Heart-Beating Manifolds: Why Transformers Need a Living Geometry with Agentic Manifolds - a new recursive reasoning model (RRM)

We introduce a computational framework for active geometry in transformer embedding spacesusing emergent Boltzmann dynamics and uniform idempotent right quasigroups (IRQs). ReplacingBoltzmann’s classical derivatives with emergent algebra operations, we construct dynamiccurvature tensors that evolve through semantic flow. This establishes a living manifold that canadapt, reason, and detect contradictions through its geometric evolution. Our approach providescurvature-based measures of semantic coherence and truth, illustrating how transformer embeddingscan form geometries that both encode and refine understanding.1 IntroductionTransformer embeddings inhabit high-dimensional, dynamic spaces that encode evolving semanticrelationships. Traditional Riemannian geometry assumes smoothness and static differentiability,which fails to capture these non-equilibrium structures. We address this by extending Boltzmann’smicroscopic-to-macroscopic framework into the domain of semantic embeddings, where curvaturebecomes living actively evolving under algebraic dynamics.2 Architecture-Agnostic ApplicabilityA central feature of the emergent Boltzmann curvature framework is its architectural agnosticism.The method operates purely on the embedding space geometry, independent of the underlyingneural architecture. It requires only access to vector representations {xi} and their semanticinteractions.