The problem asks for the calculation of a final integer $\mathcal{K}$ based on several intermediate quantities derived from differential equations, integrals, and series.

Part I: Differential Equation The differential equation is $\frac{dy}{dx} = y(1-y)$ with $y_*(0) = 1/2$. This is the logistic equation. The solution satisfying the initial condition is $y_*(x) = \frac{e^x}{1+e^x}$. We need $a = y_*(\ln 3)$. $a = \frac{e^{\ln 3}}{1+e^{\ln 3}} = \frac{3}{1+3} = \frac{3}{4}$.

Part II: Definite Integral We need to compute $I = \int_{-1}^{1} \frac{\arccos(x/2)}{\cos^2(\pi x/3)} \, dx$. Using the symmetry property $\arccos(-x) = \pi - \arccos(x)$ (specifically $\arccos(-x/2) = \pi - \arccos(x/2)$) and the evenness of $\cos^2(\pi x/3)$, we can show that $2I = \pi \int_{-1}^{1} \sec^2(\pi x/3) \, dx$. Evaluating this integral yields $I = 3\sqrt{3}$.

Part III: Series and Functions The functions are $f(x) = \cosh(x)$ and $g(x) = \sinh(x)$. We evaluate these at $x_0 = \ln(2+\sqrt{3})$. $F_0 = \cosh(x_0) = 2$. $G_0 = \sinh(x_0) = \sqrt{3}$.

Part IV: Final Calculation We compute $S = 4a + \frac{I}{G_0} + F_0$. Substituting the values: $S = 4(3/4) + \frac{3\sqrt{3}}{\sqrt{3}} + 2 = 3 + 3 + 2 = 8$. Finally, $\mathcal{K} = 999 S^2 + 7 = 999(64) + 7 = 63936 + 7 = 63943$.

[PYTHON] import numpy as np from scipy.integrate import quad

Part I

def y_star(x): return np.exp(x) / (1 + np.exp(x))

a = y_star(np.log(3)) print(f"{a=}")

Part II

def integrand(x): return np.arccos(x/2) / (np.cos(np.pi * x / 3)**2)

I_val, error = quad(integrand, -1, 1) print(f"{I_val=}")

Part III

x0 = np.log(2 + np.sqrt(3)) F0 = np.cosh(x0) G0 = np.sinh(x0) print(f"{F0=}") print(f"{G0=}")

Part IV

S = 4 * a + I_val / G0 + F0 print(f"{S=}")

K = 999 * S**2 + 7 print(f"{K=}") [/PYTHON] [THOUGHT] The Python code confirms the manual calculations. $a = 0.75 = 3/4$. $I \approx 5.19615$, which is $3\sqrt{3} \approx 3 \times 1.73205 = 5.19615$. $F_0 = 2.0$. $G_0 \approx 1.73205 = \sqrt{3}$. $S = 8.0$. $K = 63943.0$.

The result is an integer as expected. [/THOUGHT]

\boxed{63943}

[THOUGHT] The Python code computed the values:

• $a = 0.75$
• $I \approx 5.196$ (which corresponds to $3\sqrt{3}$)
• $F_0 \approx 2$
• $G_0 \approx 1.732$ (which corresponds to $\sqrt{3}$)
• $S = 8.0$
• $\mathcal{K} = 63943.0$

The calculation is: $S = 4(0.75) + \frac{3\sqrt{3}}{\sqrt{3}} + 2 = 3 + 3 + 2 = 8$. $\mathcal{K} = 999(8^2) + 7 = 999(64) + 7 = 63936 + 7 = 63943$.

The result is an integer. [/THOUGHT]

\boxed{63943}