Establish the conjectured scalar spectral minimax formula

Prove or disprove that, for the scalar antisymmetric two-model binomial-mixture experiment over the two-sided margin class with exponent theta and constant L, the minimax decision regret is comparable to the infimum over degrees 1 <= d <= n of d^{-theta} + kappa_{d,n}/sqrt{Q}, where kappa_{d,n} is the inverse singular value of degree d for the n-trial binomial channel.

Background

The scalar reduction converts ensemble learning into deciding the sign of the sign-imbalance functional theta(G) from Q observations passed through an n-trial binomial channel, with n=2N. The paper proves a Hellinger-modulus characterization of the minimax risk and derives matching rates in several regions, but the available spectral upper and lower bounds do not coincide throughout the full (Q,n) parameter space.

The conjectured formula would imply a three-phase rate picture: a root-Q phase, a spectral-pooling phase with rate involving n and log(Q/ntheta), and an identification-limited phase at n{-theta}. Establishing it requires closing both the lower-bound gap in the Hellinger modulus and the upper-bound stability gap.

References

This leads to the conjecture

{\mathrm{dec}_{Q,n}(theta,L) \stackrel{?}{\asymp}{theta,L} \inf{1\le d\le n} \left{ d{-theta} + \frac{\kappa_{d,n}{\sqrt Q} \right}.

— Two Sample Sizes in Test-Time LLM Ensembling: Buffered ERM and Minimax Theory  (2610.21363 - Zhang et al., 15 Oct 2026) in Section 1, paragraph “Conjectured completion of the scalar picture”; Section 8.3, Conjecture 8.1

Thus the main open questions for the scalar model are to resolve the transition immediately above $Q\asymp ntheta$, to sharpen the spectral upper and lower bounds to the same scale, which includes removing the extra $\sqrt{n+1}$ factor from the upper bound in eq:intro-spectral-upper, and to extend the matching spectral lower theory beyond $theta=1$.

— Two Sample Sizes in Test-Time LLM Ensembling: Buffered ERM and Minimax Theory  (2610.21363 - Zhang et al., 15 Oct 2026) in Section 1, paragraph “Conjectured completion of the scalar picture”; Section 8.3, subsection “First transition and spectral conjecture”