Abstract: 

We establish a non-definability result in the philosophy of representation. When a twelve-element cycle is represented by equatorial qubit states, no function of the state that respects the symmetries of the representation can recover process-level role distinctions that depend on traversal orientation. In the embedding Z₁₂ ⊂ U(1) ⊂ SU(2), the element U = i·σ_x ∈ SU(2) fixes the Pauli-eigenstate subgroup C₀ = {0,3,6,9} setwise and exchanges the two non-identity cosets C₁ and C₂ of C₀. Distinct processual roles for these cosets therefore require an orientation parameter τ that the symmetric kinematic structure does not supply.

The result is illustrated by two independently developed systems: the cyclic grammar of the Chinese Yi-xue formal system, whose twelve branches carry a traditional three-tier classification, and the equatorial orbit of the Bloch sphere in single-qubit quantum mechanics. We use the terms process-ontological and state-ontological as methodological labels for descriptive strategies, not as metaphysical claims. Under an explicit identification of the branches with equatorial qubit states, the four cardinal branches correspond to C₀ and the other two tiers to its cosets. This is a partial isomorphism in the sense of partial structures, and the processual roles of the cosets lie outside the shared structure.

The non-definability result is relative, not an absolute incommensurability claim: once τ is supplied, the mapping is complete. It does not depend on any interpretive identification of the roles, and it bears on the representational limits of state-based formalisms for constitutively directed process grammars.