This paper establishes the geometric foundation underlying Unified Recursion Theory by constructing an intrinsic metric on informational state space based on distinguishability. Rather than introducing new dynamics or physical laws, the work formalizes how informational distance, curvature, and stiffness arise from the ability to statistically distinguish neighboring states. Using information-theoretic distance measures, the paper shows that informational states form a curved manifold whose local geometry determines admissible recursive updates. Informational stiffness is identified with curvature in this space, providing a precise geometric meaning to quantities previously introduced in URT. Recursion is shown to follow geodesic structure under energetic and entropic constraints. This geometric formulation serves as a structural bridge between the foundational recursion and ORM framework and later applications across physics, biology, and cosmology. The paper remains strictly prior to emergent spacetime, universality, and cosmological extensions, supplying the mathematical substrate on which those developments rely. URT PAPER FAMILY This work forms part of the Unified Recursion Theory (URT) research program, which develops a cross-domain framework for physical evolution based on constrained informational recursion and an energy–entropy proportionality law. Each paper in the series is self-contained, while collectively establishing the theoretical structure across quantum, geometric, biological, cosmological, and particle-level domains. Related URT works available on Zenodo: FOUNDATIONAL PAPERS 1. Unified Recursion Theory — Core Framework (URT Core) DOI: 10.5281/zenodo.17642761Record: https://zenodo.org/records/17642761 2. Informational Field Theory in Strong Curvature (IFT-SC) DOI: 10.5281/zenodo.17850379Record: https://zenodo.org/records/17850379 3. Dynamical Evolution of the Informational Stiffness Field (ISW Theory) DOI: 10.5281/zenodo.17860533Record: https://zenodo.org/records/17860533 RESOLUTION PAPERS (PHYSICAL PARADOXES) 4. Informational Recursion and the Dissolution of the Black Hole Information Paradox DOI: 10.5281/zenodo.17868662Record: https://zenodo.org/records/17868662 5. ORM and the Quantum Measurement Problem (ORM) DOI: 10.5281/zenodo.17881944Record: https://zenodo.org/records/17881944 BRIDGING / CONSTRAINT PAPER 6. Distinguishability Geometry in Informational State Space DOI: 10.5281/zenodo.17957062Record: https://zenodo.org/records/17957062 Provides the geometric foundation for informational state space.Underpins the emergence of spacetime, efficiency universality, and landscape geometry. THEORETICAL EXPANSION PAPERS 7. Emergent Spacetime from Informational Recursion DOI: 10.5281/zenodo.17885555Record: https://zenodo.org/records/17885555 8. λ-Universality Across Scales (λ-UAS) DOI: 10.5281/zenodo.17934065Record: https://zenodo.org/records/17934065 9. Free-Energy Landscape Geometry in Unified Recursion Theory DOI: 10.5281/zenodo.17940995Record: https://zenodo.org/records/17940995 BIOLOGY / COMPLEXITY PAPER 10. URT in Biology: Efficiency, Folding Funnels, Replication Fidelity, and Molecular Motor Dynamics DOI: 10.5281/zenodo.17945209Record: https://zenodo.org/records/17945209 COSMOLOGICAL EXTENSIONS 11. Cyclic Cosmology from Informational Recursion DOI: 10.5281/zenodo.17955043Record: https://zenodo.org/records/17955043 12. Antimatter as Inverse Recursion: Temporal Operator Asymmetry and Matter–Antimatter Imbalance in Unified Recursion Theory DOI: 10.5281/zenodo.17955043Record: https://zenodo.org/records/17955625