The Sector Rule of the Curl on the Klein Bottle and Its Solid Counterpart – the sector behaviour of the derivative-tensor slices in two and three dimensions

At the meeting point of the tensor-invariant system (grad/jad/dev) and the Möbius–Klein transform framework, a sharp statement can be formulated: on the Klein bottle, the curl of a genuine vector field lives in the other sector (it is a twisted scalar), while div stays wholly in the even sector, and dev, though a genuine tensor, splits component by component — its diagonal component even, its off-diagonal one odd. What sets the curl apart is the “pseudotensor birthmark” of the tensor system: rot is a product of the epsilon -sorter, and epsilon changes sign under the orientation-reversing deck transformation. The parities of div and of the dev components come from the index parities alone. Consequence: in the Helmholtz decomposition the scalar potential lives in the even sector, but the stream function lives in the twisted sector. The chain of claims is checked at machine precision on the covering torus, with a control. One dimension up the picture refines rather than repeats: in three dimensions the curl is again a twisted (axial) field and switches sector as a whole, but its component pattern is the mirror image of the polar pattern — the component along the mirrored axis is even, the two perpendicular ones odd. This is a pointwise statement, and it carries over to the solid Klein bottle. The computation itself runs on a closed flat model, and it is there that the vector potential of the Helmholtz decomposition is shown to follow the same pattern; that statement is global and is not claimed for the solid bottle. The parity rule behind all this is classical; its explicit form for this geometry is taken from the companion paper, whose verification is cited by locator. The building blocks – the orientation line bundle, twisted forms and the fact that Hodge duality takes forms to twisted forms – are classical, and so is their slice-by-slice reading; what this report adds is the explicit statement of the rule for the two dimensions side by side, and the controlled numerical check of the two-dimensional case: no new theorem, but a usable rulebook.