Where positivity can come from in the mirror sector: the domain-wall geometry, delimited (SQG Companion — Paper B)

The lattice construction of the companion papers obtains a sign-free measure at even flavour number from an antiunitary Θ acting on a four-dimensional bipartite Hamiltonian. Chirality, by contrast, is naturally supplied by a five-dimensional domain-wall geometry. This note determines whether the two can be combined, and finds that they cannot. Three independent obstructions are established: 1. The only lattice-local Krein metric that renders the Shamir determinant real is J = \gamma_5 \otimes R_5. Because R_5 exchanges the walls, no interaction supported on a single wall is protected. The obstruction is structural: the four-dimensional kinetic block forces G \propto \gamma_5, while wall localisation requires an operator that anticommutes with \gamma_5; the intersection is empty. 2. The spectrum of the full five-dimensional operator is not real in a generic gauge background, so no positive metric exists and the PT-type rescue is closed. Determinant reality survives only through conjugate pairing. 3. The single positive propagator the geometry provides — the Furman–Shamir transfer matrix in the fifth direction — is hermitian and positive definite, yet carries complex matrix elements in position space and therefore supports no positive-weight world-line sampling. The structural reason is identified: world-line positivity in the flavour-determinant scheme rests on the absence of free-fermion propagation from the propagator, which the five-dimensional route necessarily reinstates. Two corrections to previously recorded results are stated, together with the updated status of the sextic conjectures: the proposed edge-factorisation proof is closed, and the pointwise tree conjecture has been refuted at a degree-three vertex, leaving the grouped statement as the only surviving form.