Rough paths below the Young threshold: an exact scale calculus and the locality phase transition at one quarter

Abstract

Since Young's 1936 theorem, irregular integration has been organized around the threshold 1/2: above it the path determines the integral, while below it higher-order data are needed. For fractional Brownian motion, H = 1/4 is the threshold for the canonical Gaussian enhancement, although geometric rough lifts exist for every H > 0. We prove that H = 1/4 is instead the exact threshold for measurable locality. For d-dimensional fractional Brownian motion with independent components, d at least 2, if 0 < H <= 1/4, no positive-measure Borel set supports even one finite off-diagonal second-level coordinate satisfying Chen's relation and measurable interval by interval from path increments. No moment, Holder, geometricity, stationarity, or scaling assumption is imposed. If 1/4 < H <= 1/2, every full-law local rough-path lift in the standard Holder range is automatically square-integrable and hence classified without an assumed L2 condition; for H > 1/2, every finite-step enhancement with natural graded Holder bounds is uniquely the Young signature. We also introduce an exact scale calculus below classical differentiability. Matched dyadic differences recover normalized derivatives with sharp O(epsilon^2) error, exact localized inversion, and lossless reconstruction. Multiplication and smooth functional calculus transport exactly to scale coordinates, while the corona quotient yields an exact universal derivation. Reinserting the Holder amplitude and using Fourier-normal ordering produces strong geometric lifts for every positive input regularity and every lower rough exponent, with explicit ultraviolet rates and stability. Thus rough lifts exist below one quarter, but no measurable lift on a positive-measure domain can be interval-local there. The results separate existence from local recoverability and show that nondifferentiability does not destroy exact differential information.