That would seem to raise the possibility that European Union citizens could enter Britain indirectly through Ireland, perhaps undermining control over immigration. But the document hints that European citizens would probably be allowed to enter Britain freely and directly from Europe even after Brexit, though they might face some restrictions on their right to work or to claim welfare payments as people from outside the European Union do today.

The bigger problem is trade. The European Union’s chief Brexit negotiator, Michel Barnier, warned last month that “frictionless trade” would not be possible if the United Kingdom left the bloc’s economic arrangements.

Wednesday’s document calls on the European Union to agree to a series of waivers for small businesses and farmers, to avoid the need for them to complete customs formalities.

But that is only part of the problem. Even if that were agreed, larger companies would surely face higher costs. The British government is hazy on this point, talking about setting up simplified customs procedures and applying technologies — so far unspecified — to track goods, reduce bureaucracy and prevent costly delays.

British officials say there is so far no estimate of the increased cost that some businesses would face. Stephen Martin, the director general of the Institute of Directors, a business lobby group, described the document as a “significant step forward,” while adding that it “throws up even more questions about how much flexibility and imagination will be needed to overcome some very fundamental challenges.”

John Bruton, a former Irish prime minister, said the document failed to address the need for tariffs to be collected by Ireland on some goods imported into the European Union. “Brexit is going to increase the cost of doing business,” he told the BBC.