Metric Transport and Exact Contraction Coefficients for Jensen–Shannon and Related Midpoint Divergences [Erratum: See Note on Channel-Only Universality]

A unified metric-transport framework for strong data-processing inequalities of midpoint divergences under depolarizing and erasure channels. All claims fully proved. **MT.pdf** — Main paper. Transport class TMT, universal depolarizing contraction formula, binary threshold, interior optimality for binary-input channels, Jensen–Rényi exit via Hankel obstruction. **A1.pdf** — Admissibility and well-posedness of TMT. Membership proofs for triangular discrimination and arithmetic–geometric divergence. Full Sturm certificate. Sharpness counterexamples. **A2.pdf** — Jensen–Rényi SDPI on the ternary simplex. Peano-kernel representation, sharp positivity threshold, exact factorization theorem, two-branch conjecture. **A3.pdf** — Proof of the two-branch conjecture. Symmetry-breaking exclusion confines maximizers to edges and the two-equal stratum. **A4.pdf** — Strict positivity of the alternating-branch derivative at crossings. Pure analysis via unimodal factorization and a sharp integer inequality. **A5.pdf** — Unimodality of the auxiliary factor on the two-equal stratum. Unique critical point and correct vertex-limit computation. **A6.pdf** — Universal erasure law. Dimension-free pointwise contraction identity for all TMT divergences. Elementary entropy proof for quantum JSD. Strict separation from chi-squared SDPI. ERRATUM (March 2026): The formula η = (1−p)²/s, the TMT framework, factorization principle, and Jensen–Rényi boundary (α_H ≈ 1.134) are all correct. However, the "binary threshold" / "enhancement" claims (Theorem 4.1(a)–(b): η_D > (1−p)² for d_eff ≥ 3) are misleading: channel-only η_χ² also equals (1−p)²/s by the same χ²-SDPI lemma (Lemma A.3). The formula is universal for all f-divergences under depolarizing channels, not specific to TMT. A6 claim "Strict separation from chi-squared SDPI" for erasure channels requires independent verification.