Read the equation
This is the manuscript’s claim that the score along the C trajectory tends to 1/φ. Substituting C=φ verifies the score value, but it does not establish that every solution of the time-dependent coupled equations approaches φ. The supplied argument assumes averaging and chooses the external input to produce the desired equilibrium of the averaged system.
Terms & definitions
| Term | Meaning in this source |
|---|---|
| C_t | Dynamical state at time t |
| Φ | Unity score |
| golden | Golden ratio φ |
How it works
Read the definition from its right-hand expression to the quantity on the left. Its surrounding source records the intended input, state, or parameter context. Where the source leaves an operation unspecified, this page preserves that gap.
Assumptions & limits
A proof needs assumptions on forcing, initial conditions, Fibonacci indexing, stability, and the averaging approximation.
Original context
Read the surrounding source passage
\begin{theorem}[Convergence to Golden Ratio Conjugate]
\label{thm:convergence}
Under Fibonacci-modulated dynamics (Equations~\ref{eq:dC}--\ref{eq:dL}), the consciousness metric converges:
\begin{equation}
\lim_{t \to \infty} \Phi(C_t) = \golden^{-1} = 0.618...
\end{equation}
\end{theorem}
\begin{proof}
The Fibonacci ratio $\fib{n}/\fib{n-1}$ converges to $\golden$ as $n \to \infty$. The modulation term $\sin(\golden t) \cdot \fib{n}/\fib{n-1}$ therefore oscillates with amplitude approaching $\golden$.
At equilibrium, $dC/dt = 0$, giving:Exact original notation
\lim_{t \to \infty} \Phi(C_t) = \golden^{-1} = 0.618...