At lunch on the fourth day, England were 249 for three in their second innings -- a lead of 429 runs after left-handed opener Cook had extended his overnight 88 not out to 130.
The most any side has ever made in the fourth innings to win a Test is West Indies' 418 for seven against Australia in Antigua in 2002/03.
And the corresponding record for New Zealand is the 325 for four they posted against Pakistan in Christchurch in 1993/94.
Cook, still only aged 28 and in his 92nd match at this level, extended his England Test century record, with a cover-drive off Tim Southee, his 15th four in 152 balls.
Together with Jonathan Trott, he put on 134 for the second wicket.
Trott, dropped on 40 when he gave a difficult chance to first slip Ross Taylor off a hard-hit reverse sweep against part-time off-spinner Kane Williamson, was 76 not out at lunch.
First innings century-maker Joe Root was 20 not out on his Yorkshire home ground.
Now it seemed as if only rain, which washed out all of Friday's first day, could spare New Zealand from defeat.
Their cause was not helped today when left-arm seamer Trent Boult, who took five first innings wickets, was declared unfit to bowl after aggravating a side strain yesterday.
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