Critical inverse-energy conjecture and optimal explicit embedding

Prove or refute that there is an absolute constant c>0 such that, for every n and every pair of distinct integer partitions λ,μ of n, the critical dyadic energy of the realizable Ferrers-staircase difference v_{λ,μ} satisfies 𝓔(v_{λ,μ}) ≥ c‖v_{λ,μ}‖₁²/√n; determine whether the explicit coordinate F^(1/4), or another explicit coordinate, has class-optimal Euclidean distortion Θ(n^(1/4)) and O(n)-time computability.

Background

The paper introduces the critical dyadic coordinate F^1/4 on Ferrers staircase functions and proves an unconditional distortion bound O(n^1/4(log n)^1/4). The remaining logarithmic factor arises from estimating the octave-localized total variation by Cauchy–Schwarz.

Conjecture 5.18 asserts the stronger inverse-energy inequality that would remove this factor and, by Corollary 5.20, would yield an explicit deterministic O(n)-time embedding with class-optimal distortion Θ(n^1/4). The authors verify the conjecture on sign-coherent, balanced, and geometrically distributed octave-spectrum sub-cones, but leave the general tall, sign-incoherent, flat-spectrum regime unresolved. They also separately ask whether some explicit coordinate other than F^1/4 can achieve the optimal order.