Averaged C-balance assumption

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The source replaces the time-dependent oscillation with its mean and treats L as constant in that step. If this averaging is justified and the sine mean is zero, the relation reduces to E=C* when α≠0. Correlation between a changing L(t) and the sinusoid must be addressed before extending that conclusion to the original trajectory.

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Original context

The error is $O(\golden^{-2n})$, converging exponentially.

\textbf{Equilibrium Analysis.} At equilibrium, all time derivatives vanish. From Equation~\ref{eq:dC}:
\begin{equation}
0 = \alpha(E^* - C^*) + \beta L^* \langle \sin(\golden t) \rangle \cdot \golden
\end{equation}

The time-average $\langle \sin(\golden t) \rangle = 0$ over complete periods, yielding $E^* = C^*$.

From Equation~\ref{eq:dE}:
\begin{equation}
0 = -\gamma C^* + I^* \cdot (1 + \langle \cos(\golden t \cdot \fib{n}) \rangle)

0 = \alpha(E^* - C^*) + \beta L^* \langle \sin(\golden t) \rangle \cdot \golden

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