The correct answer is option 1: Jan. 4, 2003.

The Political Council of the Nuclear Command Authority (NCA) was established on January 4, 2003. The NCA is the highest decision-making body responsible for all policy-related matters pertaining to nuclear weapons and their deployment in India. It consists of members from the political leadership and the military, including the Prime Minister, the Defense Minister, and the National Security Advisor.

Option 2, Jan. 5, 2003, is incorrect, as the Political Council of NCA was set up on January 4, 2003, not January 5th.

Option 3, Jan. 6, 2003, is also incorrect. The NCA`s formation took place one day earlier, on January 4, 2003.

Option 4, Jan. 7, 2003, is not the correct answer. The establishment of the Political Council of NCA occurred on January 4, 2003, not January 7th.

It is important to note that the correct answer is based on the given information. If this information is incorrect, then the answer may also be incorrect.