Nonzero Euler minors in smooth generic-plane families. The order is exactly d²−1, and the smooth-point argument constructs the geometric irreducibility needed by the characteristic-zero closed-form theorem.

theorem CCDN.exists_positioned_genericPlane_euler_minor_charZero {K : Type u_1} {σ : Type u_2} [Field K] [CharZero K] [IsAlgClosed K] [Finite σ] (k : ℕ) (f : MvPolynomial (Option σ) K) (hf : Irreducible f) (hd : f.totalDegree = k + 2) : ∃ (a : Option σ → K) (b : Option σ → K) (hb : b none ≠ 0), (MvPolynomial.eval b) ((MvPolynomial.homogeneousComponent f.totalDegree) f) ≠ 0 ∧ (eulerClosedFormCoefficientMatrix k (genericPlanePolynomial (normalizedHypersurfaceCoordinates a b hb f))).HasNonzeroMinor ((k + 2) ^ 2 - 1)

Over an algebraically closed field of characteristic zero, every irreducible polynomial f of degree k + 2 admits an affine coordinate change, given by a and b with b none ≠ 0 and the top-degree form of f nonvanishing at b, after which the Euler closed-form coefficient matrix of the generic plane section has a nonzero minor of order (k + 2) ^ 2 - 1.