Uniform Mixing in Barrington Branching Programs

We study the output distribution of a fixed valid balanced transposition-template Barrington branching program for a point function on uniformly random inconsistent inputs. We prove exact target-independence by a combinatorial flip bijection and negligible statistical distance from uniform on S_5 by finite-group Fourier analysis. A regular simplex describes the level-2 obstruction in the standard representation; the commutator structure breaks it at level 3. An exact integer certificate gives the uniform contraction constant 0.0355, and induction amplifies the contraction at higher levels. We also identify dimensional obstructions beyond S_5: the maximum level-3 norm is 1 for S_6 and S_7, although some configurations contract, and every level-3 configuration has norm 1 for n at least 10. The present analysis does not decide S_8 and S_9. Fresh independent terminal-output samples are exactly target-independent and remain negligibly close to uniform for polynomially many samples. These properties survive target-independent post-processing.