This is genuine 2D measurement-based computation: an actual two-qudit circuit U = ∏ CZ_dlink·(J_d(αA)⊗J_d(αB)) carried out entirely by measuring a 2D qudit cluster one qudit at a time; no logical gate is ever applied to the data. Two horizontal rails give single-qudit gates; vertical CZ_d links between them supply the entangler. Generalized from qubits to d-level systems, d = 2, 3, 5. No φ: measurement × entanglement physics.
The upper chart is the control. The entangling power of the realized circuit (the entanglement of U applied to a product input) is nonzero with the vertical links (up to the maximum ln d) and identically zero without them: strip the links and the cluster factorizes into two independent single-qudit one-way computations. The link is doing the 2D work.
The lower chart is the surprise of the one-way model. Every run measures the cluster in a random order of outcomes, so the leftover byproduct (the local generalized-Pauli correction read off the measurement record) is different on every branch (the scatter). Yet after that classically-controlled correction the output matches the target with fidelity 1.000, on every single branch. The existence of such a local correction, for every outcome, is the one-way computation theorem. Computed live in your browser via WebAssembly.