Semantic Density Effect (SDE): Maximizing Information Per Token Improves LLM Accuracy

Input length and the position of relevant information are widely cited as the primary causes of degraded LLM long-context performance. Here, we study lexical density -- the rate at which a context introduces distinct information -- as a third, largely overlooked factor that systematically reduces the effective context window of LLMs. We quantify the impact of lexical density on open-weight LLMs (9B-685B) using three "find-the-needle" style benchmarks with identical length (~12k tokens) and controlled needle position, but increasing density of information. We observe a sharp performance collapse in higher-density benchmarks: models that are near-perfect in sparse contexts drop below 60% retrieval score on denser ones. To rule out task-type confounds, we vary and control the density within each benchmark while keeping all other properties unchanged. Reducing density generally restores performance, especially in the high-density regimes where degradation appears. These results show that effective context capacity is a function of lexical density, with direct implications for real-world LLM systems operating on compact, information-rich inputs.

Evolutionary prompt optimizers such as GEPA suffer from prompt bloat: each iteration appends rules and caveats, producing prompts up to 3$\times$ longer yet no more accurate. We trace this to three deficiencies - incomplete error observation, limited search diversity, and unreliable selection - and propose ESPO (Error-Structured Prompt Optimization), which decomposes prompt optimization into three phases: Diagnose clusters all training errors into structural patterns in one round; Propose generates candidates via four complementary strategies with independent biases; Select applies bootstrap stability selection. On seven public NLP benchmarks - Tweet, MMLU, GSM8K, HotpotQA, ScoNe, HoVer, and PUPA - ESPO improves average accuracy by $+$3.76 pp over the state-of-the-art (74.67% vs 70.91% for GEPA), matching or exceeding GEPA on every dataset while producing prompts 47% shorter (1,004 vs 1,878 chars) and faster at inference. Cross-model experiments across four additional student models (Gemma 3 12B, Mistral 14B, Qwen3 32B, Claude Haiku 4.5) show ESPO yields the best average accuracy on every model tested, with the largest gap on Qwen3 GSM8K (15.00% $\to$ 91.40%). A generalization bound (Appendix) grounds each phase in a corresponding term of the test-time gap, and the ablation confirms a key prediction: adding diversity without bootstrap selection actually hurts performance ($-$1.20%).

Large language models process prompts by propagating activations through dozens of layers before generating a response. We ask whether the task-relevant information contained in an instruction prompt can be compressed into a single activation vector and re-injected into the model, replacing the original token sequence? We show this is achievable using a learned weighted sum of activations extracted at an intermediate layer and injected at an early layer of the target LLM. The compressed vector preserves task-relevant information, incurring an accuracy drop of under $2\%$ relative to full prompt processing. Beyond its practical implications, including reducing per-query computation for fixed instruction prompts without reprocessing the original token sequence, our analysis reveals structure in the activation space of LLMs: (i) mid-layer representations transfer meaningfully to early layers, suggesting a degree of cross-layer compatibility in how information is encoded; (ii) a single activation vector encodes a quantifiable and recoverable amount of semantic information; (iii) a weighted sum of activations is a robust representation compressor.