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 + Σᵣ₌₀ⁿ⁻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 reciprocal-pair insertion is governed not by ordinary row-lowering powers but by a flagged Jacobi-Trudi expansion with decreasing row alphabets m, m − 2, and m − 4. This produces flagged determinants indexed by row-choice vectors s ∈ {0,1,2}^3. The main result proves that, for m ≥ 2n − 1, G_n(e^a, e⁻ᵃ, e^b, e⁻ᵇ, 1ᵐ⁻⁴) ∈ R>0[u,v]. The proof reduces the three middle coefficients to orbit-reservoir identities and shows that each admits a positive hook-content skeleton together with a canonical residual polynomial with positive coefficients. The paper also proves positivity of the first fully mixed three-pair gate, establishing the first layers of a reciprocal-pair positivity hierarchy. This work supplies a finite algebraic framework for a future Jensen-Hermite bridge toward Laguerre-Pólya phenomena. It does not claim a proof of the Riemann Hypothesis, a general Jensen-polynomial hyperbolicity theorem, or a Laguerre-Pólya-class criterion.