Quantized Curvature Reconstruction via SU(2) Generators: Extending Absolute Unified Mathematics of Quantum Gravitation

Mercury’s relativistic perihelion precession, estimated at ~1.95 billion arcseconds over 4.54 billion years, corresponds to nearly 1,500 extra orbital revolutions. This anomaly, unexplained by Newtonian mechanics, remains a benchmark for testing Einstein’s general relativity. Clifford algebra offers a deeper representation of the geometric symmetries of the Riemann curvature tensor, providing a basis for reformulating gravitational field equations within quantum frameworks. This study extends the Absolute Unified Mathematics of Quantum Gravitation framework introduced in 2024–2025. An integral algebraic framework is developed by embedding SU(2) basis elements (𝑖𝜎) in the off-diagonal components of the metric tensor, where they generate spacetime symmetries and correspond to spinor structures. This demonstrates that Pauli matrices act as algebraic building blocks of spacetime geometry, enabling systematic study of curvature and symmetries. Newton’s gravitational potential and Poisson’s equation are derived through modified Christoffel symbols, linking algebraic operators to geometric curvature. The study further reconstructs the Riemann tensor and analyzes Pauli matrices in the spherical basis. The findings show that embedding SU(2) generators within the metric tensor reconstructs curvature structures and symmetries consistent with Einstein’s framework, while extending toward quantum–gravitational interactions. Mercury’s orbital precession anomaly is interpretable through these algebraic structures. In spherical coordinates, the operators (𝜎,𝜎 ,𝜎∅) represent spin projections along local spherical directions. Since these directions vary with 𝜃 and ∅, the matrices exhibit orientation dependence rather than radial scaling. Along a fixed ray, changes in radial (𝑟) do not alter direction, leaving the matrices unchanged. This directional dependence of spinors eliminates singularities in the framework, reinforcing its algebraic consistency. Embedding SU(2) generators within the metric tensor establishes a rigorous pathway for reformulating Einstein’s field equations in terms of quantized curvature tensor fields, providing a testable foundation under extreme quantum–gravitational conditions. This reformulation enables quantum gravity applications in quantum computing software.