Topological Consistency Conditions in Modal Triplet Theory: Internal Indices, Charge Lattices, and Holonomy

Topology-only obstructions are separated from representation checks and from physical source theorems. Chiral multiplicity is an index on the compact internal six-manifold X6, not on physical spacetime. For a special-unitary bundle over X6, the spin-Dirac index is one half of the integrated third Chern class. The current q79 branch constructs smooth non-pullback bundles with integrated third Chern class plus or minus six, and therefore index plus or minus three, but its holomorphic gerbe lift, balanced Hermitian-Yang-Mills connection, and differential Bianchi representative remain open. Abelian charges are integer characters with a separately declared physical normalization. A linear difference-charge construction is only a reconstruction after observed charge differences are inserted. A separate finite-branch theorem is stronger: on the selected chiral carrier, anomaly constraints select the primitive relative-charge vector 6Y=(1,-4,2,-3,6,0) and the faithful Standard Model gauge group quotient by Z6. We prove the resulting anomaly checks, the correct character criterion for a bare Majorana bilinear, the distinction between topological, holomorphic, equivariant, flat, and holonomy trivializations, the conditional common-principal-symbol result for photon and graviton cones, and the correctly normalized one-loop beta coefficients. Peccei-Quinn dynamics, particular baryon/lepton operator exclusions, physical family realization, and neutrino ontology remain conditional or open at the precisely stated gates.