A Borsuk–Ulam Argument for Representational Alignment

Representational alignment, defined as correspondence between distinct representations of the same underlying structure, is usually evaluated using coordinate-level similarity in high-dimensional spaces, together with correlation-based measures, subspace alignment techniques, probing performance and mutual predictability. However, these approaches do not specify a baseline for the level of agreement induced solely by dimensional compression, shared statistical structure or symmetry. We develop a methodological framework for assessing representational alignment using the Borsuk-Ulam theorem as a formal constraint. Representations are modeled as continuous maps from a state space endowed with a minimal symmetry into lower-dimensional descriptive spaces. In this setting, the Borsuk-Ulam theorem provides a lower bound on the identification of symmetry-paired states that must arise under dimensional compression. Building on this bound, we define representational alignment in terms of shared induced equivalence relations rather than coordinate-level similarity. Alignment is quantified by testing whether distinct models collapse the same symmetry-related states beyond what is guaranteed by topological necessity alone. The resulting metrics are architecture-independent, symmetry-explicit and compatible with probe-based comparisons, enabling controlled null models and scale-dependent analyses. Our framework supports testable hypotheses concerning how alignment varies with representation dimension, compression strength and symmetry structure, and applies to both synthetic and learned representations without requiring access to internal model parameters. By grounding alignment assessment in a well-defined topological constraint, this approach enables principled comparison of representations while remaining neutral with respect to the semantic or ontological interpretation of learned features.