Wilson-line Moduli, One-loop CP Locking, and the Boundary Origin of the CP-odd Invariant in Rank-2 Orbifold Gauge Theories

Can the geometry of a five-dimensional gauge theory on an interval — the boundaryprojections and the gauge sector alone — supply both the Wilson-line moduli and the CP phasethat the CCBS mechanism requires? This paper answers that question for the rank-2 simplegroups, and the answer separates cleanly into what the geometry does and does not provide. An exhaustive scan establishes a strict no-go: no pair of commuting involutive boundaryprojections in su(3), so(5) or g2 leaves exactly two commuting zero modes of the fifth gaugecomponent. For g2 the result is unconditional, since the group has a single conjugacy classof Klein four subgroups. At the kinematic level the obstruction is evaded onceholonomy-eaten modes are counted, and two candidate physical moduli survive in everyadmissible configuration. Among the three groups SO(5) is then singled out by a messengercriterion: the weights of its vector representation under the surviving Cartan pairreproduce the required messenger charges natively, which neither su(3) nor g2 does. The central result is negative and structural: the one-loop Wilson-line vacuum is locked atCP-conserving points, by four mechanisms that are independent of one another — rootalignment for B2, a quaternionic projective lift that renders the spinor representationinert, a trivial center for g2, and center-image protection for folded su(3) triplets.Boundary deformations of single-harmonic form do not unlock it. The CP phase must therefore enter through the boundary. Constructing the exact defectdeterminant for bulk fermions with brane-localized mixing yields an identity showing that,at the level of that determinant, charge-conjugation acts as a discrete flip of one brane'scoupling pair. The CP-odd datum is accordingly a discrete boundary orientation, whosesignature is a doubled periodicity of the spectrum. The associated transport invariant isbuilt from crossed inter-brane channels only, vanishes for the CP-even boundary class, andis maximal precisely at the locked vacuum — a coexistence that the doubled cover explains —with a geometric transport efficiency of order unity for infrared modes. The relative energyof the two orientations is shown to be an ultraviolet datum, which removes the naivedomain-wall obstruction at the cost of leaving the sign of the asymmetry unpredicted. Three apparent failures — the strict no-go, the locking, and the absence of spontaneous bulkCP violation — are thereby recast as complementary properties of one construction: theinterval geometry supplies the moduli, the charge segregation and the transport, while theCP-odd datum is boundary data. Numerical methods and two retracted intermediate results aredocumented in appendices.