Agentic AI systems routinely transport conclusions across biological, clinical and financial contexts, and the emerging safeguard is local verification: checking at each step that the entity is representable in the chosen tool, that parameters are compatible, and that outputs cohere with the plan. We prove this class of safeguard is structurally incomplete. Modelling a covering of context space by its nerve and evidence by a real-valued 1-cochain, an agent chaining evidence performs path integration: its conclusion is path-independent if and only if the cochain is exact, and disagreement between valid reasoning paths is exactly the holonomy of a first Cech cohomology class. Hodge decomposition partitions evidence conflict into a gradient part (calibration), a curl part (local inconsistency, visible at triple overlaps) and a harmonic part. Our central result is that no family of simplex-supported consistency checks can distinguish omega from omega+h for harmonic h, which nonetheless generates non-zero disagreement between valid paths; detection requires a statistic on a cycle basis. The resulting procedure, Ksetra, estimates by coboundary projection and gates abstention on the harmonic component, which we give a mechanism: it arises from effect modification combined with overlap-specific population composition, and vanishes to machine precision when effect modification is absent. The degrees of freedom of an evidence network partition into calibration, coherence and transport, yielding an exact F-test for the existence of a global claim; we quantify its distortion under unequal precision and supply the precision-whitened form that restores exactness. Foreign exchange, where the arbitrage-free null makes the cochain exactly a coboundary, serves as a calibration bench: the test is correctly sized, fires on loop arbitrage, and ignores triangular arbitrage.
Multimodal large language models (MLLMs) have raised new privacy challenges. On the data side, user-provided inputs often include unpredictable sensitive information; while on the downstream task side, model reasoning depends on rich visual context that may itself be privacy-sensitive. Existing privacy protection methods, however, rely on predefined sensitive categories and fixed obfuscation strategies, struggling to tackle such challenges in MLLMs. To address this dilemma, we propose Anchored Privacy Drifting (APD), a training-free method that drifts privacy-sensitive elements toward semantically equivalent alternatives while anchoring contextual cues to the source image. To systematically evaluate this dual objective of privacy protection and contextual preservation, we introduce AdaptShield, a comprehensive benchmark covering 22 privacy categories, which combines conventional privacy metrics with MLLM-based assessments of contextual utility. Extensive experiments show that our method achieves balanced improvements in both privacy sanitization and content retention, with average gains of 10.4% on textual categories and 8.5% under MLLM-based evaluation across four MLLM series, i.e., Qwen2.5, Qwen3, InternVL3, and InternVL3.5.