Large language models (LLMs) increasingly fuse heterogeneous inputs in ubiquitous systems. Yet, how LLMs implicitly allocate authority when sensor measurements and user claims conflict remains unexamined, raising critical reliability concerns for deployments where physical sensing must retain priority. Unlike explicit traditional fusion, LLMs bury authority allocation within learned representations. We discover this allocation is severely format-dependent: numerical sensor data fails to integrate into answer-relevant model directions, allowing natural-language claims to dominate the final decision, a phenomenon we term \textbf{Authority Inversion}.To diagnose and mitigate this, we develop a geometric framework of context integration, introduce two computable audit metrics, specifically the Context Integration Ratio (CIR) and Authority Alignment Index (AAI), and propose Geometric Authority Calibration (GAC), an inference-time layer-level intervention to suppress misplaced user authority. Evaluating four models (4B to 35B parameters, three architectures) across four datasets totaling 576 conflict instances reveals extreme inversion: on numerical tasks, models exhibit near-zero sensor trust (AAI = -0.805, Cohen's d = -2.14), unaffected by model capacity. Validating our geometric framework, theory-guided causal injection flips 80.2\% of incorrect decisions (vs. <0.4\% for random controls). Practically, GAC improves HAR accuracy from 0 -- 1.6\% to 21.9 -- 27.5\%, outperforming prompting baselines. Ultimately, authority allocation in LLM-mediated systems must be explicitly audited and application-specifically configured rather than left implicit.