Generic order-barrier condition for three or more patterns

Prove that for modern Hopfield retrieval dynamics with three or more stored patterns, there generically exists a point x_0 at which Df(x_0)f(x_0) does not belong to the span of f(x_0), thereby establishing the stated order barrier for all scalar reparametrizations of the relaxed attention family.

Background

The paper proves that if there is a point x_0 with Df(x_0)f(x_0) outside the one-dimensional span of f(x_0), then no scalar parameter map theta(Delta t) can make the relaxed family second-order consistent. It establishes this nondegeneracy for two linearly independent patterns, but leaves the corresponding genericity claim for N >= 3 patterns unresolved.

References

The hypothesis holds for $N = 2$ whenever the patterns are linearly independent (Lemma~\ref{lem:genericity}); for $N \ge 3$ we expect it to hold generically but do not prove it.

— Basin-Preserving Discretizations of Modern Hopfield Retrieval Dynamics: Energy Cells, Dissipation, and the Attention Limit  (2608.21304 - Villatoro, 21 Aug 2026) in Proposition 4.2, Section 4.2; Appendix A

The open problem left by this section is therefore not existence but combination; a scheme achieving simultaneously second order, exact dissipation of $E$ for every step size, a monotone interpolant, linear implicitness, and one softmax evaluation per step.

— Basin-Preserving Discretizations of Modern Hopfield Retrieval Dynamics: Energy Cells, Dissipation, and the Attention Limit  (2608.21304 - Villatoro, 21 Aug 2026) in Section 4.3, paragraph following Theorem 4.4; reiterated in Section 7