Growth stops paying for a school at 150 students. Unless it runs two curricula.

A Jewish day school runs two academic programs in every grade. General studies and Judaic studies, side by side, from pre-K on. Two sets of classes, two sets of teachers. That costs money.

What follows seems obvious. The school carries a burden. Run it the way any expensive institution is run. Hold the line. Control costs. And do not imagine that adding students will fix anything.

That last part is not folk wisdom. Decades of research on school size say it, and for most schools it is true. Past a certain size, growth makes a school more expensive per student. But every school in that research runs one curriculum.

A cost you pay whether or not the seats are full is a cost that rewards filling seats. Double it and you double the reward. The thing that makes these schools expensive is the thing that gives growth somewhere to go.

The first seats forty on one floor. One kitchen line, one section of servers, on the clock from five whether anyone comes or not.

The second seats eighty across two floors and opens both every night. One room cannot do the other's job. Two lines. Two sections. Same lights. On a slow Tuesday it looks like a badly run business.

Now fill them. Each additional guest is nearly free; the staff is already standing there. But the first restaurant runs out of room at forty. Guest forty-one means opening upstairs and hiring a second line.

The second does not run out until eighty. Everywhere in between, the two businesses move in opposite directions. Same street, same food, opposite arithmetic.

That is the shape this story tests. Not that Jewish day schools are cheap. They are not. That growth stops helping them much later.

Everything so far is a claim about level: what a school spends at a given size. It is the only claim the field has ever made about the dual curriculum.

Underneath it sits a second claim about a different thing. Not what a school spends. What happens to that number when the school grows.

Level and slope are independent. Two schools can spend the same per student this year and be heading in opposite directions. One is nearly full. The other has two half-empty tracks and two hundred students of runway.

The field has only ever argued about level. That is the mistake. A level premium tells a school what it costs to be what it is. A slope tells it what to do next.

The level premium is settled. Seventeen percent more per pupil, 2.5 more staff per 100 students, at ordinary class sizes and ordinary seniority. It clears every test by a distance. The curve is a different matter.

Under the method most papers use, the gap between the two curves looks decisive: p = 0.002. Under the randomization test, the same data returns p = 0.23. Random labels produce a gap this big about one draw in five.

That is not a technicality. Robust standard errors overstate certainty when one group is small and a few of its members sit at the extremes. Here the group is 24 schools. Drop the three with the most extreme enrollments and the conventional p-value moves to 0.10. Restrict the range to 150–600 students and the shape difference disappears.

The paper adopts randomization as its standard. Under it, the slope difference is not confirmed in any of the four designs.

What the four designs do is agree. One year of curves, two years pooled, the staffing split, the within-school panel — every one puts the Jewish slope below the comparison slope, and the difference lands in staffing, where the explanation requires it. A second year of data moved the classical p-value from 0.20 to 0.04.

So: a well-supported hypothesis, not a fact. It is a rare one with a price on it. Only about one in three identified Jewish day schools files financial data. Roughly forty more would settle it. The field can buy this answer, and it costs a survey response.

Four limits travel with all of it.