Banya AI 7: The Rotation-Operator Gate - Reading the Operation Plane as Euler Rotations for Depthless Logic Composition

This paper starts from two expressive limits of the bi-token gate of Part 3. The sigmoid gate's range of 0 to 1 expresses only passing and blocking and cannot express negation (a sign flip) in one hop, and the operation-plane components act diagonally, each adjusting only its own channel, so there is no rotation. This paper proposes the rotation-operator gate, which reads the operation plane as Euler rotation angles. The grounding is the factorization of the fractional derivative (differentiation generalized to a real order): in the frequency domain, (iω)^α splits into the product of the amplitude |ω|^α and the Euler rotation e^(iαπ/2), and it is the rotation that carries the logic. In the rotation gate, negation becomes a primitive operation of 180 degrees, the composition of operations becomes angle addition, the norm is preserved, and the adjoint (the transpose operation that flows gradients backward) comes out in closed form as the conjugate rotation R(-θ). In particular, the angle gradient closes as the two-dimensional cross product of the incoming credit and the rotation result, so the forward-pass input never needs to be stored. This adjoint was verified against finite differences with an inner-product identity difference of 0 and a median gradient ratio of 1.000000. The measurements were made with depth-excluded discrimination, restricting depth to one block so that the gate is the only remaining variable. On a task of reading three kinds of numbers and stating the mod 3 state of the running sum, teaching only 14 of the 27 length-3 combinations and hiding 13, an ablation contrast that swaps only the gate on the same one-block engine splits the ranking. Self-feedback accuracy on the hidden combinations is 100 percent on all three seeds for the rotation gate and the tanh gate, 90 percent for the sigmoid gate, 62 percent with the operation plane blocked, and 49 percent for a standard one-layer transformer (chance 33 percent). At the never-taught length 6, the rotation gate is highest at an average of 82 percent with one seed reaching 100 percent, while the standard one-layer collapses to chance. Convergence is also fastest for the rotation gate at an average of 317 steps, with the smallest variation across seeds. Since mod n addition is isomorphic to the discrete rotation group that divides the circle into n parts, the algebra of this task is rotation itself, and the reason this discrimination is deliberately biased is that no order-3 operation exists in a real scalar gate. We state explicitly the limitation of a small synthetic task and that training at the published model size remains a next task. This record contains the English edition (_en.pdf) and the Korean original (_kr.pdf). Part 7 of the Banya AI research series, derived from Part 3. Part 1: https://doi.org/10.5281/zenodo.21385447. Part 2: https://doi.org/10.5281/zenodo.21385516. Part 3: https://doi.org/10.5281/zenodo.21385676. Part 4: https://doi.org/10.5281/zenodo.21385699. Part 5: https://doi.org/10.5281/zenodo.21385717. Part 6: https://doi.org/10.5281/zenodo.21385740. All measured figures are reproducible with the public code package: https://ubmscoin.github.io/banya/banya_ai_paper_code/ (code Apache-2.0). The reproduction probe trains from scratch on a small synthetic task; no model checkpoints or corpus downloads are required.