# An Adjoint-Sensitivity Framework for Lost-in-the-Middle Phenomena in Causal Residual Transformers

arXiv: https://arxiv.org/abs/2607.17696 · HTML: https://mlsanity.com/paper/2607.17696/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 19 of its 25 references that are in the map.
- First posted on arXiv 20 Jul 2026.

## Builds on (most cited first, 19 of 19)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [RoFormer: Enhanced Transformer with Rotary Position Embedding](https://mlsanity.com/paper/2104.09864/) (2021)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Lost in the Middle: How Language Models Use Long Contexts](https://mlsanity.com/paper/2307.03172/) (2023)
- [Axiomatic Attribution for Deep Networks](https://mlsanity.com/paper/1703.01365/) (2017)
- [Efficient Streaming Language Models with Attention Sinks](https://mlsanity.com/paper/2309.17453/) (2023)
- [Quantifying Attention Flow in Transformers](https://mlsanity.com/paper/2005.00928/) (2020)
- [Robust Learning with Jacobian Regularization](https://mlsanity.com/paper/1908.02729/) (2019)
- [Stochastic Modified Equations and Dynamics of Stochastic Gradient Algorithms I: Mathematical Foundations](https://mlsanity.com/paper/1811.01558/) (2018)
- [Found in the Middle: Calibrating Positional Attention Bias Improves Long Context Utilization](https://mlsanity.com/paper/2406.16008/) (2024)
- [Stabilizing Equilibrium Models by Jacobian Regularization](https://mlsanity.com/paper/2106.14342/) (2021)
- [A Mean-Field Optimal Control Formulation of Deep Learning](https://mlsanity.com/paper/1807.01083/) (2018)
- [Continuum Attention for Neural Operators](https://mlsanity.com/paper/2406.06486/) (2024)
- [On uniform-in-time diffusion approximation for stochastic gradient descent](https://mlsanity.com/paper/2207.04922/) (2022)
- [Lost in the Middle at Birth: An Exact Theory of Transformer Position Bias](https://mlsanity.com/paper/2603.10123/) (2026)
- [A Theory of First Order Mean Field Type Control Problems and their Equations](https://mlsanity.com/paper/2305.11848/) (2023)
- [A Mean-Field Analysis of Multi-Head Self-Attention under Cross-Entropy Training](https://mlsanity.com/paper/2606.10469/) (2026)
- [A First-Order Mean Field Control Analysis of Transformer Layers under Cross-Entropy Training](https://mlsanity.com/paper/2606.23235/) (2026)

## Abstract (arXiv)

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 L1, 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.