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The honeycomb magnet α-RuCl₃ is proximate to the Kitaev Hamiltonian. In an in-plane magnetic field B||a (at low temperatures T) a quantum spin liquid (QSL) is sandwiched between the antiferromagnetic (zigzag) state (0 to 7.3 T) and a spin-polarized state (B>11.6 T). High-resolution measurements of the thermal conductivity martix κ_ij (from 0.3 to 6 K, with B||x, -▽T||x and x||a) have uncovered oscillations in κ_xx. The amplitudes are maximal in the QSL regime. Above 11.6 T, they collapse abruptly to zero, wheres below 7 T, they survive as a weak remnant to 4 T. The oscillations are periodic in 1/H_a (H_a is the component of B along a) with a break-in-slope near 7 T. Across samples, phase and periodicity are closely reproducible, with the largest amplitudes observed in the purest crystals (benchmarked by the plateeau value of κ_xx at 12 T and 1 K). The oscillations suggest Landau level quantization despite the rigorous absence of free electrons (the charge gap is 2 eV). An intriguing possibility is that the oscillations arise from a quantized neutral Fermi surface in the QSL state. The planar thermal Hall conductivity κ_xy is also of strong interest. When B||b, we confirm that κ_xy is zero (to a resolution of 20 μK at 0.4 K) in accord with mirror symmetry. If B||a, however, an unusual Onsager-type planar thermal Hall conductivity emerges above ~3 K and grows very rapidly to ~10 K. We do not observe a quantization plateau at any T. Instead, the steep variation with T implies that the heat carriers are bosonic instead of fermionic (this rules out quantization). We show that κ_xy arises from spin-excitation edge-modes driven by a Berry curvature Ω. A fit to bosonic modes yields an energy scale (1 meV) that varies with B in agreement with sharp modes seen in spin resonance experiments. Moreover, the overall magnitude of κ_xy yields a Chern number C~1 consistent with the edge-mode boson model.