Vorticity-Strain Geometry and Nonlocal Pressure Torque in High-Reynolds-Number Turbulence: Three Exact Relations and a Six-Substrate Empirical Study

This paper states three exact algebraic relations for the part of the vortex-stretching vector transverse to the vorticity. For unforced incompressible Euler flow that part is the rate of change of the vorticity direction, so the vortex-line angular rate equals the alignment-weighted standard deviation of the strain eigenvalues which is described by the Mehrhoff Variance Identity (MVI). The Mehrhoff Eigenframe Identity (MEI) relates the alignment weights to the eigenvalue gaps, and the Mehrhoff Projection Identity (MPI) factorises the alignment between that vector and the pressure-Hessian force. All three are proved and reproduced within 10⁻¹² of their scale on 1,620,000 voxels from six Johns Hopkins Turbulence Database substrates, the Mehrhoff Eigenframe Identity (MEI) on trace-free eigenvalues. To the author's knowledge none appears in this form in the literature reviewed; the body names the classical result behind each. In vortex tubes the nonlocal pressure term cancels about 60% of the local torque term in isotropic datasets and 45% in boundary-layer datasets, as a regression slope. The isotropic minus boundary-layer cancellation difference is largest near the low-z face of the crops. Beyond 128 cells it is +0.013 (95% interval −0.001 to +0.032) with the periodic-crop pressure and +0.051 (+0.036 to +0.072) with the stored pressure, mainly through the Hessian stencil. The pressure-alignment difference beyond 128 cells changes with the stored pressure from −0.087 to +0.014 (+0.003 to +0.026). The differences depend on intensity, Hessian-trace residual, position, operators and pressure source, and cannot be attributed to flow type, which is confounded with simulation type. The nominal bootstrap intervals exclude variation between simulations. For the pressure alignment they are too narrow in four of the six datasets. No claim is made about global regularity.