logical manifold is a concept which may prove useful in nuanced debates. to explain it, I need to shortly cover topological manifolds.
a topological manifold is a structure that looks 'flat' or 'straight' (euclidean) when viewed from a small distance but may not be euclidean as a whole.
think of the Escher's stairs, a three‑dimensional manifold:
look closely at each step in the picture. they all look typical, even boring. what's weird is that you could not construct such staircase, despite there being nothing wrong with any particular part. the staircase is locally euclidean.
now back to logical manifolds.
a logical manifold is a statement which lays out a seemingly logical argumentation but is wrong as a whole.
whether a statement is a logical manifold is subjective. it may also change over time, so it's not useful to categorize arguments in this way. rather, it is a description of the recipient's relation towards the statement. they may feel that something is off but be unable to point out big flaws in any of the intermediate steps. in this respect, a logical manifold is locally logical.