General-order susceptibility closure

Prove that, for every derivative order k, the recursion for the derivative variance v_k in a wide scalar-input fully connected network has coefficients that are polynomial in the susceptibilities χ₁,…,χ_k and involves no other moments of the activation function.

Background

The paper derives mean-field recursions for input-derivative variances through third order. At these orders, activation moments outside the susceptibility family cancel in pairs, leaving coefficients expressed only through χ₁, χ₂, and χ₃. The authors conjecture that the same closure mechanism persists at every derivative order.

A general proof would need to show that the input-independent variance fixed point supplies, at each order, the identities required to cancel Stein-correction terms against cross terms involving adjacent derivative orders. The conjecture is explicitly unresolved because the authors establish the cancellations only through order three.

References

Conjecture 1. For every k, the recursion for vk obtained from (9) has coefficients that are polynomial in the susceptibilities χ1, . . . , χk and involves no other moment of the activation. The evidence is orders one through three, where every non-susceptibility moment cancels in pairs, and the pairings are between a squared term and the cross term that shares its activation factor. We have not established the general case, but the cancellations have a common cause which is worth stating, because it identifies what a general proof would need.

— Critical initialization destabilizes higher input derivatives in wide scalar-input networks  (2609.09244 - Singh et al., 8 Sep 2026) in Conjecture 1, Section 4.5, “The recursion closes on the susceptibilities”

And the input is scalar throughout; for a multidimensional input the jet becomes a collection of mixed partial derivatives, and whether the closure of Corollary 2 survives that generalization is not something the present argument settles.

— Critical initialization destabilizes higher input derivatives in wide scalar-input networks  (2609.09244 - Singh et al., 8 Sep 2026) in Section 8, Discussion