To explain how the human brain makes this perceptual decision, we can model the process using a Bayesian Inference Framework. The brain must determine the likelihood of the event state $E \in \{ \text{Bounce}, \text{Pass} \}$ given the visual evidence $\mathbf{V}$ and auditory evidence $\mathbf{A}$. The posterior probability is modeled as:
$$P(E = \text{Bounce} \mid \mathbf{V}, \mathbf{A}) = \frac{P(\mathbf{V}, \mathbf{A} \mid E = \text{Bounce}) P(E = \text{Bounce})}{P(\mathbf{V}, \mathbf{A})}$$
Where the joint likelihood is determined by the temporal alignment ($\Delta t$) and visual impact parameter ($b$). We define the temporal likelihood function as a Gaussian distribution centered around zero delay, representing the tuning of multisensory neurons in the superior colliculus:
$$L(\Delta t) = \exp\left( -\frac{\Delta t^2}{2\sigma_{\text{temp}}^2} \right)$$
Where $\sigma_{\text{temp}} \approx 100 \text{ ms}$ represents the standard standard deviation of the temporal binding window. Similarly, the spatial likelihood scales inversely with the impact parameter $b$:
$$L(b) = \exp\left( -\frac{b^2}{2\sigma_{\text{spatial}}^2} \right)$$
In our synthesizer architecture, the collision sound is generated dynamically via the **Web Audio API** using a percussive additive synthesis loop. When the collision event triggers (accounting for the temporal offset $\Delta t$), the engine instantiates an additive signal path:
$$S(t) = \left[ A_1 \sin(2\pi f_{\text{low}} t) e^{-\frac{t}{\tau_{\text{low}}}} + A_2 \sin(2\pi f_{\text{high}} t) e^{-\frac{t}{\tau_{\text{high}}}} + A_3 \cdot \eta(t) e^{-\frac{t}{\tau_{\text{noise}}}} \right]$$
Where $f_{\text{low}}$ is determined by the "Collision Tone Freq" slider, $f_{\text{high}} = 2.5 \times f_{\text{low}}$, and $\eta(t)$ represents white noise to simulate high-frequency structural crackle. The decay parameters are configured with $\tau_{\text{low}} \approx 80 \text{ ms}$ and $\tau_{\text{high}} \approx 15 \text{ ms}$, creating a realistic, hard impact sound characteristic of solid wooden or glass spheres.