Exact dimension-free third-moment stability constant

Prove that, for every integer n≥1 and every coefficient vector a∈R^n satisfying ∑_{i=1}^n a_i^2=1, the normalized Rademacher sum satisfies E|∑_{i=1}^n a_i ε_i|^3 ≤ E|(/√n)∑_{i=1}^n ε_i|^3 − (5√3−6√2)∑_{i=1}^n(a_i^2−1/n)^2.

Background

The paper proves a dimension-free quadratic stability estimate at the third moment with the smaller universal constant 1/100. It also establishes a stronger constant, 5√3−6√2, in dimension three, where equality is attained by a coefficient vector whose squared coordinates are (1/2,1/2,0), up to signs and permutations.

The optimal dimension-free constant C_3{\mathrm{opt}} is bounded between 3(5√2−7)/16 and 5√3−6√2. The authors identify the dimension-three value as a natural sharpness candidate but do not determine whether it is optimal in all dimensions; the conjecture asserts that it is.

References

We conjecture that, for every $n\ge1$ and every $a\inn$ with $\sum_i a_i2=1$, \begin{equation}\label{eq:conjecture} {\sum_{i=1}n a_i_i}3 \le {\overline S_n}3 -(5\sqrt3-6\sqrt2)\,\Delta_n(a). \end{equation}

eq:conjecture:

${\sum_{i=1}^n a_i_i}^3 \le {\overline S_n}^3 -(5\sqrt3-6\sqrt2)\,\Delta_n(a). $

— Fourth-Moment Geometry of Rademacher Sums  (2608.17802 - Gao et al., 18 Aug 2026) in Conjecture, Section 6, following Remark 4.1 (the conjecture is labeled \label{conjecture})