Unforced Navier–Stokes Seven-ray stress geometry, pressure fluxes and fourfold solenoidal detectors

We study exact algebraic and analytic restrictions on compact stress corrections in a proposedfinite-time unforced three-dimensional Navier–Stokes construction on the periodic torus. Adirect meridional symmetric-divergence inverse is combined with an exact one-sided seven-ray stress cone, sharp dual flux bounds and a moving-frame connection identity. The conehas a constructive jointly sharp axial–azimuthal hexagonal projection. Its normalized scalarray functional admits the sharp unrestricted lower bound 27μ/4 when all seven coefficientsare at least μ. For anisotropically shrinking fourfold-equivariant carriers, incompressibilityselects a nonaxisymmetric axial angular-advection profile. A compact, exactly solenoidaldetector couples to this profile with positive coefficient and yields the conditional concentrationrequirement δτ τ 1/2+hMτ ≥ c > 0 for an actual unforced correction under explicitly controlledsimilarity profiles. Exact rational-π integrals certify a nonzero kinematic channel. Theseresults establish constraints and admissible algebraic structures, not a source-matched nonlinearcorrection or a finite-time unforced Navier–Stokes singularity.