Close the exponent gap for general mergeable states

Determine whether a mergeable state that is not a net histogram can close the exponent gap of d/2 between the general lower bound and the net-histogram upper bound for deterministic conformal-shift summaries over a d-dimensional weight face.

Background

The paper studies deterministic, mergeable summary states that conservatively approximate exact conformal shifts for post-hoc ensemble weight vectors. It derives a general lower bound with exponent 1+d/2 in the approximation parameter and an upper bound with exponent 1+d for the specific class of net-histogram states used by CALQA.

The authors explicitly leave unresolved whether a different mergeable state construction, outside the net-histogram class, can attain a better exponent and thereby eliminate the d/2 gap between these bounds.

References

For general states the exponents are $1+d/2$ (Theorem~\ref{thm:lb}) against $1+d$ (Theorem~\ref{thm:tq}); whether a mergeable state that is not a net histogram closes this gap of $d/2$ is open.

— CALQA: Conformal Screening of Forecast-Ensemble Configurations from One Mergeable State  (2610.13249 - Liu et al., 9 Oct 2026) in Section 5, paragraph “Net sizes and constants” (following Theorem 5.4)