We Found Toroidal Topology Emerging in a Neural Epistemic Manifold

Preliminary evidence that the 5D epistemic space of AletheionV2 converges toward toroidal topology as predicted by Directional Relational Manifolds theory — and how we measured it. The Setup We've been building AletheionV2 — a decoder-only LLM where every token generates not just a vocabulary logit, but a full epistemic tomography : aleatoric uncertainty (q1), epistemic uncertainty (q2), calibrated confidence, intentionality vector, and cognitive state. These 5 scalars live on a learned 5D Riemannian manifold. The underlying theory — Directional Relational Manifolds (DRM) — predicts that stable DRMs naturally converge to toroidal topology . Specifically, the manifold should have the homological signature of a torus T²: H1 = Z² (two independent loops) H2 = Z (one cavity) We decided to test this empirically. Here's what we found. The Experiment We applied Riemannian Voronoi tessellation to the 5D epistemic vectors generated by a 1M parameter AletheionV2 model, then ran persistent homolog