Complete positivity for the Feigenbaum–Grabner–Hardin family f_w

Prove that all forms f_w in the family introduced by Feigenbaum, Grabner, and Hardin (as in their Proposition 5.1) are completely positive, i.e., have nonnegative Fourier coefficients.

Background

In the context of the 24‑dimensional inequalities, the paper identifies its form F as a constant multiple of f_{16} from the Feigenbaum–Grabner–Hardin family. Those authors proved complete positivity for f_w up to weight 94 and conjectured it for all members of the family.

The present paper gives an algebraic proof of positivity for the specific case tied to f_{16}, but the general conjecture for all f_w remains open.

References

The authors already proved that the functions $f_w$ are completely positive for $w \leq 94$ Remark 6.3, and they conjectured that all the forms in the family are completely positive Conjecture 1.

— Algebraic proof of modular form inequalities for optimal sphere packings  (2406.14659 - Lee, 2024) in Section 6.1, “Easy” inequality (Remark following Corollary 6.2)