The Pairing Rulebook of the Brillouin Klein Bottle. Mirror circle, forbidden s-wave, and half-integer phase winding: the sector selections of Bogoliubov–de Gennes theory on non-orientable momentum space, with the Klein transform as the native mode machinery

Abstract

A Brillouin zone can be a Klein bottle: under Z2 gauge fields the algebra of crystal symmetries becomes projective, a nonsymmorphic glide reflection appears in momentum space, and the fundamental domain is a Klein bottle instead of a torus — the experimentally realized Brillouin Klein bottle (BKB) program [3–5]. So far the line has developed the insulator side; this article opens the superconducting side: the rulebook of Bogoliubov–de Gennes theory and pairing channels on the BKB, with the framework's Klein transform (KT) as the native mode machinery [1]. Three theorems. (T1) The particle–hole-invariant set of the BKB consists of two isolated points and an entire circle — the “mirror circle” — on which the PH constraint acts in a deck-twisted form; instead of the four TRIM points of the torus, a new type of invariant manifold appears. (T2) Pairing selection: the deck parity of the condensate defines two channels; in the even channel the uniform s-wave gap lives, in the twisted channel the constant gap is forbidden — the half-integer lattice has no zero mode — and the minimal singlet is cos kx: a PDW-type modulation at half the folded-zone momentum. (T3) The gap function of the twisted channel must carry, on the invariant loops, either a node or a half-integer phase winding — and the node set of the minimal twisted gap is exactly the mirror circle. The theorems are verified on a BdG toy model by a deterministic, counter-checked script (24/24). The article makes no material predictions; its contribution is the rulebook and its machine checkability.