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

We study a discrete dynamical system on the three-strand braid group B_3 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 S_3 and a framing vector f in Z^3. 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 A_2 root lattice under the order-3 Coxeter rotation, equivalently an Eisenstein integer modulo multiplication by the cubic unit subgroup. A Weyl reflection of the A_2 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 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 RP^3 topology and Skyrmion physics within the Topological Inversion Model (TIM).