EA://INTEL — When the Fix Breaks Convexity

We often imagine that fixing a corrupted system means projecting it back onto its valid state space to minimize the distance traveled. The intuition works for Euclidean geometry where the projection is unique, but it fails when the region of allowed states is non-convex. In our work on the Cure’s repair operator, we ran into exactly this problem: the set of admissible meanings does not form a convex shape, so a single shortest path to correctness might not exist.

This specific claim, CURE-REPAIR-DETERMINISM, was withdrawn because it assumed a metric distance formulation that requires convexity for uniqueness guarantees. The project itself noted that projection onto a non-convex admissible region is multivalued, meaning the repair operator can return multiple distinct outcomes depending on which “valley” you drop into. We do not hide this failure; we treat it as input for redesign rather than a verdict that nothing works.

The current path forward leans on non-metric formalizations of minimal change, utilizing preorders instead of distance metrics to handle order-theoretic constraints without needing the geometry to bend smoothly. This allows us to revisit the repair function with new theoretical footing where rational revision from a faithful preorder can yield results without invoking Hilbert’s theorem in its original form.

If you are wondering whether Universal Language implies a fixed number of primitives or sorts as some earlier iterations suggested, that question was retired because the known sets behave as generating sets rather than bases, making cardinality a property of the presentation, not the theory itself.

What would it take to construct a repair operator that remains deterministic even when the admissible region is demonstrably non-convex?