Built to Scale

Every model looks good at its chosen size. The real question is whether it survives being resized. Double the community and the surplus doubles. But so does the number of members who need covering. Halve it and the reverse. Does the finish line drift as the community grows or shrinks?

If graduation time ballooned with scale, the model would only describe one lucky size. If it collapsed, small pilots would look impossibly fast and mislead. What we want to see is a flat line. That would be evidence that the mechanism is proportional. It would mean a pilot of a few hundred predicts a city of two thousand.

Below, the model runs (seed 12345) at community sizes from 250 families to 2,000. That is an eightfold range. The curve is the year everyone reaches full coverage.

The line barely tilts. From 250 to 2,000 families, the graduation year moves by well under a year. The community reaches independence in roughly the same time whether it is a large household network or a small town. Bigger builds a proportionally bigger economy, with more businesses and more jobs. But it does not take proportionally longer.

The neutrality is built in. The community's investable capital is surplus per member times the number of members. The coverage that capital has to fund is full spend times the number of members. Both sides carry the same factor of "number of members," so it cancels. What's left is how fast the businesses compound relative to how much coverage each member needs. That has nothing to do with headcount.

Larger communities do get one small structural advantage. With more members, the pooled capital reaches the $100,000 needed to fund a business a little sooner and more smoothly. So the very earliest months are slightly more efficient. That is the small downward tilt you can see. It is a bonus for scale, not a penalty.

Scale-invariance is what lets a pilot mean something. A first club of a few dozen families is not a toy that will behave unrecognizably once it grows. It is a faithful, if smaller, instance of the same engine. Its results carry upward. That is a precondition for building in the real world. Start small, prove it, and grow without the rules changing under you.

The same holds across the model's other randomness. The only random element is which members take on new roles, and when. So the results sit in a razor-tight band across different seeds. The graduation year moves by weeks, not years. No seed fails to graduate everyone. The finish line is a property of the mechanism, not of any lucky run.