With the World Cup around the corner, New Zealand want to go in on the back of a successful series, albeit in a different format.
A Sri Lankan win, and with it the squaring of the series, does not fit that bill.
New Zealand batted an age on this ground in their second innings 11 months ago to save a test against India, and win that rubber.
A repeat, in good batting conditions, is within Williamson's scope. Sufficient support can produce a challenging fourth innings target for the Sri Lankans.
Williamson is closing on a ninth test hundred. If he gets there it will be his fifth in 19 test innings, spread over four opponents.
His concentration is admirable. With watchfulness his byword, he was impressive and luck smiled on him yesterday - dropped at 29, a return catch to spinner Rangana Herath, and at 60 when Nuwan Pradeep spilled a hook, after a couple of juggles, at fine leg off Dhammika Prasad.
He had already batted more than five hours, driving elegantly and working the ball about the field skilfully.
Watling, cutting energetically and scampering hard, is in need of a strong innings; his last of substance, an unbeaten 66 in Trinidad, was nine innings ago.
He was resourceful and conscientious yesterday over just short of three hours in the middle.
New Zealand didn't help their case earlier in the day.
Both openers Tom Latham and Hamish Rutherford, having got set, then got out, one chasing a wide delivery, the other upper-cutting straight to third man.
Ross Taylor lasted seven balls and three wickets had gone for four in 30 balls.
Think cats and pigeons.
Brendon McCullum and Jimmy Neesham fell lbw, both after umpiring referrals after lunch, but Sri Lanka's delight was dimmed slightly in the last two hours as the two Ws knuckled down.
But this pitch is as good as it's been for batting in the match, which must frustrate New Zealand, who could have been shutting the door on Sri Lanka's hopes with a better batting performance yesterday.
"When you bowl right areas and hit the seam most of the time, you can get rewards," Prasad said optimistically last night.
Minds will instead turn today to how much New Zealand must be ahead when the fourth innings starts. The ground record is Pakistan's 277 for three in 2003.
Latham thought 200 could be a tough chase
But if Kumar Sangakkara's eye is still in from Sunday, it'll likely need to be some way north of that.