Bundle Selection Rules, Anomaly Lines, and Charge Lattices A Conditional Topological Layer for Modal Triplet Theory

Abstract

Topology can constrain a four-dimensional effective theory, but only after the global gauge group, field representations, internal bundles, zero-mode spaces, coefficient bundles, and contraction maps have been specified. This paper replaces an earlier topology-only argument by that typed statement. For a fixed global gauge group, characters of its U(1) factor form an integer lattice; rational hypercharges arise after a normalization choice and quotient compatibility conditions. This lattice does not by itself select the observed matter representations. Gauge anomalies are encoded by the determinant or Pfaffian line of a family of chiral Dirac operators over background-field space, rather than by the determinant of the matter bundle on spacetime. Local anomaly is detected by curvature and global anomaly by holonomy; cancellation requires a compatible local equivariant trivialization. For effective operators, we give the exact tensor-product line classes of the Yukawa, Weinberg, QQQL, u-c u-c d-c e-c, and singlet-Majorana monomials. The two baryon-number violating examples are Standard Model gauge singlets and, in nonsupersymmetric SMEFT, have mass dimension six. They are excluded only if a declared realization supplies an additional bundle, symmetry, cohomology, or overlap obstruction. We prove a sufficient bundle-selection theorem and explain why passing its tests does not guarantee a nonzero coupling. Current Modal Triplet Theory (MTT) has exact finite results for a selected chiral representation, its anomaly table, the faithful (SU(3) x SU(2) x U(1))/Z6 group, and a unique anomaly-free shared hypercharge direction within the chosen finite completion. Those results do not yet select the physical compactification endpoint or prove that every dangerous operator is absent. The result is a rigorous realization-by-realization selection framework, not a universal topology-only derivation of the Standard Model.