Mirror matter and the self-duality of S(U(2)× U(3))

We give an explicit root-datum isomorphism showing that the Standard Model gauge group GSM=S(U(2)× U(3))=(SU(3)× SU(2)× U(1))/Z₆ is Langlands self-dual. Among the quotients by subgroups of Z₆, only the Z₆ quotient has this property. On the product theory GSM× GSM' we define a mirror duality MP= swap∘ P∘(S⊗ S), with S treated as a formal map, since S-duality is expected to be exact only in the N=4 theory. MP fixes the couplings on the locus τ'=1/τ, which at θ=0 gives g₃g₃'=g₂g₂'=4π and gYgY'=12π (g₁g₁'=20π in grand-unified normalization). The construction does not require a self-dual group; the self-duality makes the electric-frame group of the mirror sector equal to GSM, at the price of a charge-conjugation twist in the dictionary. If per-sector S-duality is an exact equivalence, MP is the standard mirror parity written in a dual frame of the mirror sector; if it is not, as the running couplings and the $B+L$ anomaly of the chiral Standard Model indicate, MP is not a symmetry. The duality maps ordinary Wilson lines to mirror 't~Hooft lines; for example, the electron's line goes to the minimal pure-hypercharge monopole line of the mirror sector. In the mirror's S-dual frame, MP forces the mixed θ-angle of that frame to vanish but leaves the kinetic mixing in that frame free. We work out the consequences for mirror-matter phenomenology: the running of the couplings, the selection rules on the operators that couple the two sectors, and neutron--mirror-neutron oscillations. On the Hitchin moduli spaces of the two sectors, the duality acts, at the level of lattices, as the exchange of the sectors composed with fiberwise T-duality.