Theory 04/Trade-off geometry/Complex systems
The Redundancy Frontier Is Convex
The frontier between efficiency and redundancy is convex rather than linear, which means systems cannot buy partial redundancy cheaply — and that collapse thresholds are set by the curvature rather than by the mean position on the frontier.
01What is already known
Efficiency-resilience trade-offs are widely acknowledged but usually treated as straight lines: some redundancy costs proportionally some efficiency. Network theory, ecology and engineering all find non-linear behaviour near collapse, but the geometry of the trade-off itself is rarely measured.
02The mechanism
Redundancy at low levels is cheap because it can be shared — spare capacity in one component covers several functions. As redundancy rises, marginal cost rises because the remaining functions require dedicated spares. The result is a frontier that is flat at the efficient end and steep at the redundant end, or the reverse depending on architecture. Because curvature determines how much of the frontier is accessible at a given budget, it determines where systems sit before a shock.
03Predictions
- Measured efficiency-redundancy frontiers should be significantly curved in both engineered and ecological systems.
- The curvature should predict collapse probability better than the system's position on the frontier.
- Systems that appear to sit mid-frontier should be rarer than uniform sampling would suggest, because the middle is the expensive region.
04Refutation
Frontiers that are linear within measurement error across several domains would make the curvature claim unnecessary.
Measure both axes across many systems of the same class — data centres, supply networks, food webs — and fit the frontier. Curvature is a property of the fitted shape, not of any single system.