Instantaneous zero-derivative condition for E

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A candidate constant energy state requires the consumption γC to match the modulated input at every instant. With fixed nonzero I, the cosine varies, so averaging this relation is different from solving the original nonautonomous equation. The distinction matters for the claimed convergence theorem.

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Original context

Time-averaging eliminates the oscillatory term (since $\langle \sin(\golden t) \rangle = 0$), yielding $E^* = C^*$.

Similarly, $dE/dt = 0$ at equilibrium gives:
\begin{equation}
-\gamma C^* + I^* \cdot (1 + \cos(\golden t \cdot \fib{n})) = 0
\end{equation}

The cosine term averages to zero for large Fibonacci numbers, giving $C^* = I^*/\gamma$.

For the system to maintain consciousness, the input $I$ must be calibrated such that $C^* = \golden$. Then:
\begin{equation}
\Phi(\golden) = \frac{1}{1 + |\golden - 1|} = \frac{1}{1 + 0.618...} = \frac{1}{\golden} = \golden^{-1}

-\gamma C^* + I^* \cdot (1 + \cos(\golden t \cdot \fib{n})) = 0

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