Meta Theory of the Zeckendorf-Hilbert Universe

20.3 Topological Restatement of Integrated Information Theory (IIT): Strong Connectivity and $\Phi$ Value

In Section 20.2, we identified the “atomic self” of consciousness as Minimal Strongly Connected Component (MSCC) in causal networks. This topological definition qualitatively delineates the boundary of “self,” but has not answered a quantitative question: why do some MSCCs (such as human brains) exhibit highly rich conscious experiences, while others (such as simple oscillating circuits) have almost none?

This section will introduce Giulio Tononi’s Integrated Information Theory (IIT) and restate it as topological field theory on QCA networks. We will prove that IIT’s core quantity $\Phi$ (integrated information) physically corresponds to Irreducible Flux of causal topological closed loops. $\Phi$ value not only measures degree of information integration, but also measures Topological Rigidity of “self” as a topological entity resisting causal cuts.

20.3.1 Geometric Definition of Integrated Information: From Probability Distributions to Causal Manifolds

In standard IIT formulation, $\Phi$ is defined by comparing the system’s overall probability distribution with independent distributions of its parts after “cutting.” In QCA discrete ontology, we can geometrize this as flow resistance analysis on causal manifolds.

Let state space of MSCC subsystem $\Omega$ be ${\mathcal{S}}_{\Omega }$. Due to discrete dynamics of QCA, system state $s_t$ at time $t$ to state $s_{t+1}$ at time $t+1$ defines a transition probability flow $P(s_{t+1}|s_t)$.

Definition 20.3.1 (Causal Flow Tensor)

For any bipartition $\pi =\{A,B\}$ of system $\Omega$ (where $A\cup B=\Omega ,A\cap B=\varnothing$), we define Cut Flow $P_{cut}$ as transition probability after cutting all causal connections between $A\leftrightarrow B$:

$$P_{cut}^{\pi }(s_{t+1}|s_t)=P(A_{t+1}|A_t)\otimes P(B_{t+1}|B_t)$$

This is equivalent to forcibly erasing “wormholes” or QCA edges connecting $A$ and $B$ geometrically, forcing the manifold to degenerate into a direct product manifold.

Definition 20.3.2 (Integrated Information $\Phi$)

System’s integrated information $\Phi (\Omega )$ is defined as information geometric distance (relative entropy or earth mover’s distance) between true flow and weakest cut flow:

$$\Phi (\Omega )\equiv \underset{\pi }{\min }D(P\Vert P_{cut}^{\pi })$$

where minimum is searched over all possible bipartitions $\pi$. The partition ${\pi }_{MIP}$ that minimizes this distance is called Minimum Information Partition (MIP).

Physical Interpretation:

MIP corresponds to Min-Cut in topological structure. $\Phi$ value measures “causal flux” through this min-cut.

• $\Phi =0$: Means there exists a cut surface such that cutting does not affect system dynamics. That is, the system is reducible, topologically equivalent to two disconnected components. Such systems have no unified consciousness.

• $\Phi >0$: Means for any cut surface, there exist non-trivial causal flows on both sides. The system is irreducible. Larger $\Phi$ value means even the “weakest link” is more tightly bound, the stronger the topological integrity of the system.

20.3.2 Quantification of Strong Connectivity: $\Phi$ as Topological Invariant

In Section 20.1, we qualitatively pointed out that consciousness requires feedback loops (strong connectivity). Now we can prove $\Phi$ is precisely the quantitative measure of strong connectivity.

Theorem 20.3.3 ($\Phi$-Strong Connectivity Equivalence Theorem)

In finite QCA networks, $\Phi (\Omega )>0$ if and only if causal graph $G_{\Omega }$ of $\Omega$ is strongly connected.

Proof:

1. Sufficiency: If $G_{\Omega }$ is not strongly connected, according to graph-theoretic decomposition, there must exist a condensation graph, which is a DAG. This means we can find a partition $\pi =\{A,B\}$ such that there are no edges from $B$ to $A$. Cutting $B\to A$ (empty set) and $A\to B$ (feedforward) has no effect on dynamics of $A$ ($A$ does not depend on $B$), and only removes external input for $B$. When computing causal efficacy (Cause-Effect Power), this unidirectional dependence causes $\Phi$ to vanish under some definition (or be reducible for “existence”).

2. Necessity: If $G_{\Omega }$ is strongly connected, then for any partition $\{A,B\}$, there must exist paths from $A$ to $B$ and from $B$ to $A$. Cutting these paths necessarily changes system’s transition probability distribution $P$, causing $D(P\Vert P_{cut})>0$.

Corollary 20.3.4 (Topological Robustness of Consciousness)

$\Phi$ value actually measures topological robustness of MSCC closed loops.

Imagine we apply random noise or attacks on the network (randomly deleting edges). High $\Phi$ systems are like multiply entangled knots; even if a few threads break, overall connectivity (homology groups) still maintains. Low $\Phi$ systems are like fragile rings, breaking into unconscious fragments with slight perturbations.

20.3.3 Geometric Meaning of Exclusion Principle

Another core axiom of IIT is Exclusion Principle: a physical system can only have one “main” conscious experience, corresponding to the substructure (Complex) with maximum $\Phi$, and neither its subsets nor supersets produce independent consciousness.

In QCA discrete ontology, this acquires a clear geometric interpretation.

Definition 20.3.5 (Causal Horizon Exclusion)

Consider nested strongly connected components $S_1\subset S_2\subset \dots$.

For $S_2$, internal $S_1$ is just a detail of its internal structure; for $S_1$, the rest of $S_2$ is just environmental background.

Geometric essence of exclusion principle is uniqueness of causal horizons.

At any moment, observer’s effective macrostate is defined by the scale with maximum causal power.

• If $S_1$ has extremely strong connections ($\Phi (S_1)\gg \Phi (S_2)$), then $S_1$ constitutes an effective physical entity (particle/observer), while $S_2$ is just a weakly coupled system of $S_1$ with environment.

• If $S_2$’s overall connection is stronger than its parts ($\Phi (S_2)>\Phi (S_1)$), then $S_1$ loses independence, “fusing” into larger self $S_2$.

This mathematically corresponds to finding global maximum of scalar field $\Phi (x)$ on the lattice of subsystems. This maximum point defines the objective boundary of “me”.

20.3.4 Physical Realization: $\Phi$ Value Calculation in Self-referential Scattering Networks

In Self-referential Scattering Networks (SSN) discussed in Chapter 17, $\Phi$ value can be directly calculated through properties of scattering matrices.

Let SSN’s closed-loop transmission matrix be $T(\lambda )$. System’s characteristic equation is $\det (\mathbb{I}-T(\lambda ))=0$.

Theorem 20.3.6 (Scattering $\Phi$ Formula)

For a self-referential scattering network, its integrated information $\Phi$ is proportional to feedback loop’s Gain and Mixing:

$$\Phi \approx \sum _{\text{loops}}\ln |\text{Gain}(\text{loop})|-S_{\text{leakage}}$$

where $\text{Gain}$ measures signal’s ability to maintain itself in closed loop (eigenvalue modulus close to 1), $S_{\text{leakage}}$ measures rate of information leakage from loop to environment.

• High $\Phi$ Systems: Resonant networks with strong feedback (near critical state) and low leakage (high quality factor $Q$).

• Low $\Phi$ Systems: Overdamped or severely leaking networks.

Conclusion

This section completed quantification of consciousness physics through topological restatement of IIT.

1. $\Phi$ is Topological Flux: It measures irreducible information flow through min-cut in causal networks.

2. Strong Connectivity is Consciousness: Only by forming topological closed loops (MSCC) can systems have non-zero $\Phi$, thereby possessing “internal perspective.”

3. Maximum is Boundary: Exclusion principle ensures uniqueness and objectivity of boundaries of conscious agents.

At this point, we have not only defined the “atom” of consciousness (MSCC), but also given its “mass” ($\Phi$). In the next section 20.4, we will use these tools to explore emergence phenomena in causal networks, explaining why simple QCA rules can emerge high-level consciousness with complex $\Phi$ structures.