The small-Hurst limit of the expected maximum of fractional Brownian motion

Unrefereed research candidate. In one sentence: as fractional Brownian motion becomes very rough, its expected maximum on [0,1] grows like √(2/H). Fractional Brownian motion BH is the standard model of a rough random path; small Hurst parameter H means very rough. Bisewski, Dębicki and Rolski (Queueing Systems, 2022) conjectured that √H times the expected maximum MH = E sup[0,1] BH has a finite positive limit as H → 0. This paper proves that the limit is √2, with convergence of the rescaled maximum in every Lp. More generally, a Gaussian field whose canonical metric is asymptotically dH on a compact Ahlfors s-regular space has limit √(2s), and every such space carries such fields; for compact Euclidean sets the upper box dimension gives an upper bound. The all-time maximum with linear drift has the same constant. On a uniform grid with n intervals and H log n → c the sampled maximum times √H tends to an explicit function F(c), equal to √c up to c = 1/2 and then rising to √2, so a grid captures half the continuum mean at c = 1/2 and all of it only when H log n → ∞; quasi-uniform nets in regular spaces obey √s F(c/s). The paper also proves MH ≤ √(3 log 2 / H) for every 0 < H ≤ 1 with a consequent bound for Pickands' constant, and gives numerically evaluated two-sided bounds at fixed H and a Monte Carlo check of the grid law, which show that the first-order formulas converge slowly. Hierarchical Gaussian (GREM/CREM) maxima, Gaussian comparison and concentration are classical; the vanishing-exponent limits, the dimension law and the grid crossover are the proposed contribution. The archive contains the manuscript (PDF, Markdown source and generated LaTeX), figures, the bibliography, three verification scripts with retained outputs (exact rational checks of the finite constructions, numerical evaluation of the fixed-H comparison bounds, seeded Monte Carlo), the dated literature audit, a rating-free internal review summary, the superseded first version, CI and SHA-256 sums. Assurance: written proofs; numerical tables are floating-point evaluations and Monte Carlo illustrations, not interval certificates; internal AI-assisted review only (not peer review, independent reproduction or formal verification). Creator: Anonymous. Text, data and figures CC0-1.0; code MIT.