Dynamical Evolution of the Informational Stiffness Field: Wave Propagation in Curved Spacetime develops the time-dependent structure of the informational stiffness field introduced in the Unified Recursion Theory (URT) framework. Building on the static formulation established in Informational Field Theory in Strong Curvature (IFT-SC), this paper derives the linear dynamics of small perturbations of the stiffness field and shows that these disturbances propagate as informational scalar waves (ISWs) on a fixed spacetime background. The analysis introduces a decomposition of the stiffness field into a stationary background sigma_0(x) and a small perturbation delta_sigma(x,t). Consistency with the URT Core Framework and general covariance restricts the propagation operator to the covariant d'Alembertian. The resulting wave equation describes massless scalar propagation with no geometric backreaction: ISWs evolve on the spacetime metric but do not modify it. Thermodynamic damping arises from the URT recursion-efficiency law, which links the local magnitude of the stiffness field to irreversible update cost. This produces a damping coefficient Gamma(x) that suppresses wave amplitude in regions of high stiffness. In low-curvature environments, waves propagate freely; near horizons, redshift dominates but damping remains weak; deep inside compact objects, stiffening leads to overdamping and the extinction of informational disturbances. This confirms dynamically the finite informational depth previously identified in IFT-SC. The paper also identifies observational signatures of ISWs, including fractional modulations of recursion efficiency (delta_lambda / lambda), curvature-dependent damping patterns, and geometric scattering effects. These signatures provide a clear, falsifiable path for empirical testing across astrophysical and high-coherence laboratory systems. This work completes the dynamical sector of the stiffness field and establishes the foundational wave mechanics required for subsequent analyses of informational flow, black hole evaporation, and measurement processes within the URT framework. 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. Discrete Admissible Regimes in Unified Recursion Theory: Operator Closure, Constraint Topology, and the Necessity of Five Operators DOI: 10.5281/zenodo.18148192Record: https://zenodo.org/records/18148193 3. Informational Field Theory in Strong Curvature (IFT-SC) DOI: 10.5281/zenodo.17850379Record: https://zenodo.org/records/17850379 4. Dynamical Evolution of the Informational Stiffness Field (ISW Theory) DOI: 10.5281/zenodo.17860533Record: https://zenodo.org/records/17860533 RESOLUTION PAPERS (PHYSICAL PARADOXES) 5. Informational Recursion and the Dissolution of the Black Hole Information Paradox DOI: 10.5281/zenodo.17868662Record: https://zenodo.org/records/17868662 6. ORM and the Quantum Measurement Problem (ORM) DOI: 10.5281/zenodo.17881944Record: https://zenodo.org/records/17881944 BRIDGING / CONSTRAINT PAPER 7. 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 8. Emergent Spacetime from Informational Recursion DOI: 10.5281/zenodo.17885555Record: https://zenodo.org/records/17885555 9. λ-Universality Across Scales (λ-UAS) DOI: 10.5281/zenodo.17934065Record: https://zenodo.org/records/17934065 10. Free-Energy Landscape Geometry in Unified Recursion Theory DOI: 10.5281/zenodo.17940995Record: https://zenodo.org/records/17940995 BIOLOGY / COMPLEXITY PAPER 11. URT in Biology: Efficiency, Folding Funnels, Replication Fidelity, and Molecular Motor Dynamics DOI: 10.5281/zenodo.17945209Record: https://zenodo.org/records/17945209 COSMOLOGICAL EXTENSIONS 12. Cyclic Cosmology from Informational Recursion DOI: 10.5281/zenodo.17955043Record: https://zenodo.org/records/17955043 13. 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