Lab 8: Cognition — Equations

Purpose#

To formalize the cognitive dynamics of triadic systems, including recursive awareness, symbolic resonance, and dimensional feedback loops.

Key Equations#

1. Recursive Cognition Index#

$$ C_n = \sum_{i=1}^{n} R_{i,i+1,i+2} \cdot \log(D_i) $$ Where ( R_{ijk} ) is triadic resonance and ( D_i ) is dimensional depth.

2. Symbolic Resonance Function#

$$ \Sigma(t) = \alpha \cdot \sin(\omega t) + \beta \cdot \cos(\phi t) $$ Models symbolic cognition as a harmonic blend of phase and frequency.

3. Feedback Loop Amplification#

$$ A_n = \frac{C_n^2}{\sum_{j=1}^{n} D_j} $$ Amplifies recursive cognition based on dimensional load.

4. Cognitive Phase Shift#

$$ \Psi(t) = \int_{0}^{t} \Sigma(t) \cdot R_{ijk} , dt $$ Tracks phase evolution of symbolic cognition.

Symbol Legend#