Modified Gravity Theory II: Observer-Relative General Relativity—Order-Closed Historical Metric Self-Coupling and Causal Reclosure

This monograph develops an observer-relative, order-closed formulation of modified general relativity. The Lorentzian carrier and the local Einstein–matter equations retain their ordinary covariant form. For a declared receiving root, however, the complete unknown is a receiver-rooted historical diagram containing the inherited metric and matter variables, retarded source anchors, relational positions, concrete causal paths, typed transports, endpoint frames, propagator and boundary data, and any declared classical material or constitutive variables. An intrinsic reconstruction equation is coupled to the local residual, so the causal domain, anchors, paths, transports, measure, source, receiving frame, and boundary history are determined by the same closed structure. This mechanism is called order-closed historical metric self-coupling. Coordinate changes, diffeomorphism covariance, and endpoint Lorentz gauge compare representations within one receiving-root orbit. A change from a root \(A\) to a root \(B\) is instead a branch-labelled partial reclosure: the target causal past, predecessor ledger, retarded anchors, paths, transports, propagators, boundary data, and complete target residual must be reconstructed on the target branch. A posterior common object for independently closed roots exists only when the declared path-indexed comparison and descent equations are satisfied. The Lorentzian carrier, each receiver-rooted historical lift, and any posterior common descent are therefore distinct typed objects. At a fixed receiving root, finite predecessor ideals and selected linear extensions provide the update algebra and inherited-spacetime recursion. The analytic formulation distinguishes concrete-path transport, fixed-root tangent evolution, local PDE solution operators, terminal-record closure responses, conditionally defined mother solution operators or resolvents, and reduced historical kernels by their source, target, graph domain, gauge, boundary, and causal-support conditions. The fixed-root constructions established here include Lorentzian receiver geometry, receiver-local Einstein–matter variation, inherited-history updates, local causal solution operators, a finite terminal-record closure inverse, and typed block reductions on declared compatible domains. When the compatible factors exhaust a fixed active field–record stage and the exact return hypotheses hold, the resulting constrained mother realization has a conditional two-sided inverse. For a same-root \(C^1\) background family, an exact pullback to fixed Banach spaces gives its total inverse variation, including the moving causal, boundary, path, frame, measure, initial-history, and record-chart contributions. Path-family independence, target-root reclosure, and common descent are governed by their corresponding compatibility and residual equations. Keywords **modified gravity; observer-relative general relativity; order-closed historical metric self-coupling; causal reclosure; receiver-rooted histories; Einstein–matter systems; Lorentzian geometry; retarded transport; causal propagators; Volterra resolvents; mother solution operators; common descent**