Six-coordinate minimizers beyond the four-dimensional endpoint

Determine whether, for d_{\max}(4)<\tau\leq d_{\max}(6), the minimizer of the Rademacher-sum functional K_n(\omega)=\mathbb{E}|\sum_i\omega_i\epsilon_i| on the constraint set of proper unit vectors at distance \tau from the two-coordinate Khintchine extremizer has the conjectured equal-coordinate six-dimensional form, with the two specified branches (r,r,r,r,s,s,0,\ldots,0) and (r,r,t,t,t,t,0,\ldots,0).

Background

Theorem \ref{thm:four} proves that, up to the distance d_{\max}(4), the sharp minimizers of K_n on the distance-constrained proper chamber are four-coordinate vectors with coordinates grouped into equal pairs.

The authors then propose a continuation beyond the four-coordinate endpoint. They conjecture that the minimizer remains supported on six coordinates and changes between two equal-block configurations at the stated transition value. The subsequent results establish only local minimality and global minimality in a neighborhood of d_{\max}(4), so the full conjecture is unresolved.

References

For $d_{\max}(4)<\tau\leq d_{\max}(6)$, the minimizer of $K_n(\omega)$ on $\mathcal{C}_n(\tau)$ has the form

\omega=(r,r,s,s,t,t,0,0,\ldots).

More explicitly, the minimizer is conjectured to be

(r,r,r,r,s,s,0,\ldots,0), \qquad d_{\max}(4)\leq\tau\leq\sqrt{2-\frac{46}{\sqrt{1107},

and

(r,r,t,t,t,t,0,\ldots,0), \qquad \sqrt{2-\frac{46}{\sqrt{1107}\leq\tau\leq d_{\max}(6),

where

r=\frac{2-\tau2}{2\sqrt2}, \qquad s=\sqrt{\frac12-2r2}, \qquad t=\frac12\sqrt{1-2r2}.

— A Two-regime Khintchine Inequality and an Improved Bound on the Degree-1 Fourier Weight for Linear Threshold Functions  (2608.27908 - Fang et al., 28 Aug 2026) in Conjecture 1 (labelled \texttt{conj:six}), Section 2, immediately before Section 2.1