The Scale-Shape Divide in Bitcoin and Ether Perpetuals: Roughness States, Quarticity, and Bipower Residuals

Abstract

This paper asks whether rolling roughness estimates that label volatility states also rank the shape of near-term tail variation. The design freezes the January 2020–July 2026 Binance BTCUSDT and ETHUSDT tape, 9,560 six-hour origins per contract, and the three functionals used in companion work. Forward 8-hour and 24-hour outcomes are reconstructed from one-minute sufficient statistics. At eight hours, a one-standard-deviation increase in Ĥᴵ ⱽ is associated with 0.186 log points more realized quarticity in Bitcoin and 0.166 in Ether; the corresponding Ĥᴾᴬ effects are 0.177 and 0.157. Origin-level 14-day block-cluster inference preserves all four effects, whereas Ĥᴾⱽ remains null. The accounting identity log RQ = 2 log RV + log NRQ locates the association: it loads through realized-variance scale. Log normalized quarticity, the raw bipower-residual share, and a conservative RQ-scaled significance-filtered residual all satisfy prespecified equivalence tests. Recursive raw-quarticity forecasts improve on both historical and state benchmarks in Bitcoin, but Ether does not reproduce the short-horizon gain; normalized-tail and residual-event results are still less stable. Roughness therefore reaches short-horizon fourth-moment scale in these contracts without furnishing a common tail-shape coordinate.

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