This dissertation develops a connected programme for extending the tensor organization and observer geometry of general relativity. Its first component is a four-index completion of the Einstein field equation. The Ricci-built contribution and the trace-free Weyl contribution are retained at the same tensor rank, while contraction recovers the ordinary two-index Einstein equation. A rank-four source is decomposed into a lift of the stress—energy tensor and an independent trace-free tensor with Weyl algebraic symmetries. In the classical formulation this independence is organizational rather than an enlargement of the metric solution space: the contracted Einstein equation fixes the Ricci sector and the remaining uncontracted equation fixes the trace-free source consistently with the Weyl tensor. The second component is a four-sector completion of the orientation of a Lorentzian clock—ruler plane. The four connected non-null domains share the same null boundary and are labelled by the phase cycle 1, i, −1, −i. Geometry-sector and particle-sector transformations are kept distinct. Ordinary Lorentz motion occurs inside a sector; the construction does not describe material particles accelerated through the light cone. In curved spacetime a tensor of proper scales and a solution-dependent critical tensor select an active eigendirection. Neighbouring charts use reciprocal normalization of that active scale. Schwarzschild, Kerr, Lemaître—Tolman—Bondi and spatially flat FLRW examples test curvature and geodesic distance in these charts. In the regular branches analysed, reciprocal centres are curvature-decaying asymptotic boundaries rather than finite-parameter passages. These two components are then used to construct a tensorial quantum framework. Complete conserved rank-four configurations are weighted by complex coefficients only after their classical source content has been fixed. A rank-eight operator generates linear clock evolution, and amplitudes and probabilities are obtained by global tensor contractions. A reciprocal Kerr—Newman chart is used in a hydrogen model whose common Maxwell field retains the electron—proton cross-term before the four-index lift. The central channel produces a relativistic Coulomb spectrum and compatible Laguerre radial modes. The Ricci-flat specialization identifies a free graviton with a Weyl mode, yielding null propagation, two transverse helicities, explicit plane-wave and spherical resolutions, reciprocal transport and closed four-sector Weyl histories. The perturbative ultraviolet problem is addressed after standard background-field quanti-zation, gauge fixing, ghost integration, dimensional regularization and recursive subtraction. Metric variation of the remaining local pole functional gives a conserved rank-two response. Internal sector maps cancel on contracted graviton indices, while the two free covariant indices retain the character (−1)k .