Conjugate-doubled two-layer fermion systems: a sign-freedom theorem, and why it does not lift the odd-generation obstruction

At odd generation number the sign problem of the Spin(10) mirror sector, written in flavour-singlet composite variables, lands on the complex vector-spinor 144, and Companion I of this series named one remaining check: whether a matter–mirror conjugation can pair the composite determinant when the interaction resolves the internal index. We settle this check, with a negative answer for the physical problem and a positive mathematical result along the way. Theorem. Any decoupled two-layer fermion problem on a bipartite lattice in which (A) the second layer's one-body matrix is the complex conjugate of the first, (B) the first satisfies the chiral condition η h† η = −h, and (C) the inter-layer pairing is real, symmetric and sublattice-preserving, has two anticommuting Majorana-time-reversal symmetries (T² = −1, the layer exchange; T² = +1, layer sign times sublattice sign) and is sign-free by the theorem of Li, Jiang and Yao. The theorem accounts for a full catalogue of admissible and forbidden terms, including five cases predicted in advance and confirmed. What it implies. Without pairing the weight of such a system is exactly |det(1+B₁)|²: the second layer is the conjugate copy of the first, and the sign-free model is the doubled ensemble of the one-layer problem. Condition (A) cannot be met by an interaction that acts on the mirror alone; a gapping interaction shared by the two layers gaps the matter layer as well. This is the positivity–selectivity tension of the lattice paper (Paper A), now in Majorana form, and it means that the construction does not provide a sampler for the physical, mirror-selective odd-generation problem. For that problem we prove that a gauge channel alone gives a real (not positive) weight, and we estimate the average sign on small lattices: mild for the gauge channel alone, severe once the density repulsion is included. All criteria marked "pre-registered" were fixed in writing before the corresponding runs. Verification and simulation scripts are included (companionII_skripte.zip). Derivations, code and drafting in collaboration with Claude (Anthropic).