Information Geometry(Observer-Centric 3D Curved Surface and 4D Chiral)

Conventional information geometry relies solely on the internal parameters (theta, eta) of probability distributions as independent criteria to define space curvature and distance, functioning as an 'intrinsic geometry.' In contrast, my paper expands the abstract probability spaces of statistics and information theory into a novel mathematical and physical framework by integrating them with the observer's spatiotemporal position, geometric curvature, and four-dimensional chiral gauge field theory. 1. Establishment of Observer-Centric Dynamic Information Geometry: Beyond the limitations of traditional information geometry—which treats probability distributions as static—this framework introduces the observer's viewpoint coordinate system (theta_{obs}) and intervention function (Omega_{obs}). This reflects the observer's active measurement agency and dynamically controls information loss and risk via asymmetric KL divergence. 2. Fusion of the Three-Dimensional Spherical Manifold (S^3) and Four-Dimensional Chiral Mirror Field Theory: The abstract probability space is extended onto a three-dimensional spherical manifold (S^3), further coupled with a fourth spatial axis (w), the chiral mirror inversion operator (mathcal{M}_{chiral}), and the Hermitian coupling operator (Gamma_{chiral}) to construct a four-dimensional energy functional and time-dependent equations of motion (probability conservation dynamics). By strictly separating real information weight functions (Psi_{3D}) from complex state amplitudes (psi_{3D}), it establishes a geometric foundation capable of accommodating quantum interference effects. 3. Time-Dependent Wave Dynamics and Engineering Extension: Transforming probability spaces from static states into time-dependent wave equations and density-conserving dynamics successfully bridges statistical geometry with quantum field theory. It presents standard governing equations applicable to practical industrial domains, including sensor sensitivity measurement, asymmetric risk control, and wave propagation simulations for aerospace, robotics, and defense.