Flagged Reciprocal-Pair Positivity for Boundary-Absorbed Schur-Minor Gates

This paper extends the boundary-absorbed Schur-minor positivity mechanism introduced in the preceding work from endpoint principal specialization to reciprocal-pair deformations. The central object is a signed Schur gate G_n = (n − 1)T + Σ_{r=0}^{n−2}(C_r + D_r − 2A_r), where A_r, C_r, D_r, and T are three-column Schur families. The preceding paper showed that the local defects C_r + D_r − 2A_r are not individually positive, and that positivity is instead restored by the boundary reservoir (n − 1)T. The present paper proves that this boundary-absorbed positivity survives two independent reciprocal-pair insertions. For two reciprocal pairs, written u = cosh(a) − 1 and v = cosh(b) − 1, the paper shows that…