Main result. The paper proves an asymptotic Γ‑Bayer–Macrì–Toda (Γ‑BMT) inequality on every smooth polarized complex threefold. Set D = 1/2·K_X (half the canonical class). There exist a positive number t0 and a fixed correction class Γ in N^2(X)_Q such that for every ν^D_{b,w}‑semistable object E with w − 1/2·b^2 ≥ t0 and ν^D_{b,w}(E) = b one has ch^{3}_{D+bH}(E) ≤ (w − 1/2·b^2)/3 · H^2·ch^{1}_{D+bH}(E) + Γ·ch^{1}_{D+bH}(E). Here ch^i denotes the i‑th Chern character component (a standard topological invariant of E). Both t0 and Γ can be chosen independently of the parameters (b,w) and of E. (F1, F11)

How it was proved. The proof reduces the problem to small “tilt‑stability” parameters. It adapts the technique of Bayer–Macrì–Stellari (BMS16) to a recent Cheng–Feyzbakhsh inequality. This yields the same leading coefficient predicted by the original BMT conjecture, but with a fixed correction term Γ. The paper uses the tilt‑stability parameterization from Feyzbakhsh–Thomas (FT23). (F2, F3, F10)

Concrete features. The leading coefficient in the inequality was improved to match the BMT conjecture. The improvement comes at the cost of the explicit Γ correction. The paper gives explicit choices for t0 and Γ in its lemmas and theorem statements. (F3, F11)

Applications. If the canonical class is numerically proportional to the polarization (K_X ≡ λH for some rational λ), then the polarized threefold (X,H) satisfies Feyzbakhsh–Thomas’s condition (wBG). (F4)

Calabi–Yau consequence. For polarized Calabi–Yau threefolds (K_X ≅ O_X and H^1(X,O_X)=0) the asymptotic Γ‑BMT result implies a reduction of generalized Donaldson–Thomas (DT) invariants. The paper first reduces positive‑rank DT invariants to invariants counting sheaves pure in codimension one; see Corollary 5.1. It then expresses those rank‑zero (codimension‑one) invariants in terms of rank‑one invariants; see Proposition 5.5 and Theorem 5.6. Concretely, the paper states J(v) = P_v(J(β_1),…,J(β_m)) where ch_0(β_i) = 1. The paper states J(v)=P_v(J(β_1),…,J(β_m)). (F5, F6)