Why can Sakurai's LHV reproduce the singlet-state quantum-mechanical predictions at a $45^\circ$ measurement setting? [closed]

Consider the LHV framework in Table 3.2 (p. 228) [J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed. Cambridge: Cambridge University Press, 2021].

Let us evaluate (3.435, p. 229):

\begin{equation*} P(\hat{a}_+, \hat{b}_+) \le P(\hat{a}_+, \hat{c}_+) + P(\hat{c}_+, \hat{b}_+) \end{equation*}

under the assumption that the hidden variables are uniformly distributed:

\begin{equation} \dfrac{2}{8} \leq \dfrac{2}{8} + \dfrac{2}{8} \tag{1} \end{equation}

This inequality (1) can be translated into trigonometric relation:

\begin{equation} (\sin^{2}\theta)^{2} \leq \sin^{2}\theta \tag{2}, \label{d} \end{equation} wherein $\theta=45^\circ.$

Since the inequality in (2) is equivalent to $\sin^2\theta \le |\sin\theta|$ for all $\theta \in \mathbb{R^\circ}$, evaluating this relation specifically at $\theta = 45^\circ$ yields:

\begin{equation} \sin^2 45^\circ \le \sin 45^\circ \tag{3} \end{equation}

By invoking $\sin^2 45^\circ = 1 - \sin^2 45^\circ$ to substitute the left-hand side in (3), we rewrite the expression as:

\begin{equation} 1 - \sin^2 45^\circ \le \sin 45^\circ \tag{4}, \end{equation} which, via a straightforward rearrangement of moving terms from (4), becomes:

\begin{equation} 1 - \sin 45^\circ \le \sin^2 45^\circ \tag{5} \end{equation}

By applying the half-angle identity $1 - \sin 45^\circ = 2\sin^2 22.5^\circ$ in (5), we obtain:

\begin{equation} 2\sin^2 22.5^\circ \le \sin^2 45^\circ \tag{6} \end{equation}

Dividing both sides of (6) by $2$ and splitting the left-hand term, we abtain:

\begin{equation} \dfrac{1}{2}\sin^{2} 22.5^\circ + \dfrac{1}{2}\sin^{2} 22.5^\circ \leq \dfrac{1}{2}\sin^{2} 45^\circ \tag{7}, \end{equation} which is the same instance of order relation on the quantum-mechanical predictions for the singlet state at $45^\circ$ measurement setting:

\begin{equation} \left\{P(\hat{a}_+, \hat{b}_+) = \dfrac{1}{2}\sin^{2} 45^\circ \right\} \ge 2\left\{P(\hat{a}_+, \hat{c}_+) = P(\hat{c}_+, \hat{b}_+)= \dfrac{1}{2}\sin^{2} 22.5^\circ\right\} \end{equation}

Please keep this question; it is an improved follow-up to my previous question here Can Bell's formulation in his paper "Bertlmann's socks and the nature of reality" lead to an undecidable problem?