The Curl Spectrum on Non-Orientable Geometry: Sector Doublets, Vanishing Helicity, and Beltrami Pairs

Abstract

The spectral theory of the curl operator on closed 3-manifolds is a developed and active field — but exclusively on oriented manifolds, because the definition of curl requires an orientation. This article opens the non-orientable column using the sector calculus. The key is a one-line algebra: the deck transformation of a non-orientable quotient anticommutes with curl. Three theorems follow. (T1) On a non-orientable flat 3-manifold there is no single-sector Beltrami field: for lambda != 0 the equation curl B = lambda B has neither an ordinary nor a purely twisted solution. (T2) The spectrum organizes into sector doublets: on the cover the spectrum is symmetric under lambda <-> -lambda, every curl^2 eigenspace splits exactly half-and-half between the even and odd sectors, and curl is the intertwiner between the two — chirality is not a quotient observable. (T3) The total helicity of every sector-pure field vanishes identically; the Woltjer–Taylor helicity constraint trivializes on non-orientable domains, and the relaxed states are the sector-paired curl^2 eigenspaces. On the model of the flat Klein 3-manifold (K^2 x S^1) the first eigenvalue table reads lambda^2 = 1, 2, 3, 4 with dimensions 12, 24, 16, 12, split exactly in half everywhere. Every claim is verified by a deterministic, counter-checked script (21/21). The theorems originate in the framework's sector-switching curl theorem and the component rule of the solid Klein bottle.