The first full day of the cease-fire passed on Tuesday with no notable violations, but an atmosphere of deep mistrust prevailed in Syria’s war-ravaged areas, residents and monitoring groups reported. The skepticism was fed partly by what appeared to be delays in expediting United Nations aid deliveries to the northern city of Aleppo and other combat zones where civilians have long been deprived of food and medicine. Relief for these areas is an important component of the cease-fire agreement.  The United Nations mediator in the conflict, Staffan de Mistura, told reporters in Geneva that there had been a “significant drop in violence” since the cease-fire plan took effect at sundown on Monday. But he said United Nations relief trucks bound for Aleppo, lined up at southern Turkey’s border with Syria, had not received assurances of safe travel. Mr. de Mistura also said the Syrian government had still not provided required authorizations for deliveries to other locations, “but we are eagerly hoping and expecting the government to issue them very soon.”  The divide between Mr. Kerry and Mr. Carter reflects the inherent conflict in Mr. Obama’s Syria policy. The president has come under increased fire politically for his refusal to intervene more forcefully in the five-year civil war, which the United Nations says has killed more than 400,000 people, displaced more than six million and led to a refugee crisis in Europe. But keeping large numbers of American ground forces out of Syria has also created space for Russia to assume a greater role there, both on the battlefield and at the negotiating table.  The result is that at a time when the United States and Russia are at their most combative posture since the end of the Cold War, the American military is suddenly being told that it may, in a week, have to start sharing intelligence with one of its biggest adversaries to jointly target Islamic State and Nusra Front forces in Syria.  “I remain skeptical about anything to do with the Russians,” Gen. Philip M. Breedlove, who recently stepped down as NATO’s supreme allied commander, said Monday in an interview. “There are a lot of concerns about putting out there where our folks are.”  In an email, Peter Cook, the Pentagon press secretary, said: “As Secretary Carter has said, Secretary Kerry has worked tirelessly to try and ease the suffering of the Syrian people and bring about a diplomatic resolution to the Syrian civil war. If fully implemented, this arrangement could advance those important goals.  The Defense Department, he added, “will carry out its responsibilities, but it’s really up to Russia and the regime to comply with the terms and finally start doing the right thing in Syria.”  Chief among Pentagon concerns is whether sharing targeting information with Russia could reveal how the United States uses intelligence to conduct airstrikes, not just in Syria but in other places, which Moscow could then use for its own advantage in the growing confrontations undersea and in the air around the Baltics and Europe.  But to Mr. Kerry’s inner team of advisers, the Pentagon approach was reflexive Cold War-era thinking. For all the other tensions with Russia, they believed that the Russians themselves did not want to get mired in Syria and would cooperate up to a point.  The two countries have worked together in a limited way in the past, although that cooperation has been more at the negotiating table than on the battlefield. American and Russian diplomats worked together toward the Iran nuclear deal. They also sought, unsuccessfully, to reach a lasting agreement to denuclearize the Korean Peninsula, but North Korea has since tested a number of nuclear weapons.  When Russia entered the Syrian conflict last year to support Mr. Assad against the Islamic State but also against American-backed rebels, the two countries agreed to have talks to de-conflict their military activity. But those talks have been limited, focusing primarily on making sure that Russian and American planes do not get in each other’s way in the skies over Syria. A Pentagon official said that by and large, the “de-confliction” talks had worked out, with some exceptions.  Meanwhile, the two militaries have adopted an increasingly combative stance toward each other in other places.  Russia’s invasion of Crimea and the ensuing fight over eastern Ukraine led the Obama administration to substantially increase the deployment of heavy weapons, armored vehicles and other equipment to NATO countries in Central and Eastern Europe, a move that administration officials said was aimed at deterring Russia. This year, Russian planes have taken to buzzing American warships in the Baltic Sea. And American intelligence officials said in July that they believed the Russian government was behind the theft of emails and documents from the Democratic National Committee.  “From a Pentagon perspective, the U.S. military is the one that, around the world, is on the receiving end of Russia’s military misbehavior,” said Derek Chollet, a former assistant secretary of defense in the Obama administration. “There is just generally a lack of trust.”  It is a measure of the sensitivity of the agreement — for Washington, for Moscow and for an array of other countries and opposition groups — that the State Department has not released text of the agreement with the Russians, or even a fact sheet summarizing it.  Reporting was contributed by Eric Schmitt and Julie Hirschfeld Davis from Washington; Anne Barnard from Beirut, Lebanon; Nick Cumming-Bruce from Geneva; and Rick Gladstone from New York.  Follow The New York Times’s politics and Washington coverage on Facebook and Twitter, and sign up for the First Draft politics newsletter.  A version of this article appears in print on September 14, 2016, on page A1 of the New York edition with the headline: Syria Agreement Widens U.S. Rift. Order Reprints| Today's Paper|Subscribe
Defense Secretary Ashton Carter opposed the plan for U.S. and Russian military collaboration against ISIS, which Secretary of State John Kerry supports.