The projective loop

An odd three-relation cycle can fail to return on the base configuration while closing on a connected double cover. The placement triangle lifts to a six-cycle, and coherent transport can require twelve steps before every marked datum returns.

This is the mathematically disciplined form of the twisted odd-gear intuition: the obstruction lives in projective lifting and holonomy.

The central return −I preserves determinant orientation. It therefore cannot simply be renamed chirality or identified with a Pfaffian-sheet reversal. The negative result is productive: it tells the programme that orientation needs an independently odd datum.

Write an arrow on a strip of paper, give the strip a half twist, and join the ends. After one circuit the arrow returns reversed; after two it returns fully. A projective double cover records this without pretending that the paper moved through an extra physical dimension.

The programme's three-placement loop behaves similarly. Local transitions can be consistent while their product returns a central sign. Lifting the triangle to a six-cycle makes the sign bookkeeping single-valued, and a twelve-step marked transport can return every decoration.

Multiplying every vector by minus one changes its sign but preserves the orientation of an even-dimensional frame. Handedness instead changes under a reflection with negative determinant. This is why a central -I holonomy cannot simply be called chirality.

The distinction is not pedantry. A physical selector must couple to something that actually changes under the proposed exchange. If the candidate datum is even, no amount of suggestive language makes it an orientation source.

A topological label can remember how a loop was prepared, but a localized laboratory mode also needs a place to live. The charge-three Higgs sector has a precise level-three BF limit, yet its apparent phase vacua are gauge copies and cannot make a domain wall. Adding a neutral \(\mathbb Z_2\) field supplies an explicit kink and one localized even bosonic reader.

This cleanly separates memory from hardware: BF theory carries the sector label; the gapped reader carries non-topological dynamics; and the wall's origin remains an external microscopic question.