A thin elastic sheet that grows non-uniformly develops intrinsic curvature — the metric prescribed by growth may not be realizable as a smooth surface in three-dimensional space. When the prescribed Gaussian curvature is negative, the sheet ruffles. When positive, it forms a dome. The classical incompatibilities (Gauss and Mainardi-Codazzi-Peterson) capture most cases.
A new obstruction appears in positively curved growing sheets that the classical conditions miss (arXiv:2603.21112). When the growth metric generates sufficiently strong positive curvature, a geometric horizon forms — a boundary beyond which no isometric embedding exists. The sheet cannot achieve a stress-free configuration because the shape it needs to assume is geometrically impossible. It's not that the material is too stiff or the constraints too rigid. The target shape literally does not exist in three-dimensional Euclidean space.
The frustrated material responds by forming periodic dimple structures resembling d-cones — point-like concentrations of curvature arranged in a pattern. The pattern is not a material instability but a geometric inevitability: the sheet must concentrate its incompatibility somewhere, and the dimples are the most economical arrangement.
The obstruction has topological origin. It's not a matter of degree — “almost compatible” sheets don't gradually become “slightly stressed.” Either the isometric embedding exists or it doesn't. The transition is discrete.
The structural insight: frustration in elastic sheets is not always about energy. The classical picture — the material wants a shape, constraints prevent it, stress results — assumes the target shape exists. This new incompatibility means the target itself is impossible. The sheet is not prevented from reaching its goal; its goal is incoherent. The stress doesn't arise from obstruction but from impossibility.