We develop an adjoint-sensitivity framework for positional influence in causal residual Transformers and separate unconditional analytic results from conditional boundary-shape conclusions. The principal unconditional theorem is the residual-to-depth-flow estimate for layer controls converging in $L^1$, complemented by a finite-token-to-Volterra attention estimate that explicitly controls the first cells near the causal endpoint. We define a normalized adjoint-energy influence density and derive its exact evolution along full-batch gradient flow. The adjoint admits an exact generator-term decomposition into residual transmission, nonlocal Volterra, and local channels, including all covariance cross terms. Causal masking can amplify early-position sensitivity and residual identity paths can transmit a right-localized terminal bias, but neither mechanism alone forces a U-shaped profile. We therefore state boundary advantages under independently checkable energy, correlation, and local-channel bounds; these conditions are sufficient rather than necessary. Finite-token influence balancing, positional reweighting, and task-aligned observability are presented as diagnostics or regularizers with explicit differentiation requirements, computational costs, and limitations. Controlled simulations illustrate that each intervention controls its designated surrogate, while observability balance or outer-loop reweighting need not monotonically reduce the influence-based Lost-in-the-Middle diagnostic.
Electronic health records now routinely exceed 100,000 tokens per patient. Yet large language models exhibit the lost-in-the-middle (LitM) effect: information near the center of a long context is retrieved less reliably than information near the edges. In clinical use this is not benign: the single most consequential fact in a note can sit at its center. We term this the clinical lost-in-the-middle (CLitM) problem, give its first systematic characterization using MedAlign, and compare context-selection strategies as remedies. Across 2,196 instruction-response pairs and six language models, we observe a 21.9 percentage-point gap between peak accuracy (59.5%, 95% CI [46.3, 71.0], 20-30% decile) and trough accuracy (37.6% [23.2, 52.5] at 70-80%); 67.8% of reference answers fall between the 10th and 90th percentiles of the EHR timeline, inside the CLitM trough. We introduce Query-Conditioned Clinical Suppression (QCCS), a lightweight query-conditioned selection gate, and evaluate it against BM25, BM25 with section-header filtering, dense retrieval, and cross-encoder reranking (N=83 held-out instructions). With Qwen2.5-7B-Instruct (16k context), QCCS outperforms all five comparators under LLM-as-judge scoring: for middle-position instructions QCCS reaches 16.7% versus BM25 3.3%, cross-encoder 0.0%, dense 0.0%, and full context 6.7%; overall QCCS reaches 25.3% versus at most 3.6% for retrieval-only comparators. This advantage is not explained by retrieval recall: at k=20, BM25 retrieves the gold evidence sentence in 98.8% of instructions (QCCS 34.9%), yet retrieval arms stay at most 2.6% accurate even when they retrieve it, whereas QCCS reaches 25.0% even when it does not. In this proof-of-concept evaluation, query-aligned context selection predicts EHR instruction-following accuracy better than gold-sentence retrieval recall.