ABERDEEN TOWNSHIP, N.J.—The government's civil-fraud allegation against Goldman Sachs Group Inc. centers on a deal the firm crafted so that hedge-fund king John Paulson could bet on a collapse in U.S. housing prices.

It was a dizzyingly complex transaction, involving 90 bonds and a 65-page deal sheet. But it all boiled down to whether people like Stella Onyeukwu, Gheorghe Bledea and Jack Booket could pay their mortgages.

They...