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lower signs respectively. For small α we can write ap- proximately 

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We define the spin-voltage V$_{X}$ (or V$_{Y}$ ) as the differ- ence between the voltages measured in the parallel and the anti-parallel configurations by X- (or Y-) directed in- jecting and detecting magnets. This is expected to be twice that measured in the parallel configuration using a setup like the one shown in Fig. 1(a) [10] which is ex- actly how the oscillatory signals V$_{X}$ in [1] are measured, although the spin-valve signal V$_{Y}$ is measured the same way our V$_{Y}$ is defined. It is shown in Appendix A (sup- plementary information) that for a point injector located at x = 0 and a point detector located at x = L , as- suming ballistic transport, the voltage signals for X- and Y-directed magnets can be written as ( B : constant) 

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where s ≡ k$_{Y}$ /k$_{0}$ = k$_{Y}$ $_{/}$√$_{2}$$_{m}$ $^{∗}$E and θ$_{L}$ = 2 m $^{∗}$αL/ $^{2}$. The contributions from different E , k$_{Y}$ all act ‘in parallel’ giving a voltage equal to the average. At low tempera- tures we can average the contributions from all transverse wave-vectors k$_{Y}$ over the Fermi circle ( E = E$_{F}$ ) to write 

Interestingly, the results obtained from the integration in Eq. 6 looks almost exactly like the single cosine re- sult in Eq. 1 that describes the experimental observa- tions. This can be understood by noting that the argu- ment θ$_{L}$$_{/}$√$_{1}$ − s 2 has a stationary point at s = 0 [11] and we can use the method of stationary phase to write approximately, 

As shown in Appendix B (supplementary information) these approximations describe the results from the exact integration quite well for θ$_{L}$ ≳ 2 π which is true for the range of α and L involved in the experiment. This should help answer some of the concerns raised in a recent com- ment [12]. 

Let us note that the simple results presented above are made possible by our assumption of periodic boundary conditions (PBC) in the y-direction making k$_{Y}$ a ’good quantum number’ like E . Most of the prior work, on this topic [13], uses hardwall boundary condition (HBC) which does not seem to permit the simple decoupling of different transverse wavevectors ( k$_{Y}$ ) due to non-trivial ‘boundary scattering’. We have checked numerically that the use of HBC does not change the conclusions described above in a significant way although some details are dif- ferent. Furthermore, one could argue that since the ac- tual boundaries in the experimental structure of [1] are 

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