Why is the ovenbird — potter and faithful, the seemingly ideal totem — absent from every Jivaro myth, as Lévi-Strauss himself noted in The Jealous Potter? This paper proposes a structural answer. From two declared commitments — every representation is Imaginary, bounded by an inaccessible Real and Symbolic; a measurement is the local meeting of two variances, what is observed and what is judged — the space of representation is modelled as the simplest one-sided closed surface, the cross-cap. Gluing local measurements step by step reconstructs a two-faced sphere, and the global failure of that reconstruction — one tour returns with the sides exchanged — is an obstruction in the strict sense. The myth of the goddess Nightjar performs exactly that tour; the ovenbird, which would close the model without remainder, is foreclosed: a cold society cannot adopt a new symbol without ceasing to be itself. Mathematics repairs its impossibilities by the same double inversion — √2, i, the quaternions — but adopts the new symbol. The structure is minimal — a two-element group acting on a two-element fibre — and the surface carrying it is unique. Three refutable conditions are stated.