Sector-Switching Curl on the Klein Bottle — numerical verification report

Verification of the sharp statement formulated at the meeting point of the tensor-invariant system (jet/jad/dev) and the Möbius-Klein transform framework: on the Klein bottle the curl of a genuine (sigma-equivariant) vector field lives in the other sector (it is a twisted scalar), while div and dev stay in sector. The cause is the pseudotensor character of the system: the curl is a product of the epsilon contraction, and epsilon changes sign under the orientation-reversing deck transformation (det dsigma = -1). As a consequence, in the Helmholtz decomposition the scalar potential lives in the even sector while the stream function lives in the twisted sector — the complete spectral description of a genuine vector field requires the bases of both sectors. Method: a 64x64 grid on the covering torus, band-limited random fields, spectral (FFT-based, exact) differentiation, grid-exact sigma action, sector projectors P±. Result: 24 of 24 tests pass at machine precision, largest error 7. 8e-16; the control (a non-equivariant field) leaks an O (1) fraction into the "forbidden" sector, so the result is a consequence of equivariance rather than a numerical artefact. A deterministic verification script accompanies the report (kleinᵣotₛectorᵥerify. py, fixed seed).