Topology-Only Constraints in Modal Triplet Theory: Indices, Anomalies, and Line-Bundle Triviality

We establish the Tier-1, topology-only consequences of Modal Triplet Theory (MTT) using standard tools of index theory, group representation theory, and line-bundle topology, without invoking internal metrics, harmonic analysis, or spectral gap assumptions. On a smooth, oriented spin four-manifold equipped with three determinant line bundles obeying a flux-balance condition, we derive a series of exact results. These include: chiral family numbers from a Dirac index formula; a sharp topological criterion distinguishing Dirac from Majorana masses; absence of the SU(2) global (Witten) anomaly for three families; exact Standard Model hypercharges from difference-charge assignments; cancellation of all local gauge and mixed gravitational anomalies per family; a Peccei–Quinn–like mechanism for strong-CP relaxation with integer domain-wall number; generic forbiddance of many baryon- and lepton-number–violating operators by line-bundle triviality; equality of photon and graviton propagation speeds in the GR-compatible effective field theory; and a holonomy determinant / phase sum rule implied by canonical bundle trivialization. All results follow purely from topology and representation bookkeeping and are independent of geometric moduli or dynamical assumptions. They provide a rigorous, metric-free foundation on which higher-tier geometric and calculational results in MTT are built.