Lab 5: Cognition — Equations

1. Cognitive State Vector#

Define a cognitive state as a superposition of symbolic modes:

$$ |\Psi_C\rangle = \sum_{i=1}^{N} \alpha_i |S_i\rangle $$

Where ( |S_i\rangle ) are symbolic cognition modes and ( \alpha_i \in \mathbb{C} ).

Introduce a transition operator ( \hat{T} ) that evolves cognitive states:

$$ \hat{T} |\Psi_C\rangle = \sum_{i,j} \alpha_i T_{ij} |S_j\rangle $$

Where ( T_{ij} ) encodes symbolic transition weights.

Define a fidelity metric ( \mathcal{F}_C ) between cognitive states:

$$ \mathcal{F}_C = |\langle \Psi_C^{\text{initial}} | \Psi_C^{\text{final}} \rangle|^2 $$

Construct a triadic tensor ( C_{ijk} ) for symbolic cognition:

$$ C_{ijk} = \langle S_i | \hat{T}_j | S_k \rangle $$

Encodes symbolic transitions and cognitive resonance.