Model-guided directed evolution seeks to identify high-fitness protein variants under limited oracle budgets. Protein language models (PLMs) provide rich representations for this task, but existing methods either rely on task-agnostic zero-shot scores or perform search in high-dimensional embedding spaces where surrogate modeling and uncertainty estimation can be sample-inefficient. We propose the Linear Fitness Subspace (LFS) hypothesis: mutation-induced fitness variation encoded by PLMs is concentrated in a low-dimensional linear subspace of residue-level representation changes. This view explains why zero-shot PLM scores are often useful but incomplete: they behave as fixed, task-agnostic projections, whereas few-shot supervision can recover assay-specific fitness directions. Building on this observation, we introduce Subspace-Guided Evolutionary Search (SGES), which learns the LFS from a small initial sample and performs surrogate modeling, uncertainty estimation, and acquisition directly in the learned subspace. Across 10 core ProteinGym assays and an extended set of 87 assays, SGES improves fitness prediction, budgeted search efficiency, computational cost, and biological interpretability over zero-shot PLMs and recent ML-guided protein optimization baselines. These results suggest that PLMs encode protein fitness in a geometry that is more linear, lower-dimensional, and more directly exploitable than previously assumed.

Automated homework assessment requires not only recognizing student answers but accurately locating complete answers and intermediate reasoning steps in noisy, multi-page handwritten work. We address the missing evaluation setting of *page-aware, two-level answer-region grounding*: given homework page images, a model must localize complete answer regions and their ordered step-level subregions. We introduce **HG-Bench**, comprising 500 human-annotated K–12 homework samples curated from a 1,489,278-image source pool, with hierarchically linked question- and step-level boxes. Its page-aware protocol separately measures complete-answer localization (ℱ_A) and step-level decomposition (ℱ_S^𝜇), distinguishing structured spatial grounding from visible-text parsing. Across frontier closed-source and open-weight VLMs, no zero-shot system exceeds 55.22% on ℱ_A or 48.22% on ℱ_S^𝜇, whereas a GLM-4.6V 9B reference fine-tuned on approximately 10k in-domain examples reaches 74.97/72.26. These results expose step-level handwritten grounding as a concrete capability gap and provide a reproducible benchmark, evaluation protocol, and trained reference for future work on automated homework assessment.