We will next demonstrate the impossibility of processes of the form (32313). Immediately after the first collision, in the frame of 1, 2 is moving away from 1 and 3 is moving toward 1. Let the velocity of 3 be w and the velocity of 2 be v exp( iα ) . We view the next collision, (2), in the frame of particle 1. After the collision (2), 2 needs to have its velocity shifted by a phase of π so that it changes from an outgoing particle to an ingoing particle. The collision matrix for collision (2) is: As mentioned, we need to have the velocity of 2 shifted by a phase of π . This requires that the term 2 bw sin( θ/ 2) exp[ i ( θ + π ) / 2] / ( a + b ) have opposite phase from v exp( iα ) and be larger in magnitude than v . For the phase of the correction term to be right, we need θ = 2 α + π . Plugging this into Eq. A11 gives the velocities as: and for the velocity of particle 2 to be turned around, we need: We will analyze the third collision by working in the frame of 1 before the collision and by working in the frame of 2 after the collision. That way, before the collision, 3 is moving away from 1 and 2 is moving toward 1 and after the collision, 1 is moving away from 2 and 3 is moving toward 2. At the moment of the third collision, 2 replaces 1 at the origin. Thus before the collision, 3 is moving away from the origin and after the collision it is moving toward the origin. So we need the collision to change the phase of particle 3 by π . The matrix for collision (3) in which we work in the frame of 1 before the collision and the frame of 2 after the collision is: 31   − exp( iφ ) 0 − (1 + a exp( iφ )) / (1 + a ) 1   . (A14) v < 2 bw cos α a + b . (A13) (( v − 2 bw cos α/ ( a + b )) exp( iα ) $^{,}$−w exp(2 iα )) T (A12) ( v exp( iα ) + 2 bw sin( θ/ 2) exp[ i ( θ + π ) / 2] / ( a + b ) , w exp( iθ )) $^{T}$. (A11)   1 2 b a + b sin( θ/ 2) exp[ i ( θ + π ) / 2] 0 exp( iθ )   (A10) Before the second collision, the velocities of 2 and 3 are ( v exp( iα ) , w ) $^{T}$. If we apply the above matrix to this vector, we find the velocities after the second collision: