Paper 6: NTLA-O V: Path Identity, Fundamental Groupoids, Coverings, Monodromy, and Holonomy

Path Identity, Fundamental Groupoids, Coverings, Monodromy, and Holonomy — Toward History-Sensitive Observer Transport

NTLA-O series Paper 6. Even when local data is fully compatible and glues, one kind of information remains unaddressed: how is an observer state transported from one location to another? If the endpoints are the same but the paths differ, is the resulting observational state still the same? This paper distinguishes five levels of path identity, retaining decreasingly more generative history: Raw Path, Reparameterization Class, Thin-Homotopy Class, Endpoint-Fixed Homotopy Class, and Homological Information. This is the most natural algebraic-topology interface for the original NTLA's principle that "holes are not just how many, but how connected."