Canonical Classication of Framed Closures in a Three-Strand Braid Transfer Model

We study a discrete dynamical system on the three-strand braid group B₃ in which generators act on an integer-valued framing vector through a local slot-based transfer rule. Terminal states are pairs (P, f) consisting of a permutation P in S₃ and a framing vector f in Z³. Under trace closure, strands belonging to the same permutation cycle form a single loop, and physical states are defined modulo cyclic relabeling of the starting point of each loop. We prove a complete classification theorem: two terminal states are physically equivalent if and only if a finite sector-dependent invariant I agrees. The invariant decomposes according to the cycle type of P into three sectors (identity, transposition, three-cycle), and every equivalence class admits a unique canonical representative computable by a constant-time algorithm. A key structural finding is that the three-cycle sector carries a discrete cyclic-ordering obstruction not reducible to symmetric polynomial invariants of the framing differences. The three-cycle invariant is identified with an orbit in the A₂ root lattice under the order-3 Coxeter rotation, equivalently an Eisenstein integer modulo multiplication by the cubic unit subgroup. A Weyl reflection of the A₂ lattice is shown to reproduce the weak-isospin exchange pattern on the particle dictionary while fixing all right-handed states, and a no-go theorem proves that braid words (root translations) cannot achieve this chiral selectivity. The Z₃ cyclic quotient in the three-cycle sector is interpreted as colour averaging: at the colour-resolved level the Weyl reflection provides a chiral isospin operator for all first-generation states, while the colour-averaged level describes colour-singlet observables. The analytic proofs are independently verified by exhaustive BFS enumeration of 631 terminal states within word length 10. This work provides the classified-state foundation for a broader programme connecting combinatorial braid kinematics to RP³ topology and Skyrmion physics within the Topological Inversion Model (TIM).