Last June, Mayor Bill de Blasio and the City Council speaker, Corey Johnson, stood before scores of older New Yorkers outside City Hall to celebrate the news that this year’s capital budget would include $500 million to build new housing for low-income seniors.

Yet, nearly a year later, it turns out that the money wasn’t in the budget, the plan wasn’t new and the senior housing envisioned hasn’t been built or financed.

Mr. Johnson told The Times that he thought the city had committed $500 million to the initiative. But de Blasio administration officials say that it’s all a misunderstanding and that the $500 million was the project’s total cost to build about 1,000 apartments over several years. The city’s share would be $100 million, a de Blasio spokeswoman said, and the rest would be financed by a mix of low-income housing tax credits and Section 8 dollars, from the federal program to subsidize housing for low- and moderate-income Americans.

But that’s not what people were led to believe, and the truth was revealed only after the city comptroller, Scott Stringer, asked Mr. de Blasio in March for an update on the project. In response, the de Blasio administration wrote that the $500 million wasn’t in this year’s budget.