The delivery stymied England's flow. They scored three runs in the next five overs before Gary Ballance (29) settled.
Cook received a standing ovation for his feat, as did Yorkshireman Lyth reaching his maiden half and full centuries on his home ground in his third test innings before being run out by Trent Boult for 107.
The pair demonstrated finesse. How Cook achieved such an acute angle with his leg glances would have made a good question in an NCEA maths exam. In contrast, a pull shot to the boundary off Matt Henry showed his prudence wasn't based on timidity. Lyth used the pace of the ball well behind square but was also circumspect in the front 180 degrees, steering rather than engaging in full flourishes. It was an archetypal opener's innings.
England built their total at a slower tempo than New Zealand's 'fast game is a good game' mindset. Boult, Southee and Henry delivered tidy spells but the England openers got behind the ball, played methodically and dealt with all manner of attrition.
The second new ball brought a change in the balance of power with the dismissal of two locals. Boult bowled Ballance and Southee had Joe Root caught behind by Luke Ronchi for one, his maiden wicketkeeping catch in tests.
As a steward in the corridor said in the aftermath: "Well, that's buggered t' crowd for tomorrow, 'addinit."
Craig was the adhesive that bound the New Zealand display together.
He anchored the batting counterattack with 41 not out, accentuating why he offers value at No.8. It was his fifth time getting beyond 40 and his fifth time not out in 15 test innings.
With the ball, he had an over before lunch, before bowling with control during the remaining sessions to finish with figures of 22-10-38-1, a marked contrast to his expensive spells at Lord's. He was helped by McCullum's bespoke field against the left-handers with a slip, leg slip, backward point, deep cover point, short cover, wide mid-off and mid-on, mid-wicket and square leg.
The dryness of the surface suggests it will continue to erode. Footmarks could become crucial to Craig's chances of bowling England out.
The spinner also held a sharp second slip catch to remove Ben Stokes with 11 balls remaining in the day.
"The runs this morning were crucial, then with the ball it was nice to do a job after being disappointed with the way I bowled at Lord's. I knew I could bowl better and it was just a matter of relaxing. I slipped into rhythm quicker and bowled a bit fuller whereas at Lord's I was a bit 'two lengths' and started getting hit."
Boult has the best figures of two for 63 from 23 overs.
Earlier, a 69-run haul for the last two wickets rejuvenated New Zealand's first innings. The visitors accelerated their total to 350, adding 53 runs in 50 balls to the overnight score.
Henry, with 27 off 21 balls, and Boult, 15 off 11, gave themselves room and took advantage of some errant England bowling.
As a couple of media box sages noted, the latter part of the New Zealand innings may as well have been played without stumps because they did not appear to be a target area and it was as if yorkers had been made illegal.
Stuart Broad had the best England figures of five for 109 but they came at 6.34 runs per over. The overall run rate of 4.84 saw the innings played at close to a limited overs' cadence.
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