Unsupervised representational alignment recovers a stimulus-by-stimulus correspondence from geometry alone, but the automorphism group of the stimulus geometry bounds what any such alignment can identify, before data exist. The obvious diagnostic for this degeneracy, the cheapest non-identity relabelling, ranks two published designs in the wrong order, because dense sampling creates near-duplicates whose transposition is nearly free. We turn this known invariance (Demetci et al., 2024) into a design-time diagnostic and intervention. In colour, where candidate geometries have closed form, we show that the failure is structural: sixty-four times the restart budget leaves a symmetric design unmoved while an asymmetric set at the same N recovers every time. Discriminating representational models and recovering a correspondence are essentially uncorrelated objectives (r = -0.02 over 3,000 subsets). Choosing nine colours by this diagnostic alone, without consulting any learned representation, moves all 93 model representations away from the degenerate point and cuts catastrophic alignment failures from 75% to 2% with the models, the layers, N and the solver all held fixed. The same risk arises wherever a regular design meets its candidate geometry's isometry group, including evenly spaced orientations, tones, or motion directions, and the check costs one function call before data collection.
Word alignment has traditionally been studied between sentences, but many cross-lingual tasks increasingly require correspondences across full documents. While recent multilingual embedding models can encode long inputs, we show that applying algorithms designed for sentences directly to documents leads to performance degradation. To address this, we introduce CTFAlign, a lightweight, training-free approach for document-level word alignment. CTFAlign applies a coarse-to-fine refinement strategy that restricts the alignment search space to semantically similar regions. Additionally, we introduce MDPAlign, a simpler alternative that constrains alignments by position with a main diagonal prior. Both approaches operate directly on full documents without relying on sentence segmentation or sentence alignment. We evaluate these methods across six language pairs varying in typological distance, resourcedness, and document length. Averaged over three models, CTFAlign reduces word alignment error rate from 0.412 to 0.326. These gains transfer downstream, leading to improvements in document-level translation coverage evaluation and recognition of semantic differences. We release CTFAlign as a Python package and make the code and data to reproduce our experiments publicly available.
Predicting the 3D structures of atomic systems is fundamental to advancing material science and drug discovery. While flow-matching models (, FlowDPO) have recently shown promise in this domain, their performance relies heavily on alignment with ground-truth coordinates via supervised preference learning. However, obtaining experimental labels for novel crystal phases or de novo proteins is prohibitively expensive, creating a bottleneck for structural modeling in data-scarce regimes. In this work, we propose (Atomic Policy Optimization), a fully unsupervised alignment framework that eliminates the need for ground-truth reference structures. APO adapts group-relative policy optimization to 3D atomic environments, utilizing a novel dual-reward mechanism: (i) a that reinforces the policy's dominant latent structural modes through eigen-decomposition of sample similarities, and (ii) a that enforces thermodynamic stability. Our framework enables the model to ``self-correct'' by identifying physically plausible configurations within sampled groups. Extensive benchmarks on crystal and antibody structure prediction demonstrate that APO consistently outperforms fully supervised baselines, achieving a new state-of-the-art in match rates and structural fidelity. Furthermore, we show that APO effectively straightens probability paths, significantly improving inference efficiency. Our results suggest that intrinsic physical consistency can serve as a superior guide for alignment compared to noisy, supervised coordinate matching.