The non-transverse intersection can't be destroyed.
Symplectic blenders near whiskered tori (arXiv:2603.20830): for any smooth symplectic diffeomorphism possessing a one-dimensional whiskered torus with a homoclinic orbit, arbitrarily small perturbations generate blenders — hyperbolic structures whose stable and unstable sets project onto center subspaces with greater topological dimension than the original sets.
The consequence for homoclinic orbits: non-transverse homoclinic intersections at saddle-center periodic points are persistent. The original system lives in the closure of an open set where systems exhibiting these homoclinic features are dense. You can't perturb your way out of them.
This inverts the standard picture. Non-transverse intersections are generically unstable — the slightest perturbation breaks them. But the blender mechanism creates a different kind of stability: the intersection itself may break, but nearby intersections immediately appear. The set of systems with the feature is dense in a neighborhood of the original. The crossing doesn't persist pointwise but it persists setwise — there's always a crossing nearby.
The structural principle: some fragile features are robust when measured correctly. The individual orbit is unstable. The class of orbits is stable. The blender acts as a topological amplifier — it stretches the invariant sets until they unavoidably intersect, even when the specific intersection is destroyed. Fragility at the object level; robustness at the class level. The wrong scale of measurement sees instability; the right scale sees persistence.