Bianchi permutability for sine-Gordon Bäcklund transformations ($\tan [(u_{ab}-u_0)/4]=\frac{a+b}{b-a}\tan [(u_b-u_a)/4]$)

Two compatible Bäcklund steps commute after integration constants are matched. The displayed superposition relation constructs their common output algebraically from the seed and the two one-step outputs, in the reciprocal-parameter convention of the Sine-Gordon Bäcklund transformation. It uses the angular field $u=\beta \varphi$. Smooth inverse-tangent branch continuation is required to retain the correct vacuum labels.