Holonomy as the Error Term: The Ladder of Tensor Invariants on Non-Orientable Spaces and Spacetimes

In curved space the identities of the flat ladder of tensor invariants acquire curvature as their error term. On a non-orientable space a second error term appears, the twist, and the 2 are local and global faces of one object, the holonomy. Around a small loop of a curved Möbius band the holonomy is a rotation by the enclosed curvature; around the core it is a reflection without rotation. This article carries the ladder through non-orientable spaces and spacetimes and counts what survives in each sector of the orientable double cover. On the flat Klein bottle 1 of the 2 parallel fields survives. The lost tensor-valued theorems live in the twisted sector, where they pair a twisted tensor with a twisted test field. On the projective plane the Killing fields of the sphere descend, the gradient fields of its conformal algebra are twisted, and the Korn deficit of a rotation is half that of the sphere. On a Möbius band the divergence theorem closes with a density, the Kelvin–Stokes theorem only for a twisted field. In 4 dimensions an orientation-reversing isometry exchanges the self-dual and the anti-self-dual Weyl curvature, so on a non-orientable 4-manifold the 2 halves of the jad rung form a twisted pair. For a single mode on a periodic background a time-Möbius admits a self-consistent solution only where a Floquet multiplier is minus one, and self-consistency acts as a sector filter. A Noether charge that commutes with the twist keeps its meaning, and its spectrum labels the sector; one that anticommutes switches the sector. Every statement is checked by a deterministic script with counter-checks.