What the Shadow Loses: the Double Curve of an Immersed Klein Bottle as a Spectral Blind Set. A reflection involution, a vanishing lemma, and the immersion-dependence of the blind set

A closed non-orientable surface cannot be embedded in R^3: every realization is an immersion with a double curve, where two surface points share one ambient point. On the flat Klein bottle K the reflection iota(theta,phi)=(theta,-phi) commutes with the deck map, hence descends to K with a pair of disjoint circles as fixed set. Splitting the sector calculus of the Möbius–Klein framework by iota-parity gives a one-line vanishing lemma: the iota-odd half of each sector — the half-integer theta-lattice of the invariant sector — vanishes pointwise there. The figure-8 (lemniscate) immersion places its double curve exactly there: an ordinary scalar restricted to the self-intersection equals its integer-frequency part alone, so the spectrum it can carry there is indistinguishable from that of a function on a torus. On one test field the missing part carries 49.0% of the energy — a single draw around the mode-count value 1/2. The blindness belongs to the immersion, not to the geometry: adding the invariant term eps*cos(phi) to the vertical component separates the two circles into the planes z=+-eps, so the fixed set is no longer contained in the double set, which non-embeddability keeps non-empty; Newton continuation places the relocated curve at mean distances 0.0234, 0.0694 and 0.1703 rad, and a closed-form lemma shows that it still meets the fixed set in exactly two points. A characterisation follows: a vertical deformation destroys the blindness precisely through its odd-order cosine content. The twisted sector loses the complementary half. The embedded Möbius strip provides the control. A deterministic script verifies every statement (148/148 tests, seed 20260811). For the inverse problem: a measurement confined to the self-intersection can never establish non-orientability. Of the two immersions whose admissibility in a nowhere-vanishing magnetic field is settled, the one that can occur is exactly the blind one.