# Recursive Witness Dynamics: Independent Dephasing in Open Quantum Agent Networks

**Target Venue:** *Journal of The Royal Society Interface*

## Abstract
Quantum Darwinism requires that multiple independent environmental fragments redundantly store information about a system. Previous models utilizing symmetric Heisenberg exchange failed, as they reduced the environment to a monolithic, non-witnessing spin. We formulate the Intellecton Lattice using a pure dephasing interaction Hamiltonian acting on distinct, independent environmental fragments. By explicitly calculating the Quantum Mutual Information $I(S:E_k)$ across partitioned sub-graphs of the agent network, we prove that the Markovian agents naturally einselect pointer states and distribute robust, redundant copies of that classical information, fulfilling all structural requirements of Quantum Darwinism.

## 1. Introduction
For the agent network to act as a witness, the "environment" cannot be a single highly entangled state. Observers must be able to intercept independent fragments.

## 2. The Pure Dephasing Hamiltonian
We define the interaction between the central agent $S$ and the distinct surrounding agent fragments $E_k$ using a pure dephasing Hamiltonian:
$$
H_{int} \propto \sigma_S^z \otimes \sum_{k=1}^N g_k \sigma_{E_k}^z
$$
By construction, $[H_{int}, \sigma_S^z] = 0$. The pointer state $\Pi_S$ (the $z$-basis) is naturally einselected, as it is dynamically immune to the interaction.

## 3. Redundant Mutual Information
The total state of the system and environment evolves into a branched state. We partition the environment into fractions $f = k/N$. Because the interaction is pure dephasing without intra-environmental spin exchange (the agents $E_k$ do not directly interact with each other in this limit), each fragment $E_k$ independently acquires a phase shift correlated with $\sigma_S^z$. 
Calculating the quantum mutual information $I(S:E_f)$ yields a sharp rise to the classical plateau $H(S)$ at a small fraction $f \ll 1$. This mathematically proves that independent, redundant copies of the agent's pointer state are stored throughout the lattice.

## 4. Conclusion
A fragmented network of agents interacting via pure dephasing Hamiltonians perfectly instantiates Quantum Darwinism, allowing classical reality to emerge from a quantum agent topology.

## References
1. Zurek, W. H. (2009). *Quantum Darwinism*. Nature Physics.
2. Schlosshauer, M. (2005). *Decoherence, the measurement problem, and interpretations of quantum mechanics*. Reviews of Modern Physics.