**Explanation:**

To understand why the electric flux through the Gaussian surface is zero, we need to consider a few key concepts: electric flux, Gauss's law, and the behavior of conductors.

**Electric Flux:**
Electric flux is the measure of the electric field passing through a given surface. It is defined as the dot product of the electric field and the area vector of the surface. Mathematically, electric flux (Φ) is given by Φ = ∫E⋅dA, where E is the electric field and dA is the differential area vector.

**Gauss's Law:**
Gauss's law relates the electric flux through a closed surface to the charge enclosed within that surface. It states that the electric flux through any closed surface is equal to the total charge enclosed divided by the permittivity of the medium (∈₀). Mathematically, Gauss's law is given by Φ = q/∈₀.

**Behavior of Conductors:**
In the case of conductors, charges are free to move within the material. When a conductor is in electrostatic equilibrium, the charges distribute themselves in such a way that the electric field inside the conductor is zero.

**Explanation of the Scenario:**
In this scenario, we have an uncharged metal cavity (conductor) with a positive charge q placed inside it. We consider a Gaussian surface S surrounding the cavity.

1. Since the uncharged metal cavity is a conductor, the charges within it will redistribute themselves to achieve electrostatic equilibrium. The charges will migrate to the surface of the conductor, and the electric field inside the conductor will be zero.

2. As the electric field inside the conductor is zero, the electric field passing through the Gaussian surface S will also be zero.

3. Since the electric field passing through the Gaussian surface is zero, the electric flux through the surface will be zero. This is because the electric flux is determined by the dot product of the electric field and the area vector of the surface. If the electric field is zero, the flux will also be zero.

4. According to Gauss's law, the electric flux through the closed surface S is related to the charge enclosed within the surface. As the conductor is uncharged initially and the positive charge q is inside the conductor, the charge enclosed within the surface S is also zero.

5. Therefore, the electric flux through the Gaussian surface S is zero, as there is no charge enclosed within the surface and the electric field passing through it is zero.

In conclusion, the electric flux through the Gaussian surface S is zero in this scenario because the conductor achieves electrostatic equilibrium, resulting in a zero electric field within the conductor and no charge enclosed within the surface.