Controls
P(t) = base + amplitude · sin(2π·frequency·t). Memory M(t) updates each step.
Status
Mirror stack Sk(t)
Expansion index E(t) over time
Equations
Base layer: \[ S_0(t) = f\!\big(\alpha\,P(t) + \beta\,M(t)\big) \]
Recursive mirrors: \[ S_{k+1}(t) = f\!\big(\gamma\,S_k(t) + \alpha\,P(t) + \beta\,M(t)\big) \]
with \( f(x)=\tanh(\eta x) \), and \( f'(x)=\eta\big(1-\tanh^2(\eta x)\big) \).
Memory: \[ M(t+1) = (1-\lambda)\,M(t) + g\big(P(t), S_0(t)\big) \] with \( g = \kappa\,S_0(t) \) (here \(\kappa=0.1\)).
Local stability: mapping \(T(S)=f(\gamma S + c)\) has \( |T'(S^\*)| = \big|\gamma\,f'(\gamma S^\* + c)\big| \lt 1 \).
Expansion index (proxy): \[ E(t)=\sum_{k=1}^{K(t)} c^k \,\tilde I_k, \quad \tilde I_k = 1 - \frac{|S_k-S_{k-1}|}{1+|S_{k-1}|} \in [0,1] \]
Prototype note: \( \tilde I_k \) is a simple, bounded similarity proxy. In a paper, replace with a proper mutual information estimator.