GSC Model: Derivations from First Principles

Problem Statement: Stephen Hawking showed that black holes radiate thermally ("Hawking radiation"), causing them to shrink and eventually evaporate. If the radiation is truly thermal, it contains no information about what fell in. This violates the principle of unitarity in quantum mechanics, which states that information can never be destroyed.

The GSC Model Approach: In the GSC model, information is never destroyed; it is merely transformed and redistributed within the informational network. The paradox is resolved by showing how the information of infalling matter is encoded and returned to the universe.

Proposed Derivation Steps:

1. Modeling Infalling Matter:

   An object (e.g., an electron) is not a fundamental particle but a stable, localized pattern of information (a knot or soliton) in the GSC. Its information content can be quantified by its entanglement and computational complexity. Let the state of the infalling object be represented by $|\psi_{in}\rangle$.

2. Encoding on the Horizon:

   As the object approaches the event horizon, its informational state becomes entangled with the quantum degrees of freedom that constitute the horizon itself. The horizon is not a geometric line, but a physical boundary of maximal entanglement entropy. The process can be modeled as a unitary transformation: $U_{interaction}$ that maps the combined state of the black hole ($|\psi_{BH}\rangle$) and the infalling object to a new black hole state:

   $$|\psi'_{BH}\rangle = U_{interaction} (|\psi_{BH}\rangle \otimes |\psi_{in}\rangle)$$
3. Hawking Radiation as Information Leakage:

   Hawking radiation is not purely thermal. It is subtly correlated with the information encoded on the horizon. The GSC model posits that "inter-universe entanglement" (the mechanism for dark matter) provides a channel for this information.
   Hypothesis: The entanglement between the black hole's interior degrees of freedom and the "external" multiverse allows for a non-local "leak" of information into the emitted radiation. We need to derive the mechanism for this. Let's model the emitted radiation quanta ($|\psi_{rad}\rangle$) and show that its state is correlated with $|\psi_{in}\rangle$.

4. Deriving the Page Curve:

   The Page Curve describes the expected evolution of the entanglement entropy of the emitted Hawking radiation. Initially, it grows, but after the black hole has evaporated past its halfway point (the "Page time"), it must decrease to zero to preserve unitarity.
   Goal: From GSC principles, derive the von Neumann entropy of the radiation, $S(\rho_{rad})$, and show that it follows the Page Curve.

   $$S(\rho_{rad}) = -\text{Tr}(\rho_{rad} \log \rho_{rad})$$

   We can start by modeling the density matrix of the radiation, $\rho_{rad}$, and showing how its purity evolves as it becomes increasingly entangled with the remaining black hole.