Persistent Convolution: A Topological Framework for AI Alignment Testing and Semantic Space Characterization

Language model benchmarking is a difficult task. Outcome reasoning alone does not test the model's conceptualization of language and popular open-source benchmarks are quickly saturated or ingested as training data. It is important to test the model's output, but augmenting these tests by characterizing semantic structure gives more insight to how models relate abstract concepts. However, the high dimensional embedding spaces are not easy to interpret. This work demonstrates how topological methods can be used to rigorously compare these spaces to low dimensional and interpretable baselines like ontologies and curated knowledge graphs. These multi-modal alignment tests make it possible to track model adaptations and test phrase understanding across multiple languages.

Alignment evaluation in machine learning has largely become evaluation of models. Influential benchmarks score model outputs under fixed inputs, such as truthfulness, instruction following, or pairwise preference, and these scores are often used to support claims about deployed alignment. This paper argues that deployment-relevant alignment cannot be inferred from model-level evaluation alone. Alignment claims should instead be indexed to the level at which evidence is collected: model-level, response-level, interaction-level, or deployment-level. Two studies support this position. First, a structured audit of eleven alignment benchmarks, extended to a sixteen-benchmark corpus, dual-coded against an eight-dimension rubric with Cohen's kappa = 0.87, finds that user-facing verification support is absent across every benchmark examined, while process steerability is nearly absent. The few interactional benchmarks identified, including tau-bench, CURATe, Rifts, and Common Ground, remain fragmented in coverage, and benchmark construction rather than data source determines what is measured. Second, a blinded cross-model stress test using 180 transcripts across three frontier models and four scaffolds finds that the same verification scaffold raises one model's verification support to ceiling while leaving another categorically unchanged. This shows that scaffold efficacy is model-dependent and that the gap identified by the audit cannot be closed at the model level alone. We propose a system-level evaluation agenda: alignment profiles instead of single scores, fixed-scaffolding protocols for comparable interactional evaluation, and reporting templates that make the inferential distance between evaluation evidence and deployment claims explicit.

Neural networks encode inputs as high-dimensional vectors, known as representations, that capture how models process data by encoding task-relevant structure and semantics. Representation alignment refers to the degree to which different models, layers, or training conditions produce similar representations for the same inputs, with important implications for model interpretation, selection, and robustness analysis. Existing approaches to measure alignment primarily rely on geometric properties, such as neighborhood and cluster similarity, offering limited insight into the global organization of representations. In this work, we present TopoAlign, a topology-aware framework for visually comparing model representations from a structural perspective. Leveraging mapper graphs from topological data analysis, TopoAlign jointly analyzes graphs constructed from representations of shared inputs across different models or layers. The framework supports a top-down comparative workflow: it first performs global structure alignment via joint force-directed optimization to produce coordinated graph layouts; it then identifies local correspondences through automated detection of structurally matching regions, visualized with Bubble Sets; and finally it enables fine-grained pattern inspection through motif-based queries and membrane-inspired visualizations. We demonstrate TopoAlign through case studies on language and multimodal models, complemented by expert feedback. Our results show that TopoAlign provides meaningful insights into representation structure and alignment from a topological perspective.