The reducedness conjecture for Finklerberg-Mirkovic Schubert schemes held across every case anyone checked. The geometric objects — living in the affine Grassmannian of simply-connected simple algebraic groups — appeared always to be reduced, meaning their structure sheaves contained no nilpotent elements. No hidden fuzz, no infinitesimal thickening. Clean geometry.
Besson, Hong, and Yu killed it with E8.
By computing tangent spaces of these Schubert schemes for the quasi-minuscule coweight, they exhibited a non-reduced scheme specifically when the group is of type E8 — the largest exceptional simple Lie group. Every smaller type conceals the pathology. The counterexample lives exclusively at the algebraic maximum of complexity.
This is not coincidence. E8's 248-dimensional Lie algebra provides enough room for nilpotent directions that smaller groups cannot accommodate. The conjecture survived because the test cases were too small. The geometry of A-type, B-type, even E6 and E7 — all of these are “too simple” to exhibit the failure. Only at the apex does the structure crack.
The pattern is older than this conjecture. Exceptional objects in mathematics are not decorative curiosities appended to classification theorems. They are where universally believed properties go to die. The octonions break associativity. The Monster group breaks intuitions about size. E8 breaks reducedness.
What makes the result sharp is the quasi-minuscule restriction. These are not arbitrary Schubert varieties — they are the geometrically simplest nontrivial case. The counterexample does not require exotic inputs. It requires only that the ambient group be large enough for the geometry to express what it has been hiding.
Conjectures that survive all small cases are not robust. They are untested.