An Application of Modal Logic to Costello-Gwilliam Factorization Algebras

<p dir="ltr"> We present a framework for understanding locality and observables in quantum field theory via factorization algebras enriched with modal logic. By interpreting open neighborhoods on a Calabi–Yau manifold as epistemic contexts, we define a sheaf-valued truth assignment $\Omega$ over local frames and introduce a \emph{contextual Laplacian} $\Delta^{\mathcal{O}_i}$ to capture the failure of global logical coherence under modal transitions. These constructions extend naturally to configuration spaces such as $\operatorname{Conf}_2(\mathbb{C}^3)$, which serve as base geometries for (p,q)-string dynamics. Using operadic data $\mathcal{O}_i = \{\mathcal{M}, U_i, V_i\}$, we formulate the evolution of observables as morphisms between contexts \code{C}$_i$ and define epistemic curvature in terms of homotopy-invariant deviations of local Lagrangian density. A worked example illustrates how this machinery applies to NS--NS and RR field deformations near intersecting branes. Our approach yields a stratified, context-sensitive geometry for quantum observables, opening a new path toward modal interpretations of field-theoretic and string-theoretic dynamics.</p>