Curved Boolean logic (CBL) generalizes propositional logic by allowing local truth assignments that do not extend to a single global evaluation, similar to curvature in geometry. We present a conservative, context-aware proof computation within the flat bound, with equivalent bundle and exclusive graph semantics. We formalize CBL-SAT and its underlying complexity (generally NP-complete), and present operators (CBL-AC and CBL-CONS) that eliminate contradictions early on in existing hardware. We model noise with iid, AR(1)-correlation, and adversarial boundary perturbations, and provide permutation-based significance with Benjamini-Hochberg FDR control. We provide a Colab-ready notebook (Supplementary Files) that reproduces all figures and statistics. We place CBL in relation to KCBS, CSW, and bundle frameworks, and briefly discuss the link between SAT/CSP and robustness/adapter stability in large-scale language models.