The core of political democracy is centered around the principle of electoral democracy, which is the process of choosing representatives through elections. This means that the power to select leaders and form governments lies in the hands of the citizens.

Option 1: Right to education refers to the fundamental right of every individual to receive education. While education is important for a democratic society, it is not directly connected to the process of electoral democracy.

Option 2: Freedom of speech refers to the right to express one`s opinions and ideas without censorship or punishment. While freedom of speech is a fundamental democratic right, it is not the foundation of electoral democracy.

Option 3: Right to equality refers to the principle that all individuals should be treated equally under the law. While this is an important aspect of a democratic society, it does not directly relate to the process of electoral democracy.

Option 4: Universal adult franchise is the principle that all adult citizens have the right to vote in elections. This option is the correct answer as it directly relates to the core of political democracy. Universal adult franchise ensures that all citizens have an equal opportunity to participate in the democratic process and choose their representatives.

Overall, the correct option is option 4, as it best captures the essence of electoral democracy in political democracy.