1.2 Information Island Hypothesis: “Interface-Free Processes” in QCA Networks

In the previous section, we established that dark matter is not an exotic particle, but a “background process” in the QCA network. Now, we must define this state of existence through rigorous physical language. We call this state an “Information Island”.

1.2.1 Decoupling of Topological Knots and Charge

In QCA ontology, any particle (excited state) is defined by the microscopic properties of its evolution operator $\hat{U}$.

Recalling our derivation from Chapter 5 of First Principles:

1. Mass (Inertia): Arises from non-trivial homotopy classes of $\hat{U}$ on the momentum space Brillouin zone, i.e., winding number $\mathcal{W}\in {\pi }_1(S^1)$.

   • If $\mathcal{W}\ne 0$, the particle must maintain internal vibration $v_{int}>0$, thus possessing rest mass $m_0$.

2. Interaction (Charge): Arises from the non-commutativity of $\hat{U}$ with the local gauge connection field ${\hat{A}}_{\mu }$.

   • If $[\hat{U},{\hat{A}}_{\mu }]\ne 0$, the particle acquires phase rotation when passing through the connection field, manifesting as carrying charge $q$.

Standard Model particles (such as electrons) have both mass and charge, meaning they are both topological knots and couple to the $U(1{)}_{EM}$ connection field.

Photons have no mass but propagate interactions; they are excitations of the connection field itself.

Then, logically, there must exist a third possibility:

A topological knot with non-trivial winding number ($\mathcal{W}\ne 0$) but completely commuting with the $U(1{)}_{EM}$ connection field ($[\hat{U},{\hat{A}}_{\mu }]=0$).

We define this state as a “Dark Node” or “Information Island”.

• Physical Meaning:

  • It is “heavy”: Because it must consume computational resources (internal refresh rate) to maintain its topological structure from collapsing. According to light path conservation, it has inertia.

  • It is “hidden”: When it passes through electromagnetic fields (photon sea), it causes no phase fluctuations and is not scattered by photons. It is completely transparent to light.

1.2.2 Independent Evolution of Subspaces

To formalize this concept, we assume that the local Hilbert space $\mathcal{H}$ of QCA can be decomposed into two direct product subspaces:

$$\mathcal{H}={\mathcal{H}}_{vis}\otimes {\mathcal{H}}_{hid}$$

• ${\mathcal{H}}_{vis}$ (Visible Sector): Contains all familiar quarks, leptons, and gauge bosons.

• ${\mathcal{H}}_{hid}$ (Hidden Sector): Contains dark matter degrees of freedom.

Correspondingly, the Hamiltonian decomposes as:

$$\hat{H}={\hat{H}}_{vis}\otimes {\mathbb{I}}_{hid}+{\mathbb{I}}_{vis}\otimes {\hat{H}}_{hid}+{\hat{H}}_{int}$$

If the cross-interaction term ${\hat{H}}_{int}\approx 0$ (or extremely weak, limited to gravity), then these two sectors are decoupled dynamically.

This means that at the same physical location (lattice point $x$), there may simultaneously exist a “visible electron” and a “dark particle.” They are like two beams of light at different frequencies transmitted through the same optical fiber, or two independent virtual machines running on the same computer, mutually non-interfering, except competing for the same underlying resource—spacetime bandwidth.

1.2.3 Why Is Dark Matter Stable?

If dark matter is merely some excited state, why doesn’t it decay into photons?

In standard particle physics, this requires introducing some new conserved number (such as R-parity).

In QCA, stability arises from topological protection.

Since dark matter particles correspond to non-trivial winding states in ${\mathcal{H}}_{hid}$ space, to make them disappear, their winding number $\mathcal{W}$ must be changed to 0.

However, because they are decoupled from the photon field (connection field), they cannot release energy and change topological number by emitting photons.

This is like a radio without speakers: although the internal circuit oscillates (has energy), it cannot convert this oscillation into sound waves (radiation) to release.

Therefore, these “dark topological knots” are extremely long-lived. They are “topological defects” or “primitive data fragments” left over from phase transitions in the QCA network during the early Big Bang.

1.2.4 Conclusion: Not Ghosts, but Neighbors

This section not only provides a mathematical definition of dark matter but, more importantly, changes our philosophical perspective.

Dark matter is not a ghost hidden in cosmic corners; it is here.

Right now, billions of “dark information streams” may be passing through your body.

They are “dark” not because they are far away, but because they operate on orthogonal logical channels from us.

However, they are not completely unknowable. Because although the logical channels are orthogonal, the underlying hardware (spacetime) is shared.

In the next section, we will explore how this sharing manifests through gravity—the only language of dark matter.