There were no votes cast, no startling changes in polls, no fatal gaffes. Yet last week brought something new and crucial to the often-chaotic presidential campaign: clarity.

On national security and the fight against radical Islam, assumed differences were flushed into the open between President Obama and the Republican field. For Democrats, clarity comes with the evidence that Hillary Clinton is having serious trouble making the sale.

The week started with a petulant Obama using a State of the Union lecture to come clean about his ­casual view of terrorism. After years of failing to persuade Americans he is serious about combating jihadists, we now know why: He confessed that he’s not all that worried about Islamic State or any other terror group.

He scoffed at “over-the-top claims that this is World War III” and insisted that “masses of fighters on the back of pickup trucks and twisted souls plotting in apartments or garages . . . do not threaten our national existence.”

The too-cool view revealed that he hasn’t moved very far from the time two years ago when he called Islamic State the “JV team.” It was a mistake then, as bloodcurdling events proved, and it is almost pathological that he still downplays the global reach of the group and its goal of launching mass attacks against America and our allies.

By his calculations, we should save our concern until the terror group has the ability to destroy us. That is an extremely foolish approach because waiting until then could be too late, not to mention all the innocent blood that will be shed in the meantime.

Still, at least we have clarity about how he really feels.

Fortunately, Thursday’s GOP ­debate showed without doubt that, despite other fault lines among them, the seven major candidates are united in their conviction that Obama is dead wrong. The contenders took turns blasting Obama’s failure to even acknowledge the role of Islam, and promised to destroy Islamic State instead of merely trying to contain it.

Sen. Marco Rubio provided the most electrifying version of the juxtaposition when he said of the terror army: “The most powerful intelligence agency in the world is going to tell us where they are; the most powerful military in the world is going to destroy them; and if we capture any of them alive, they are getting a one-way ticket to Guantanamo Bay, Cuba, and we are going to find out everything they know.”

That’s a difference with a distinction!

The Fox Business debate also provided clarity on the race two weeks before the Iowa caucus by showing why there are only three real contenders: Donald Trump’s muscular performance reflected his big lead nationally and in New Hampshire, while Sen. Ted Cruz was scrappy and focused in trying to protect his slim margin in Iowa. Only Rubio is consistently within striking distance of both, and is most likely to rise if the Trump-Cruz showdown leaves just one them standing.

The emerging clarity on the Democratic side comes with a twist. The dead heat in Iowa between Hillary Clinton and Sen. Bernie Sanders adds to the belief that Clinton is going backward. She’s behind in New Hampshire, and her team is touting a “firewall” in the South, suggesting they are lowering public expectations and their own goals for the first two tests.

The problems for Clinton are not new, but they are intensifying. The FBI investigation into her handling of classified information reportedly expanded to include whether corporations that contributed to the Clinton Foundation got favors from her State Department.

I’ve always believed that was the mother lode of scandal, with hundreds of millions of dollars ending up in the foundation or in the Clintons’ personal pockets. The amounts are so large, and the timing of some payments so suspicious, that it defies belief that all of them were kosher. The river of cash deserves serious law-enforcement scrutiny.

Coming on top of the widespread doubts about Clinton’s integrity and the greater enthusiasm among Sanders’ supporters, the FBI probe has the potential to fatally wound her. Even the stock-market chaos and economic slowdown work against her because of her long ties to Wall Street, and because of Sanders’ socialist screeds. Although the RealClearPolitics average of national polls has her ahead by nearly 9 points, a Sanders win in both Iowa and New Hampshire would shatter any sense of inevitability.

Indeed, Hillary already is rattled enough that she sent daughter Chelsea out on the stump to level false charges against Sanders’ health-care plans. The move had stunned Dems asking whether the Clinton campaign already is in full panic mode.

The answer is yes, and for good reason. Clarity means everyone can see that she’s in trouble.

Give NY the gift of an honest government

If Gov. Cuomo and legislators are serious about tackling corruption, they could follow the lead of the city’s ethics rules. As The Post reports, seven city firefighters were fined up to $4,000 each for accepting free tickets from the NFL to last year’s Super Bowl. The tickets violated the $50 limit on gifts that employees can take from anyone doing business with the city.

It’s a draconian rule, but if it’s good enough for civil servants, it should be good enough for their political masters. So I propose that Cuomo and lawmakers vote to live within the $50 limit — and also apply it to political donations.

Bye-bye, million-dollar backers. Hello, honest government.

Taxis running a ‘read’ light

The City Council bill that guts English requirements for cabdrivers is an example of lousy government at work.

Instead of showing they can read and write English — traffic signs, for example — cabbies would only need to pass an oral exam.

Fuming riders were wrong to assume that City Hall serves them. In truth, the bill aims to fatten the coffers of cab owners, at the expense of safety and common sense.

Arrogant reign of King Putz

It’s good to be king. Just ask Bill de Blasio.

Reacting to criticism from the principals union about how he is running the schools, The Putz responded with a royal swagger.

“If anyone says they would prefer the previous administration’s model, well, they should have voted for that,” the mayor said. “We have a different vision.”

The sneering smackdown recalls how President Obama rudely silenced Republican critics by declaring that “the election is over.”

Both men are afflicted with the mistaken impression that elections are blank checks and that winning confers unlimited power to do whatever they want, the public be damned.

In fact, political capital has to be carefully nurtured and expanded, so it’s available for emergencies and important decisions. Wasting your cache on unpopular policies and arrogant reactions to legitimate criticism is a recipe for public anger and polarization.

To see the proof, de Blasio only need look at Obama’s low standing among most Americans. Our DNA doesn’t tolerate kings.