Three-Body Holonomy as a Toy-Model Mechanism for Family Triplication and Mass Hierarchy in (1+1) Dimensions
We investigate the phenomenological consequences of a genuine three-body holonomy in a relativistic Dirac system in one spatial dimension. After removing the center-of-mass coordinate, the triple-coincidence point punctures the two-dimensional relative configuration space and permits a nontrivial $U(1)$ winding phase, whereas the pairwise Sakamoto--Munakata--Ino contact interactions give trivial net matching around this point. At fixed intrinsic parity, the six particle-ordering sectors reduce to a three-dimensional cyclic space. A Hermitian $C_3$-invariant effective mass operator admits a Peierls-type realization in which the gauge-invariant phase around the three links equals the three-body holonomy $θ_3$. Its eigenvalues are $$ M^{[k]}=M_0+2Δ\cos\!\left(\frac{θ_3+2πk}{3}\right), \qquad k=0,1,2. $$ The holonomy lifts the conjugate-channel degeneracy and can generate a parametrically light branch through cancellation between the common mass and the holonomy-induced shift. We further allow cyclic-symmetry breaking and consider a general Hermitian three-state mass matrix. Exact elimination of two heavy states by the Schur complement yields a low-energy correction containing the rephasing-invariant loop product $\mathrm{Re}(t_{12}t_{23}t_{31})\propto\cosθ_3$. Thus, even when only one branch is kinematically accessible, its effective mass can retain finite memory of the complete three-state loop. The complementary invariant $\mathrm{Im}(t_{12}t_{23}t_{31})\propto\sinθ_3$ is phase sensitive but does not alone imply CP violation. The construction provides a low-dimensional phenomenological proof of concept for family-like triplication, mass hierarchy, and infrared memory, rather than a microscopic theory of Standard Model fermion generations.