Nair (195no) and Ravi Ashwin (54no) shared an unbroken stand of 147 to carry the hosts to a teatime 582-5, 105 runs in front, as England mustered just one wicket in two sessions at the MA Chidambaram Stadium.
The challenge here for Alastair Cook’s tourists will therefore be to summon the resilience, at a time of India’s choosing unless there is an improbable late collapse, to close out a stalemate in this tough series already lost after their hosts went 3-0 up in Mumbai last week.
England had little obvious motivation to accelerate proceedings in the morning, so a first hour containing just 12 overs and 34 runs was an acceptable overture.
They would have had a wicket too, if only Stuart Broad had a review left at his disposal when Murali Vijay was given not out caught-behind on 21 but - according to technology - had made contact with a thin edge.
Instead it was to be Liam Dawson who struck with his maiden Test wicket, Murali lbw on the back foot when he failed to spot the arm ball.
By then, Nair had completed his hundred from 185 deliveries with eight fours and a six.
The fifth-wicket stand was broken at 63, and England reined the innings run rate temporarily below 3.5 an over.
Ashwin appeared in no hurry in front of his home-town crowd before lunch, taking 15 balls before he got off the mark with a single off Dawson.
But there was more urgency in the afternoon, and another instance of England regret at using up both reviews the previous evening when Nair this time could have gone for 154, caught-behind from a reverse-sweep at Adil Rashid - again given not out by umpire Simon Fry.
Ashwin had only nine from 36 balls at lunch but helped to up the tempo on the way to his 50 from another 80 deliveries.