Key pieces of special counsel Robert Mueller's Russia investigation appear to be falling into place.

In three court filings Friday, prosecutors for the first time connected President Trump to a crime involving hush money payments to a porn actress. They revealed new details about outreach from Russia early in Mr. Trump's presidential campaign. They detailed how they say two central figures, lawyer Michael Cohen and onetime campaign chairman Paul Manafort, were continually tripped up by lies.

Here are some takeaways from the latest round of court documents from Mueller's investigation:

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Early Russian outreach

Mr. Trump announced his presidential candidacy in June 2015. By that November, the Russians were reaching out about "political synergy," according to the sentencing memo for Cohen.

The court papers provide new details about one of the earliest known contacts between Russia and a Trump campaign associate. In fall 2015, Cohen was months into his work on a proposed Trump Tower in Moscow when an unidentified Russian national proposed a meeting between Mr. Trump and Russian President Vladimir Putin. This person, prosecutors say, claimed to be a "trusted person" in Russia who could offer the Trump campaign "political synergy" and "synergy on a government level."

This person sought to connect the Trump business project with the campaign, saying the meeting could have a "phenomenal" impact on the proposed tower in Moscow. There is "no bigger warranty in any project than the consent of" Putin, the person told Cohen.

Prosecutors said Cohen didn't follow up and that the meeting never occurred.

The memo also discussed Cohen's lies to Congress about the extent of the Trump Tower negotiations, which the special counsel described as a "lucrative business opportunity that sought, and likely required, the assistance of the Russian government."

The outreach is more evidence that Russia was eager to build relationships with the campaign and tried to use Mr. Trump's business as an opening.

Mr. Trump directed Cohen's crime

Prosecutors didn't mince words: The campaign finance violations Cohen committed came "in coordination with and at the direction of" Mr. Trump, according to the new filings.

Those violations stemmed from payments Cohen made to buy the silence of porn actress Stormy Daniels and former Playboy model Karen McDougal. Both women alleged they had affairs with Mr. Trump, which the White House denies. Daniels was paid $130,000 as part of a nondisclosure agreement signed days before the 2016 election.

Mr. Trump has denied knowing anything about the Daniels payment. But the filing directly contradicts that claim. It also, for the first time, directly ties Mr. Trump to a federal crime. Campaign finance law requires candidates to report any payments made to influence the election. The Trump campaign failed to report the payment at the time.

Prosecutors don't say Mr. Trump broke the law and the Justice Department has maintained that a sitting president cannot be indicted.

Public statements matter to Mueller

At least such statements did matter to Mueller when Cohen lied to Congress, and that could have implications for other episodes under investigation in the Russia investigation.

Cohen has admitted lying to Congress about how long he worked on the Trump Tower Moscow project and repeating the falsehoods to the media. But Mueller's team doesn't just consider this self-protection. It was a "deliberate effort" to publicly present a "false narrative" in the hopes of limiting the scope of the various Russia investigations, prosecutors say in the court papers.

Mueller's focus on public assertions and their impact on witnesses, lawmakers and ongoing investigations could serve as a warning to Mr. Trump.

The president also has spread falsehoods about his campaign's ties to Russia. The special counsel has questioned witnesses about a statement Mr. Trump dictated on Air Force One last year that omitted several details about a June 2016 Trump Tower meeting with a Russian attorney.

The filing suggests Mueller intends to hold witnesses accountable for the statements made privately and publicly. If the lies are meant to influence the investigation, they may factor into Mueller's investigation into whether Mr. Trump has tried to obstruct the probe.

Trump administration contacts

Despite their criminal cases, the Trump administration just can't leave Manafort or Cohen behind, according to prosecutors.

In Cohen's case, Mueller's team said he has provided "relevant and useful" information about his contacts with people connected to the Trump White House in 2017 and 2018. With Manafort, prosecutors say he also had several recent administration contacts and lied about them.

After Manafort pleaded guilty and agreed to cooperate with the government, prosecutors say he told them he had "no direct or indirect" contact with people in the administration. But that was a lie, they say.

Instead, they found evidence, including electronic documents, showing contacts with multiple administration officials. That included communication with a "senior administration official" through February 2018. Manafort also directed a person to speak with an administration official on his "behalf" on May 26.

Neither Cohen's nor Manafort's filings detail the content of the conversations or identify the officials. Manafort has contended he was truthful with Mueller's team.

Not just Cohen's word

Since his guilty plea, Mr. Trump has attacked Cohen as a liar who is telling "stories" to get a reduced prison sentence. But prosecutors revealed Friday that they're not just taking Cohen's word for it.

The information Cohen told prosecutors in seven separate interviews "has been credible and consistent with other evidence obtained" in Mueller's investigation, they note in the sentencing recommendation.

The special counsel's filing said that Cohen has "gone to significant lengths to assist the Special Counsel's investigation. He has met with the [special counsel's office] on seven occasions, voluntarily provided the SCO with information about his own conduct and that of others on core topics under investigation by the SCO, and committed to continuing to assist the SCO's investigation."

Some of that information from Cohen, prosecutors say, concerns "certain discrete Russia-related matters" at the "core" of Mueller's investigation, particularly those involving his contact with Trump Organization executives.

Mr. Trump said in a tweet on Friday evening that the filing "totally clears the president." In a tweet on Saturday morning, Mr. Trump reiterated that there was "NO COLLUSION."

'Lucrative' Moscow deal

Mr. Trump and his lawyers have played down the Trump Tower Moscow proposal. The president has said he never put any money into it and ultimately decided not to do it. But Mueller's team reveals that if he did, they believe they know the windfall.

According to Cohen's filing, the deal could have yielded "hundreds of millions of dollars from Russian sources in licensing fees and other revenues."