Weighted RT background interface for Mod(C)

Suppose a modular ribbon fusion category $\mathcal{C}$ satisfies the hypotheses of The weighted Reshetikhin–Turaev TQFT and modular-functor factorization, including its weighted-Lagrangian bordism convention and proved state-space, composition, and factorization maps. The weighted RT background interface for the present page means using those TQFT state spaces and bordism maps as a separate 2+1-dimensional mathematical background alongside the algebraic defect 2-category $\mathrm{Mod}(\mathcal{C})$. The interface asserts no canonical functor from a module category, module functor, or module-natural transformation to a physical defect, and no classification of 3+1-dimensional phases.

The actual supplier The weighted Reshetikhin–Turaev TQFT and modular-functor factorization now proves the full marked weighted-Lagrangian RT functor and geometric factorization, including presentation/coupon independence, exact cap normalization, signature–Maslov composition, cylinders, duality, disconnected and empty cases, and both sewing inverses. Its weight action is ${\kappa }^{+n}$ for the prescribed $+{\mu }_W$ composition, with the actual cap-reference correction for arbitrary real object Lagrangians. This establishes the background used in the Definition for every category satisfying the stated modular hypotheses. Full AC enters through the boundary duality/topology suppliers and countable choice through framed coupon evaluation. The algebraic $\mathrm{Mod}(\mathcal{C})$ construction remains independently defined; no canonical physical realization or phase classification is inferred.