Necessity of the fourth-order threshold

Determine whether the condition mβ>3 is necessary for a fourth-order fusion expansion of the normalized Sineβ correlation function, and characterize the behavior at and below the threshold.

Background

The main fourth-order fusion theorem is proved under the sufficient integrability condition mβ>3. The coefficient contains a factor (mβ−3) in its denominator, suggesting a boundary at mβ=3, but the paper does not establish whether this boundary reflects a genuine obstruction to a fourth-order expansion. At exceptional parameter values, cancellations may remove the apparent pole, so the necessity of the condition cannot be inferred from the displayed coefficient alone.

References

The condition $m\beta>3$ is a sufficient range for the present proof; we do not prove that it is necessary for a fourth-order expansion.

— Fourth-Order Fusion Asymptotics for $\mathrm{Sine}_β$ Correlation Functions  (2609.11543 - Fang, 10 Sep 2026) in Section 6, “Classical checks and the fourth-moment boundary”

For arbitrary $\beta$, proving local convergence eq:univ-local-convergence together with both uniform bounds eq:univ-origin-bound--eq:univ-tail-bound for general potentials remains a separate universality problem.

— Fourth-Order Fusion Asymptotics for $\mathrm{Sine}_β$ Correlation Functions  (2609.11543 - Fang, 10 Sep 2026) in Section 8.3, “A conditional criterion for arbitrary beta”; reiterated in Section 9, “Further questions”