New York ? If there is one thing we should have learned around here lately, it is that coaches and managers perform most effectively when they are down to their last paychecks and their backs are pushed squarely against the dugout or locker room wall.

The myth that a coach cannot command respect from players without an extra season or two of job security just doesn’t hold water, or an ounce of Gatorade. The longer the pacts, the quicker and nastier they are likely to end. For proof, see Larry Brown v. James Dolan et al.

Which is why the single most unfortunate word bandied about Monday at the Knicks’ practice site in Greenburgh was “multiyear.” Dolan would not say how long he had given Isiah Thomas on his extension, even though Cablevision is a publicly traded company and it might be in the interest of stockholders to know details of the pact. We don’t even know for sure if Thomas was working on the final year of his contract, only that he had been given a mandate to turn things around or exit through the back door.

Now, though, Thomas has at least two more years, which is exactly the wrong message to send to the coach, to the players and to the fans. Thomas surely earned and deserved one more year, another season on the brink. The Knicks are just as clearly a work in progress that will need to be judged again at the close of this season and then, more importantly, at the close of the next.

This can still go either way for the Knicks. They may continue to improve, earn the seventh playoff spot and give LeBron James a frenetic run in April. Or they may plateau, miss the postseason and then come back with another 38-victory year, not nearly enough. It was not a good sign Monday when the team announced David Lee was seeking a second opinion on his ankle injury, another reminder of how things can go wrong.

Thomas is fully aware of all potential pitfalls. “There’s a lot of work that still has to be done,” he told reporters.

The problem is that Dolan’s message Monday was quite the opposite, and typically hasty. Dolan was essentially telling Thomas the job is done, rewarding him in kind with that most precious commodity – indefinite time.

Dolan read at times to beat writers from a prepared script – with his favorite words “significant” and “evident” highlighted in blue marker. He spoke of all the progress he saw in a 29-34 team that is fighting for a playoff spot in the worst conference ever known to basketball.

And Dolan was right this time, to some extent. Thomas has done a good job this season, much better than what Brown managed. The players love Thomas. They play hard for him, against the best teams and then in games after tough defeats. They should. Thomas brought them here and pays them well.

“Hugs and kisses,” Eddy Curry said about the prevailing mood in the locker room. “He had a hand in bringing us all in. It was our duty (to get him a new contract).”

Thomas says he is more concerned with his “basketball legacy” than with job security, and he sounds sincere. So far, he gets a C-minus for his work in the front office, a B-plus for his job as coach. His GPA adds up to a one-year extension, not an honorary Glen Sather lifetime position.

“I didn’t want to have a failure here,” Thomas said. “Taking the job, you worry that if you can’t get it done in New York, it’s not going to be a pretty picture. I didn’t know how well the team would respond to my coaching style . . . So far, it’s been a match.”

There is a famous logic problem, in which a prisoner is told he will be executed in one of the next seven days, and that his death will be a surprise to him. The prisoner tells his executioner that he can never be surprised. If his execution is held until the final day, he will know it is on that day. If it is held to the second-to-last day, he will know it is then, because it cannot wait until the last day. And so on.

Coaches also can never be surprised. What good does it do to give Tom Coughlin or Joe Torre or Thomas an extra year, if everybody understands the second-to-last season really represents a last chance, anyway? It is wasted money.

Dolan is the very best in New York at throwing dollars around. He threw some around Monday, far more than were necessary. Thomas was sent the wrong message: Job well-done.

It isn’t even half-done.