Abstract
We present a rigorous parameter-free derivation of the scaling exponents for turbulent longitudinal structure functions, based on angular displacement spacetime framework φ(τ) and the physical upper bound of Lagrangian micro-rotation velocity (ΩFluidMAX): ζp=p/3+p(3−p)/8π²+ max (0, p−6)/4π². For the low-order regime (p≤6), the derived strict quadratic formula reads: ζp=p/3+p(3−p)/8π². This formula contains zero adjustable parameters, with the correction term stemming solely from the universal topological constant 1/(4π²) intrinsic to rotational motion. It automatically satisfies the exact Kolmogorov 4/5 law at p=3 and is intrinsic physical consistency p=3, naturally explaining from first principles the ubiquitous experimental observation that “scaling exponents exceed K41 values for p3”. Remarkably, it reduces to a form mathematically identical to Kolmogorov-Obukhov (K62) log-normal model, p= p/3 - (μ/18)·p(p−3), with the effective intermittency parameter fixed at μ= 9/4π² ≈0.228. The log-normal formulation is therefore not a competing model, but is subsumed as the p≤6 continuum approximation within the present geometric framework. We further predict that a physical phase transition occurs at p=6, corresponding to the breakdown of the continuum hypothesis. For 7≤p≤13, the scaling exponents deviate from the quadratic law due to the dominance of high-order nonlinear terms. Crucially, this high-order deviation is not a “fitting residual,” but the inevitable consequence of a hard-truncation phase transition (nonlinear breakdown): as the local micro-rotation velocity approaches ΩFluidMAX , the hard-wall truncation of rotational kinetic energy causes a δ-function accumulation in the probability density at the threshold (angular condensation), thereby exciting a linear additional term ((p−6)/4π²) in the high-order moments. A complete set of zero-parameter predicted values for p=1 to 13 is provided for independent experimental verification—the predictions for ζ1−ζ11 already agree with the existing experimental/DNS data), whereas no DNS data exist for ζ13—prior to the release of the Rλ≈2500 DNS data. This constitutes a blind test: the predictions are fixed, and the data will provide an independent verdict.