TREP–Navier–Stokes Blow-up Exclusion by One-Mouth Projective Rigidity. A Non-Circular Conditional Endpoint-Exclusion Criterion.

We formulate a non-circular TREP one-mouth projective-rigidity criterion for the three-dimensional incompressible Navier–Stokes equations. The argument is organized around a hypothetical finite-time singular point, but the theorem is stated conditionally on an explicit Navier–Stokes–TREP verification package. That package isolates the rows that cannot be hidden inside the final contradiction: guarded one-mouth extraction without an uncharged second carrier, scalar-critical quantization and compactness, restricted projective observability, recurrence versus angular-dissipation telescoping on the same stopped windows, projective rigidity, synchronized axial no-tube witness generation, primitive same-parent endpoint packing, and no-late-endpoint closure. Under this package, the primitive branch produces a non-trivial ancient vorticity profile on an inherited carrier. Nonzero projective angular defect would give non-summable recurrence on the selected branch, whereas angular-dissipation telescoping gives summability; hence the endpoint has vanishing projective defect. Projective rigidity and the divergence-free constraint reduce a nonzero endpoint to a two-dimensional-three-component profile. A positive planar mass would then generate synchronized axial witnesses in every fixed finite axial window. If those witnesses are counted by the original primitive stopped-tree budget with constants independent of the window length, one obtains a contradiction as the radius tends to infinity. The paper therefore gives an arXiv-safe endpoint reduction rather than an unconditional proof of global regularity. No one-mouth branch, projective recurrence row, angular summability row, 2D3C no-tube packing row, or endpoint budget is allowed to be introduced after compactness or inferred from the desired blow-up exclusion itself.