NSP-02: Hyperbolic Cognitive Memory with Ricci-Adaptive Curvature

Current artificial memory systems store knowledge as points in Euclidean space, destroying hierarchical relationships and suffering from catastrophic forgetting under concept drift. While static hyperbolic embeddings offer improved capacity for tree-structured data, they remain rigid against structural evolution. We present the Poincaré Cortex, a hyperbolic cognitive memory substrate that continuously adapts its local curvature via Ricci flow. By replacing computationally expensive Ollivier–Ricci curvature with effective resistance curvature (ERC), we achieve real-time updates in O (|Eₖ| log |Vₖ|) per insertion, where Eₖ, Vₖ are edges and nodes in a k-hop neighbourhood. We prove two results: (1) discrete Ricci flow converges to a stationary configuration with bounded curvature variance (Theorem 1) ; and (2) under classical thermodynamic assumptions, the energy cost of curvature adaptation is bounded by the agent's PCES score (Theorem 2). Vitality—a measure of node usage—modulates curvature evolution: high-vitality regions flatten for rapid access, while dormant nodes sink to the Dark Matter reservoir, a formal archival region with resurrection via geodesic search. During offline consolidation cycles (the Dream Cycle, analogous to sleep), the system runs Ricci flow without new data, synthesizing ghost nodes via geodesic interpolation to minimize global free energy. Crucially, monitoring curvature variance provides a geometric early-warning signal for cognitive drift and adversarial attacks, triggering safety interrupts before performance degrades. The Poincaré Cortex is the geometric substrate for the Tharine Theorem (NSP-01): the fidelity term Fᵢ = exp (−λ dH²) is computed from hyperbolic distance, and high curvature variance serves as a geometric proxy for elevated variational surprise. This is NSP-02 in the 17-paper Nexus Sovereign Physics & AGI research program, establishing the geometric memory substrate for the Tharine Theorem (NSP-01). --- The Nexus Sovereign Physics & AGI Research Program is a 17-paper series formalising the physics, geometry, and thermodynamics of artificial cognitive organisms. The series proceeds as follows: • NSP-01: The Tharine Theorem — Foundational Physics for AGI Sovereignty• NSP-02 (this paper): Hyperbolic Cognitive Memory with Ricci-Adaptive Curvature All papers in this series are published open access under CC BY 4. 0 and are synchronised to the author's ORCID profile. --- Published by Pietarien (Pretoria, South Africa) and the Nexus Sovereign Project. Pietarien is building the Nexus Ecosystem — an all-round personal and business operating system powered by autonomous agents, constitutional identity, and sovereign memory architecture. Learn more: https: //pietarien. comCorrespondence: david@pietarien. comORCID: https: //orcid. org/0009-0000-3053-6613