General Relativity as the Einstein Limit of Time-Density

This work develops a time-density approach to gravity in which general relativity emerges as a controlled Einstein limit through two independent constructions. In both, the central field is a scalar time-density quantity defined operationally via local clock-rate comparisons, rather than introduced as an auxiliary field. Pathway A (Clock-Induced Geometry) reconstructs spacetime geometry from clock-synchronization consistency on a pre-geometric manifold, yielding a disformal metric structure in which the temporal and spatial sectors are operationally distinguished through an operationally selected timelike direction. The derived isotropic-coordinate weak-field sector reproduces standard Solar System post-Newtonian behavior with γ = β = 1, and admits a covariant completion with a controlled gravitational-wave sector. Pathway B (Time-Density Coupled Gravity) treats the metric and the time-density field as dynamical partners in a covariant theory. The operational meaning of the scalar, together with an explicit Strong Equivalence Principle condition on matter coupling, selects the Jordan frame as the physical matter frame. Solar System viability is achieved through an intrinsic Yukawa-type screening mechanism of the scalar mode, so that the theory approaches the Einstein limit in the appropriate screened regime. A unifying weak-field result — the Time-Strain Hierarchy — organizes redshift, acceleration, tidal response, light bending, and the temporal Ricci projection under a single strain variable. Both pathways are compatible with the standard weak-field tests of general relativity in the analyzed regimes. The manuscript establishes weak-field consistency and Einstein-limit recovery for the time-density framework; full strong-field completion, cosmological solution space, and global observational closure are reserved for future work.