Jackie Calmes / New York Times News Service

WASHINGTON — Seventeen months after President Barack Obama signed doomsday budget legislation that was never intended to become law, the sweeping spending reductions in the measure have been imposed.

Late Friday night, Obama formally triggered spending cuts that will reach across the breadth of the federal government after he failed to persuade congressional Republicans to replace them with a mix of cuts and tax increases.

Lost in the talk of Washington’s dysfunction is this fact: On paper at least, President Barack Obama and Congress have reduced projected deficits by nearly $4 trillion over a decade — the widely embraced goal for stabilizing the national debt.

The spending cuts, which total about $1 trillion through 2023, come on top of $1.5 trillion in reductions that Obama and Congress committed to in 2011, mainly from the accord that averted the nation’s first debt default.

Nearly $700 billion more will come from tax increases on wealthy Americans, the product of the brawl in December over Bush-era tax cuts, and another $700 billion is expected to be saved in projected interest on the reduced debt.

If the latest cuts stick, the two parties will have achieved nearly the full amount of deficit reduction over the next decade that economists and market analysts have promoted.

Yet the mix comes with substantial downsides.

It does not add up to the “grand bargain” that the two parties had been seeking, because it leaves virtually untouched the entitlement programs — Medicare, Medicaid and Social Security — that are responsible for projections of an unsustainably rising federal debt in coming decades.

“This is not a result that deals with our long-term debt problem,” said Vin Weber, a Republican former congressman. “The fact we’ve gotten to a $4 trillion deficit-reduction deal without tackling entitlements is almost a bad thing,” he added, if it lulls the public and the politicians into thinking the problem is solved.

Private and government forecasters project that sequestration alone will cost about 700,000 jobs this year and will shave at least a half percentage point from economic growth.

Letters began to go out to governors Saturday, informing them of the smaller grants. And the Air Force Thunderbirds — the elite team of F-16 pilots who perform tricks at air shows — announced on its website that all of its shows have been canceled starting April 1.

Still, it will take some time before the cuts begin to make life more difficult for teachers, defense contractors, Head Start students, border patrol agents or others who rely on federal funding.