PAPER-FBT07A: Three Semiclassical Sector Classes on the Dual–Phase Torus

A central structural question of the Standard Model is why matter appears in exactly three generations. In this paper we give a mathematically sharpened flavour-sector formulation of the Fracture–Berry–Tension (FBT) programme.

The starting point is the reduced dual-phase sector of the FBT framework, modelled by a compact symplectic torus

Σ2 ∼= T2.

Its distinguished discrete symmetry is not imposed ad hoc but arises from the reduction of three primitive phase channels modulo the diagonal phase. In the minimally symmetric realisation this yields the Weyl symmetry

W(A2) ∼= S3.

The key conceptual point is that the compact torus itself carries only the real Morse skeleton, whereas the convergent oscillatory integral is defined only after passing to a suitable complexified neighbourhood X. Accordingly, we distinguish:

(i) an S3-equivariant real Morse function F : Σ2 → R, whose minima determine stable basins on the compact dual-phase torus;

(ii) an analytically continued oscillatory action S : X → C, understood in the thimble sense established in the FBT06 dual-phase quantisation bundle;

(iii) admissible Lefschetz thimbles in X, which supply the actual convergent semiclassical
integration cycles.

Within the minimal S3-equivariant admissible class, we prove that there are exactly three stable admissible sector classes. These are permuted transitively by S3, are spectrally isolated at the semiclassical level, and provide the natural geometric precursors of a threegeneration flavour interpretation. We further show that their depth differences generate an exponential semiclassical hierarchy, while saddle-mediated tunnelling yields a nearestneighbour mixing pattern.

The scope of the result is intentionally precise. We do not derive chiral zero modes, explicit Standard Model representations, or full Yukawa matrices. Rather, we establish that a unique three-fold stable semiclassical sector geometry emerges from the dual-phase structure, and that this geometry provides the correct minimal precursor for a three-generation flavour interpretation.