An AI-Assisted Solution to the Signed BAR Conjecture: Uniqueness in the Harrison–Reiman Class and a Completely-S Class Obstruction

This paper addresses whether the basic adjoint relationship (BAR) uniquely characterizes stationary distributions for multidimensional reflected diffusions, a problem open for more than 35 years. It proves finite signed uniqueness for stable Harrison–Reiman data with a nonsingular M-matrix reflection matrix. The argument uses pathwise differentiability, tangent projection at boundary faces, and smoothed resolvent test functions to establish invariance. The paper also shows that the M-matrix condition is structural: in the broader completely-S class, singular proper principal blocks generate nonzero zero-mass signed BAR tuples, with an infinite-dimensional interior subspace. The proof was discovered with assistance from ChatGPT 5.5 Pro and later verified by the authors.