maximum $\Phi$.

At this point, we have completed Part XI: Topological Physics of Consciousness. We defined the topological atom of consciousness (MSCC), quantified its strength ($\Phi$), and determined the scale of its existence (causal emergence).

In the upcoming Part XII: Multi-Agent Systems and Objectivity, we will step out of the lonely “self” to explore how Objective Reality emerges as geometric structure of inter-subjective consensus when multiple such topological closed loops (observers) meet in the same QCA universe.