Large Language Models (LLMs) are increasingly trained to abstain from answering questions they are unsure about. However, this ability is often misapplied: in real-world applications, user prompts sometimes contain elements of uncertainty, which lead LLMs to abstain even on problems they are capable of solving. We argue that LLM abstention is not only an expression of genuine uncertainty; it can also be an artifact largely shaped by prompts. We name this phenomenon *Abstention Inflation*. We add "Unknown" as an extra option for LLMs to choose from; experiments show serious accuracy drops on True/False Questions (TFQs). Replacing "Unknown" with an unrelated random word produces a similar effect. We argue that LLMs are trained to imitate the surface pattern of abstention, rather than to express genuine uncertainty. Based on ten experimental settings, we support four claims that form a progressive argument: **(C1)** *Abstention Inflation* can be triggered by the presence of an extra option, not by genuine uncertainty; **(C2)** it makes the models deny they can answer, even when they can; **(C3)** it is a later-layer output override, as the reasoning traces and mid-layer representations preserve correct answers; **(C4)** it is not stochastic noise: it results from various factors, emerges through instruction tuning, is boosted by problems' higher difficulty, and can be mitigated at larger model sizes.
Large-scale linear programs with millions of variables and constraints strain time and memory budgets. First-order methods such as the primal-dual hybrid gradient scale to this regime because they avoid matrix factorizations, but they require many iterations. Learned accelerators can reduce these iterations, yet fewer iterations do not guarantee a shorter solve. Learning inference also takes time, and running the accelerator longer trades solver time for inference time. Existing unrolled accelerators handle this trade-off poorly: each layer processes every variable and constraint, the predicted iterates still need solver refinement, and every instance runs the same layers. We therefore ask: What neural accelerator design achieves end-to-end speedups on large-scale linear programs across sizes and target accuracies? We introduce AHEAD, an accelerator that predicts step sizes for first-order linear programming solvers while keeping their update rules. Its weight-shared predictor reads compact solver statistics and rescales the primal and dual step sizes, so each warm-start iteration stays inexpensive and a short warm-start phase hands the solver a warm start. A learned handoff policy sets the number of warm-start iterations for each instance and target accuracy, after which the solver finishes under its own termination criterion. On packing linear programs with up to $10^7$ variables, AHEAD achieves $6.5$--$11.6\times$ end-to-end speedups over Gurobi's first-order solver from a cold start, including on instances of unseen sizes and distributions. The gains extend to PageRank linear programs and hold across difficulty levels and target accuracies.