Orientation-doubling positional channel

Use this in a graph transformer, sequence model, or state-space model when inputs have a natural reversal operation, such as cyclic sequences, molecular bond directions, road or trajectory graphs, toroidal grids, or bidirectional scans. For every node or token v, construct a forward code C_v^+=(f(v\cdot y_1),...,f(v\cdot y_k)) and an inverse code C_v^-=(f(v\cdot y_1^{-1}),...,f(v\cdot y_k^{-1})); on an ordinary directed graph, obtain the inverse view by reversing the canonical walk and swapping directed edge slots. Add an orientation bit b\in{0,1}, represented by a learned two-vector embedding e_b. Encode z_v=MLP([Embed(C_v^+),Embed(C_v^-),e_b]) and use z_v as the positional input to attention or message passing. Share the encoder between original and reversed examples. During training, create a reversed augmentation, exchange C^+ and C^-, flip b, and add a consistency loss lambda||z_v^{original}-z_v^{reversed}||_2^2 when the task is reversal-invariant; omit or reverse the sign of this loss for direction-sensitive tasks. The paper’s exact construction is f(a)=h(a), f(a tau)=h(a)+1, with the forward word W(a) and reversed word W(a tau) shown above. In the neural adaptation, h is represented by learned q-way logits and the additive one is implemented as the orientation embedding or a reserved binary symbol. First test on synthetic cycles and dihedral grid graphs where labels depend on direction, comparing standard positional encodings, two independent directional encoders, and this shared encoder. Measure reversal-confusion accuracy, parameter count, and task accuracy. Success is lower confusion and equal or better accuracy with fewer directional parameters.