Federal prosecutors drew some more important lines between Russia and those connected to President Trump on Friday, in a trio of filings in the Michael Cohen and Paul Manafort cases.

Beginning late Friday afternoon, we saw Cohen sentencing recommendations filed by both the Southern District of New York and special counsel Robert S. Mueller III’s Russia investigation, and a document from Mueller’s team laying out Paul Manafort’s alleged lies to it.

In all three, the plot thickened for Trump just a little bit. Below are the big takeaways.

1. SDNY: Cohen has overstated his cooperation with Mueller

This might be the biggest takeaway when it comes to the SDNY document — and its relevance to Trump. We knew Cohen never technically had a cooperation agreement with SDNY or Mueller, but he made a big public show of looking like he was atoning for his wrongs by telling prosecutors whatever they asked.

It turns out to have been too much of a show for the SDNY team.

“Cohen’s description of those efforts is overstated in some respects and incomplete in others,” prosecutors say. “To be clear: Cohen does not have a cooperation agreement and is not receiving a Section 5K1.1 letter either from this Office or the [special counsel, or SCO], and therefore is not properly described as a ‘cooperating witness,’ as that term is commonly used in this District.”

They add later: “While Cohen’s provision of information to the SCO merits credit, his description of his actions as arising solely from some ‘personal resolve’ — as opposed to arising from the pendency of criminal charges and the desire for leniency — ignores that Cohen first reached out to meet with the SCO at a time when he knew he was under imminent threat of indictment in this District. As such, any suggestion by Cohen that his meetings with law enforcement reflect a selfless and unprompted about-face are overstated.”

The prosecutors wound up requesting only a “modest” reduction in Cohen’s sentence, for which the guidelines recommend 51 to 63 months. (This is separate from the plea deal Cohen struck with Mueller last week — for lying — for which Mueller recommended his sentence run concurrent with his SDNY sentence.)

2. The government has implicated Trump in Cohen’s crimes

We knew from his plea deal, in which he admitted to eight crimes, that Cohen had implicated Trump in campaign finance violations involving the payment to Stormy Daniels. But, as The Post’s Philip Bump details, here the SDNY prosecutors also state Trump’s role in directing the payment as plain fact.

“In particular, and as Cohen himself has now admitted, with respect to both payments, he acted in coordination with and at the direction of Individual-1,” they say (again referring to Trump as “Individual-1,” as the documents last week did).

As Bump writes:

Linking Trump to knowledge of the payment and the payment to the campaign is important. One of the defenses that might have been offered by Trump is that he regularly had his attorney pay off women to keep their stories quiet. The government filing indicates that AMI and Cohen discussed the company helping to make such payoffs as early as 2014. But the references to the rationale behind the payments in 2016 and the inclusion of the phrase “at the direction” of the candidate bolsters the evidence that the McDougal and Daniels payments were not just run-of-the-mill behavior.

Given that Cohen indicated that the payments were meant to influence the election and that they came at the direction of Trump, Lawrence Noble, former general counsel for the Federal Election Commission, told The Post, “there is little question Cohen, the campaign and the candidate are liable for the campaign finance violations.”

3. Cohen was contacted by a Russian national in 2015

One potential clue for the collusion investigation has to do with a contact Cohen received in 2015 from a “Russian national” seeking “synergy” between the campaign and the Russian government.

From Mueller’s document:

The defendant also provided information about attempts by other Russian nationals to reach the campaign. For example, in or around November 2015, Cohen received the contact information for, and spoke with, a Russian national who claimed to be a “trusted person” in the Russian Federation who could offer the campaign “political synergy” and “synergy on a government level.” The defendant recalled that this person repeatedly proposed a meeting between Individual 1 and the President of Russia. The person told Cohen that such a meeting could have a “phenomenal” impact “not only in political but in a business dimension as well,” referring to the Moscow Project, because there is “no bigger warranty in any project than consent of [the President of Russia].” Cohen, however, did not follow up on this invitation.

4. Mueller appears to be keying on Russia ties in Trump’s business

The writing was on the wall for this when Mueller reached that plea deal with Cohen for lying about such matters last week; it was the best explanation for Cohen’s continued pursuit of Trump Tower Moscow being entered into the public record.

But the Mueller document makes clear Cohen has given it information about these matters, information it is interested in and called “useful.”

“Second, Cohen provided the SCO with useful information concerning certain discrete Russia-related matters core to its investigation that he obtained by virtue of his regular contact with [The Trump Organization] executives during the campaign,” Mueller’s team says.

Elsewhere in the document, they elaborate:

The defendant’s false statements obscured the fact that the Moscow Project was a lucrative business opportunity that sought, and likely required, the assistance of the Russian government. If the project was completed, the Company could have received hundreds of millions of dollars from Russian sources in licensing fees and other revenues. The fact that Cohen continued to work on the project and discuss it with Individual 1 well into the campaign was material to the ongoing congressional and SCO investigations, particularly because it occurred at a time of sustained efforts by the Russian government to interfere with the U.S. presidential election. Similarly, it was material that Cohen, during the campaign, had a substantive telephone call about the project with an assistant to the press secretary for the President of Russia.

This is Mueller laying out a rationale for why it would be fair game to look into Trump’s attempt to do business in Russia, and why it matters in the collusion investigation.

5. Manafort’s alleged lies were also Russia-focused — and deal with a big unknown

As soon as we found out last week that Manafort had allegedly lied to Mueller’s team, in violation of his cooperation deal, the question was about what. What was worth lying about for a man whose cooperation was required for the leniency he apparently sought?

Friday’s filing in that case doesn’t shed much light on what Mueller knows, but it is noteworthy how much of Manafort’s allegedly lying pertain to his business colleague in Ukraine, Konstantin Kilimnik, whom the U.S. government has said has ties to Russian intelligence.

Mueller’s team says Manafort lied about a meeting with Kilimnik and also about Kilimnik’s role in “a criminal conspiracy” to get two witnesses against Manafort to alter their testimony.

Much of the document is redacted, but the sheer volume of Kilimnik in it leads to an obvious question: Why was Manafort trying to protect him? And could this link somehow play in the broader collusion probe? Was Manafort really worried about a foreign national being in trouble, or was he worried about Mueller connecting some dots that he didn’t want connected?

Lots more questions. Not a whole lot more answers. But we have a little better idea of what Mueller is interested in, and what might be being covered up.