Abstract: 

We establish a non-definability result in the philosophy of representation: in state-space formalisms satisfying orientation-reversal equivariance, no internal function can recover process-level role distinctions that depend on traversal orientation. In the canonical minimal embedding Z₁₂ ⊂ U(1) ⊂ SU(2), the element U = i·σ_x ∈ SU(2) exchanges the two non-identity cosets C₁ and C₂ while fixing the Pauli-eigenstate subgroup C₀ = {0,3,6,9}. Assigning distinct processual roles to these cosets therefore requires an orientation parameter τ not recoverable from the kinematic state space alone. This construction is independently verifiable using standard group-theoretic methods.

The result is grounded in two independently developed systems admitting the same embedding structure: the discrete cyclic grammar of the Chinese Yi-xue formal system and the equatorial orbit of the Bloch sphere in single-qubit quantum mechanics. We use the terms process-ontological and state-ontological as technical characterisations of descriptive strategies: the former labels grammars whose primitive units are directed cyclic stage-labels, the latter labels formalisms whose primitive units are positions in a state space. These terms are deployed as methodological labels for distinct representational strategies, not as metaphysical claims about the nature of process or state as such. These two systems share independently verifiable algebraic correspondences constituting partial isomorphisms in Bueno's (1997) sense; the non-definability result is independent of any interpretive coset-role identifications.

The result supplies a precisely located, formally tractable case for the philosophy of representation: within state-description registers satisfying the stated symmetry constraints, orientation-dependent processual distinctions are not recoverable by any internal function. This is a relative non-definability result, not an absolute incommensurability claim, with implications for debates concerning the representational limits of state-based formalisms when applied to constitutively directed process grammars.