A fight is heating up around whether the Federal Reserve's war on inflation will result in a big jump in unemployment.

• On one side: Leaders of the Fed, who believe it can drive down demand in the economy without causing too much pain in the job market.
• One the other: Influential economists — including ex-Treasury Secretary Larry Summers — who say the Fed's optimal outcome hasn't happened before, and there's no reason to think it can now.

Why it matters: The fight is playing out in wonky academic papers, but the real world stakes for workers are high. At odds is not whether unemployment rises, but by how much as the Fed tightens.

At the crux of the debate is the inverse relationship between unfilled job openings and the unemployment rate. As job vacancies rise, the unemployment rate falls (and vice versa).

• Where it stands: As of May, job openings drifted lower but remained near their highest levels ever. With unemployment rate holding near a half-century low, the result is roughly two open jobs for each unemployed worker — an unprecedented phenomenon.

What they're saying: "Fighting inflation will require a decrease in vacancies and an increase in unemployment. There is no magic tool," economists Olivier Blanchard, Alex Domash and Larry Summers wrote last month.

• "In all episodes when the vacancy rate came down in a meaningful way, the unemployment rate increased substantially. There was no free lunch, and there is no reason to expect one today, " the authors say.

But Fed governor Christopher Waller and economist Andrew Figura are shooting back in a paper published Friday.

• They say because the job market is so tight, each open vacancy has a lower probability in actually resulting in a hire. So if vacancies fall, there will be a smaller impact on hiring (and in turn, a smaller impact on the unemployment rate) than in a more typical labor market.
• "We recognize that it would be unprecedented for vacancies to decline by a large amount without the economy falling into recession," they wrote. "As a result, we are, in effect, saying that something unprecedented can occur because the labor market is in an unprecedented situation."
• The authors estimate a decline in job vacancies, to 4.6% from 7%, would push up unemployment by 1 percentage point. That would still leave it below 5%, a historically low level consistent with a soft landing. They do concede that any jump in joblessness is still harmful to households.

Yes, but: There are at least two big risks, the authors admit.

• One is that supply shocks persist, and inflation expectations creep higher. That will make it "extremely challenging" for the Fed to bring down inflation, and still have a soft landing.
• The other is that households and firms pull back on spending as the economy slows and begin to layoff workers. This is already happening, though so far it's been concentrated in industries like technology.

When the entire world is on recession watch, quick-turnaround surveys of business activity take on outsized importance for knowing where things stand. The Institute for Supply Management's manufacturing data is one such beast.

• The July index, out this morning, points to an economy that is clearly slowing down, but didn't enter recession last month.

Driving the news: The headline fell to 52.8, from 53 in July. Numbers above 50 indicate expansion, so overall activity was still in growth mode in recent weeks, for all the recession chatter.

Yes, but: There were plenty of warning signs. The new orders gauge — which tends to be forward-looking — fell 1.2 points to 48, putting it in the contraction zone. The production index fell by 1.4 points, though still comfortably in positive territory at 53.5.

What they're saying: “Our markets are still holding up; however, I believe a slowdown is coming," one unnamed respondent told the organization.

• "We are cautious about going out too far with orders. Also, I believe the general market is in the beginnings of a recession.”