Mirror Duality in the SMM4 and SMM4b Models

We study the mirror duality of the companion paper, built on the Langlands self-duality of the Standard Model gauge group $G_{\mathrm{SM}}$, in the four-dimensional mirror-matter models SMM4 and SMM4b with gauge group $G_{\mathrm{SM}}\times G_{\mathrm{SM}}'$. In SMM4, reading the $SU(2)$ factors of the left-handed leptons and quarks as two halves of one gauge $SU(2)$ makes each sector exactly $G_{\mathrm{SM}}$, anomaly free and self-dual without added matter; with shared neutrinos, only the mirror parity $\mathrm{swap}\circ(P\otimes P)$ and its CP partner $\mathrm{swap}\circ(C\otimes C)$ preserve the content, and elementary quarks cannot be shared. In SMM4b with shared neutrinos the bosonic and fermionic degrees of freedom balance, at 90 per sector, as a count of broken generators and not as a supersymmetry. Sharing neutral quark condensates as well identifies only vacuum values, and it forces equal vacuum values on the shared quark type. A flow from SMM4 to SMM4b through four-fermion interactions realizes the symmetry-breaking pattern of SMM4b but not the Standard Model spectrum, and its baryon-number-violating couplings must be strongly suppressed; the strong--weak part of the duality, a formal map from the outset, is lost at condensation, and the flow contains an axion of the Weinberg--Wilczek type. Spontaneous mirror breaking at the size needed for neutron--mirror-neutron oscillations requires tuning and a cosmological history that selects a single minimum in each sector, and in the two-doublet model the tuned vacuum puts the mirror sector at $\bar\theta'=\bar\theta+\pi$, far outside the oscillation window, unless $\bar\theta$ is relaxed in each sector. In an $\mathcal{N}=2$ model with a single $G_{\mathrm{SM}}$ the mirror fermions are the superpartners of the ordinary ones, but its two-sector form cannot be exchange symmetric for an odd number of generations.