Number 04/Failure & fragility/Trade-offs
The Frontier Curvature
The quantity
The second derivative of the efficiency-redundancy cost frontier, normalised by its slope. It measures how sharply the marginal price of redundancy rises.
01Why it matters
Curvature, not position, determines how much of the frontier is accessible at a given budget, and therefore where collapse thresholds sit.
02Expected value
Curvature should be significantly positive across engineered and ecological systems, and should predict collapse better than position on the frontier.
03How it would be measured
Fit the frontier across many systems of one class and take the normalised second derivative.
04What would refute it
Linear frontiers would make the number zero and the claim empty.