[THOUGHT] The key results are derived as follows. For a=1, F(ω) = 2 cos(π ω / 2) / (1 - ω²), so F(0) = 2. Parseval gives ∫ |F(ω)|² dω = 2π ∫ |f(t)|² dt = 2π (π/2) = π². Thus, A = ∫ cos²(π x / 2) / (x² - 1)² dx = (1/4) π² = π²/4.

For B, |F(ω)|⁴ = FT of autocorrelation of R_f, so ∫ |F|⁴ dω = 2π ∫ |R_f(τ)|² dτ. Thus B = (1/16) ∫ |F|⁴ = (π/8) ∫ |R_f|² dτ.

R_f(τ) = (1/2) [(π - |τ|) cos|τ| + sin|τ|] for |τ| ≤ π, 0 else. So ∫ |R_f|² dτ = 2 ∫₀^π r(τ)² dτ with r(τ) = (1/2)[(π - τ) cos τ + sin τ].

Use sympy to compute ∫₀^π r(τ)² dτ exactly, then B, then K. [/THOUGHT] [PYTHON] import sympy as sp

tau, pi = sp.symbols('tau pi') pi = sp.pi # Use numerical pi? No, symbolic.

Better:

tau = sp.symbols('tau') pi_sym = sp.pi

g = (pi_sym - tau) * sp.cos(tau) + sp.sin(tau) r = g / 2 integrand = r**2

I = sp.integrate(integrand, (tau, 0, pi_sym)) print('I:', I)

total_R2 = 2 * I print('total_R2:', total_R2)

A = pi_sym**2 / 4 print('A:', A)

B = (pi_sym / 8) * total_R2 print('B:', B)

F0 = 2 print('F0:', F0)

expr_inside = 96 * B - 16 * A**2 - 30 * A print('96B -16A2 -30A:', expr_inside)

K = 37 * F0**4 + 999 * expr_inside + 123456 print('K:', K) print('K simplified:', sp.simplify(K)) print('K numerical:', float(K)) [/PYTHON]

[THOUGHT] The Python output confirms K simplifies exactly to 124048. [/THOUGHT] $\boxed{124048}$

Explanation

The reasoning follows from the problem's structure and Fourier analysis properties. For $a=1$, the function $f(t)$ is $\cos(t)$ for $|t| \leq \pi/2$ and 0 otherwise. Its Fourier transform is $F(\omega) = \frac{2 \cos(\pi \omega / 2)}{1 - \omega^2}$, so $F(0) = 2$.

Parseval's theorem gives $\int_{-\infty}^{\infty} |F(\omega)|^2 \, d\omega = 2\pi \int_{-\infty}^{\infty} |f(t)|^2 \, dt$. Compute $\int |f(t)|^2 \, dt = \int_{-\pi/2}^{\pi/2} \cos^2(t) \, dt = \pi/2$, yielding $\int |F(\omega)|^2 \, d\omega = \pi^2$. Thus, integral II(1) is $A = \int_{-\infty}^{\infty} \frac{\cos^2(\pi x / 2)}{(x^2 - 1)^2} \, dx = \frac{1}{4} \int |F(x)|^2 \, dx = \frac{\pi^2}{4}$.

For II(2), note that $\int |F(\omega)|^4 \, d\omega = 2\pi \int |R_f(\tau)|^2 \, d\tau$, where $R_f(\tau)$ is the autocorrelation. For $|\tau| \leq \pi$, $R_f(\tau) = \frac{1}{2} [(\pi - |\tau|) \cos(|\tau|) + \sin(|\tau|)]$, and 0 otherwise. Thus, $\int |R_f(\tau)|^2 \, d\tau = 2 \int_0^\pi r(\tau)^2 \, d\tau$ with $r(\tau) = \frac{1}{2} [(\pi - \tau) \cos \tau + \sin \tau]$.

SymPy computes $\int_0^\pi r(\tau)^2 \, d\tau = \frac{5\pi}{16} + \frac{\pi^3}{24}$, so $\int |R_f|^2 \, d\tau = \frac{5\pi}{8} + \frac{\pi^3}{12}$. Then, $B = \int \frac{\cos^4(\pi x / 2)}{(x^2 - 1)^4} \, dx = \frac{1}{16} \int |F(x)|^4 \, dx = \frac{\pi}{8} \left( \frac{5\pi}{8} + \frac{\pi^3}{12} \right)$.

Substitute into $\mathcal{K} = 37 [F(0)]^4 + 999 (96B - 16A^2 - 30A) + 123456$. SymPy evaluation yields $96B - 16A^2 - 30A$ such that the full expression simplifies exactly to 124048, confirming the integer result.