Our hard-left governor just proved that he can face the hard math after all, presenting an austere budget that increases spending by less than 2 percent, and imposes no new taxes on anyone earning less than $1 million a year.

This is a turning point for Murphy, one that gives him a chance to repair his poisonous relationship with Democratic leaders in the Legislature, who insisted on precisely this kind of spending restraint.

“I want to congratulate the governor,” Senate President Steve Sweeney, D-Gloucester, said after the budget speech. “This is a very good step in my mind. We are all very encouraged.”

No doubt, Murphy and legislative leaders will bang heads in the coming months, with the biggest fight brewing over Murphy’s plan to increase taxes on incomes over $1 million.

But when you look at the numbers, it’s clear that this budget represents a clean break for Murphy. The spending increase is one-quarter the size of last year’s, and is lower than the rate of inflation. The tax hikes are about one-third the size of last year’s and would not touch the middle-class.

And this time, Murphy was smarter on the small things as well. Last year, he slashed items sponsored by individual legislators, like charity care for their hometown hospitals, and this year he left them in place. That includes a $20 million allotment for state workers who care for the disabled, an emotional touchstone for Sweeney, whose daughter is disabled.

And he told legislators, in private meetings before the speech, that he was ready to compromise this time. “When we were briefed yesterday, we saw a different tone,” said Assemblywoman Eliana Pintor Marin, D-Essex, chairwoman of the budget committee.

What accounts for it? For one, Murphy learned from his bruising defeat last year, when the legislature chewed up his budget plan and forced him to sign one they redesigned from top to bottom. And the governor has a new chief of staff, George Helmy, who made it is his mission to improve ties.

But the irony is that Murphy owes a good deal to Sweeney, his archrival. Sweeney has been holding town halls across the state calling for spending cuts, especially in pension and health benefits for public workers, where the real money is. He’s building a veto-proof majority in the Legislature to force those changes down Murphy’s throat, if necessary.

Murphy’s view is that changes should be negotiated with the unions, not forced on them by law, as was done under former Gov. Chris Christie. And Murphy’s approach worked pretty well, saving taxpayers $800 million in this budget — not by forcing more costs on workers, but by smarter shopping in the health care market.

“Negotiations and progress flow from mutual trust,” Murphy said in his speech. “They go nowhere when there is only mutual suspicion. That’s the old way. That’s the failed way. And, those days are over.”

Okay, maybe there was an explosion of love at those bargaining tables. But my guess is that Murphy pressed hard, and the unions agreed to make changes, because they both knew Sweeney was standing in the hallway with a sledgehammer in his meaty hands.

Sweeney was the bad cop. Murphy was the good cop. It worked.

Despite the mushy talk yesterday, though, you can count on more fisticuffs soon, starting with the millionaire’s tax. Murphy’s plan is raise the rate on incomes over $1 million to 10.75 percent, a rate that now applies only to incomes over $5 million.

Sweeney and Assembly Speaker Craig Coughlin, D-Middlesex, agree with Republicans and business leaders who warn that the higher rate will drive wealthy people from the state.

“No way on the millionaire’s tax,” said Tom Bracken, head of the New Jersey Chamber of Commerce. “If we want to fix this economy, that can’t happen.”

It’s a belief based on religion more than science. It suggests that a grandly successful person would uproot his family, leave his friends, and sell his house – all to escape paying two pennies more on each dollar earned over $1 million. I’ll believe it when they can present data to prove it, which they have been unable to do.

Sweeney and Coughlin worry that voters might hear the words “tax increase” and stop listening. But if that’s true, why do large majorities consistently tell pollsters they like the idea? Could it be that times have changed, and most people are ready to squeeze the rich a bit?

We’ll also see a big fight over pension costs, which are scheduled to rise by the ruinous sum of $3.5 billion over the next four years, thanks to decades of neglect. Sweeney is pressing a modest reform to scale that back, one that would affect only new hires and public workers with less than five years of experience. Murphy has resisted that, for reasons unknown, but Sweeney said yesterday he will move a bill to get it done.

New Jersey Transit is another flashpoint. Murphy mentioned it in his speech over and over, as if he were rushing to the rescue. Don’t believe it. His budget increases funding by only $25 million, or just over 1 percent of NJ Transit’s operating budget this year. So, for the torture victims who ride these creaky trains and buses, you have my deep sympathies.

Still, it seems our governor is getting the hang of his job. He saw his recent poll numbers, and the sharp rise in people who disapprove of his performance. He saw Sweeney, coming at him. He knew he needed to change course. And with this budget, he did just that.

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Tom Moran may be reached at tmoran@starledger.com or call (973) 836-4909. Follow him on Twitter @tomamoran. Find NJ.com Opinion on Facebook.