Sharp Solid-Angle Bounds for Trace-Free Quadratic Forms on the Sphere - with an Application to Strain-Cone Geometry

How many directions can an incompressible strain stretch? This short note proves a sharp and, to our knowledge, new answer: for every trace-free symmetric 3×3 matrix — the strain-rate tensor of incompressible flow, but equally the general degree-two spherical harmonic or the indefinite Gaussian quadratic form, the fraction of directions with positive stretching always lies between 42.3% and 57.7% (exactly between 1−1/√3 and 1/√3), with the extremal configurations fully characterized. The proof is elementary (Archimedes' projection and a single Jensen inequality whose bound is independent of the spectrum), and the note develops the result into a small self-contained theory: a complete monotone ordering of the spectral family, a sharp quadratic rigidity estimate near the extremizers, explicit bounds under anisotropic direction statistics (density, total variation, and second-moment control of signed production), and a sharp extension to n dimensions with limit erf(1/√2) ≈ 0.683. Applications to vortex-stretching geometry are described but not required, the results are pure spherical integral geometry. A structured two-agent prior-art sweep across seven adjacent literatures (solid-angle geometry, Gaussian quadratic forms, nodal domains, conic geometry, turbulence strain statistics, Q-tensor theory, random matrices) found the classical machinery everywhere and this extremal statement nowhere. Paper A of the geometric regularity program for 3D Navier–Stokes; developed under multi-model adversarial review with machine-verified adjudication.