Space Between Minds: A Geometry of Perception and the Representation That Could See

This paper proposes that perceptual space has structure, that the structure is measurable, and that the distances it contains are meaningful. The Space of Perceptual Internal Models (SPIM) makes one specific formal move: it applies the Fisher-Rao metric across observers rather than within them. Each observer’s internal generative model is estimated as a probability distribution over perceptual objects, derived from odd-one-out triplet similarity judgments, and placed as a point in the statistical manifold. The Fisher-Rao distance between two observers measures the statistical distinguishability of their internal models, thus how differently they organise a shared world. At current sampling, the marginal representation used here is estimable but provably blind to relational perceptual organisation; a relational representation recovers that organisation in simulation but is not estimable on the real data, where per-observer object-pair coverage is near-disjoint. The instrument’s resolution limit is characterised precisely, and the data design that would lift it is specified. SPIM is constructed from the THINGS database (Hebart et al., 2020), using data from 1500 observers divided into three consecutive batches of 500. The instrument produces stable, replicable geometric structure: the same core-periphery density gradients appear in all three batches. A noise-null control, applying the identical pipeline to synthetic data from uniform Dirichlet distributions, produces flat geometry. The instrument does not hallucinate structure from noise. Two permutation analyses then characterise the instrument’s resolution. Both the MDS-projected geometry and the Fisher-Rao density structure are reproducible from sampling assignment alone at current data density. This does not mean the geometry is not real, since the noise-null control confirms it is not noise. It means the current sampling design cannot yet separate perceptual signal from sampling structure. The resolution condition is precisely identified: a partially orthogonal anchored design, in which all observers co-observe a shared anchor set while otherwise receiving non-overlapping objects. The paper also sketches a dynamic extension (tangent vectors, geodesic flows, mobility fields) as an outlook on how internal models evolve under shared events. Their precise definition and the empirical tools to test them at scale are left to future work. This paper builds the instrument and identifies the resolution condition required to recover the relational structure: a partially orthogonal anchored design that collects new co-observation data.