The project shows both what is possible for such housing to achieve and how difficult is de Blasio's task.

London-based David Adjaye, who worked with local housing specialist SLCE, buried about a third of the building in the steeply sloping site. A band of floor-to-ceiling glass runs around the building on the entry level, contrasting with the visual weight of the 11 floors of apartments above. An intimate entry plaza will be animated with children, visitors, and residents when all of the project's components are occupied, since entrances to the housing, school, and museum all open onto it. Kids started playing on a terrace below when school began in September.

The education spaces, bathed in daylight, take advantage of east-facing views of the Harlem River and nearby Yankee Stadium. The glass along 155th Street puts the children on display above passersby as the street drops off, a feature supported by Ellen Baxter, the director of the nonprofit developer, Broadway Housing Communities, as a cheerful advertisement of the value of prekindergarten education.

The lobby of the museum (scheduled to open in the spring of 2015) draws visitors past a small sculpture courtyard, then descends one level to a landing where a window looks into a skylit gathering space below. The corridor divides generous, high-ceilinged display space from a suite of classrooms that borrow sun from the skylight.

Mixing the mutually reinforcing early- education center and the storytelling museum “helps children succeed academically from the get-go” says Baxter. “These are resources not available in low-income communities like this one, where 70 percent of children are born into poverty.”

Inside, the architects brought touches of character and domesticity to small but functional apartments that range from studio to three-bedroom. Hardwood floors in the dwellings and stone tiles in the residential hallways survived value engineering. Residents, who are starting to move in this month, will share a planned rooftop terrace and garden. Outdated energy standards required windows that are smaller and less varied than those shown in Adjaye's renderings, affecting both the units and the building's exterior, where they advertise Sugar Hill's budget constraints and institutional character. One regulator insisted on unsightly, energy-wasting through-wall air-conditioning units—a crude way to mask the noise of 155th Street and less efficient than a central air system.

Eschewing the pink and beige bricks that usually clad affordable housing, Adjaye called for precast panels tinted charcoal gray with a striated texture and the embossed roses. The pattern is most visible in raking light, and almost disappears on dull days.

With a broken funding process endemic to “affordable” housing, Sugar Hill's benefits will be difficult to replicate at the scale de Blasio hopes to achieve. A gift from the Sirius Fund, a foundation, allowed Broadway Housing Communities to acquire the site. Sugar Hill needed 14 other funding sources from federal, state, and local governments as well as several banks deploying Federal New Market tax credits. Each contributor operates with different criteria and on different cycles, requiring an elaborate and expensive syndication process to reconcile. The education and museum programs also added to construction, financing, and regulatory complexity. Some nine city agencies will have signed off on the project before it can fully operate.

This Byzantine process is one reason why the total project cost for the 191,000 square-foot building was $89 million, only $59 million of it for construction..

The project's 124 apartments attracted a staggering 48,000 applications and Baxter's team has been filtering applicants according to exacting requirements: 40 income tiers (as low as $18,000 for a family of four, who would pay as little as $460 per month), with apartments set aside for nearby residents, city employees, the formerly homeless, and people with disabilities.

Adjaye says he likes the ambiguity of the brooding exterior, but the clumsy workmanship and irregular gaps between panels unfortunately evoke the stigmatizing low-income housing that blights too much of New York. “I wanted to see a different palette,” says Adjaye, who noted that one of his earlier projects, the Dirty House in London, had been criticized for its contrast to its surroundings, but is now embraced. “I believe the project will succeed in its use and what it offers—a very low-cost but signature building.”

James S. Russell is a New York-based architecture critic and the author of The Agile City, published in 2011.