In 1975, Wachspress conjectured that the adjoint curve of a polycon — a region bounded by conic arcs — never vanishes in the polycon's interior. For polygons (polycons with straight edges), this is true: the adjoint is positive inside. The conjecture extended this to curved boundaries.
Fifty years later, a polycon bounded by three conics disproves it.
The counterexample has elegant internal structure. Replace one of the conic boundaries with a line, and the old polycon's adjoint becomes a contact curve of the new one — a curve that touches the boundary at precisely the right points. The geometric relationship between the counterexample and simpler configurations is not accidental; it is the mechanism that forces the adjoint to change sign.
The failure of the conjecture is specific to curved boundaries. For polygons, the adjoint is a product of linear factors, each associated with a boundary edge, and the positivity follows from the geometry of the polygon. For conic boundaries, the adjoint involves higher-degree terms whose signs depend on the curvature of the boundary, and sufficiently curved boundaries can force a zero inside.
What makes the result sharp is that the counterexample is minimal: three conic arcs. Not some exotic high-degree curve — the simplest possible curved polycon beyond a triangle. The conjecture failed not at the boundary of its applicability but at its foundation. The gap between straight and curved was not a matter of degree but of kind.