A Discrete Common-Origin Model for Newtonian Gravity and Relativistic Spherical Fields: The TDRD Pythagorean Cap and Two Journeys

We construct a discrete committed-state framework, termed TDRD, connecting Newtonian gravity with Schwarzschild-equivalent spherical-field readouts under explicit working rules. Clocks, motion and optics are evaluated on the same causally ordered state. A dimensionless quadratic cap and specified clock mapping supply relativistic momentum and an exact finite-state work identity, with Newtonian momentum and kinetic work recovered at low speed. Symmetric settlement yields an exact quadratic endpoint identity for every finite partition. A three-dimensional, six-neighbour source rule supplies an inverse-radius far field and exact finite-cell flux balance. The Full Journey working axiom couples temporal, spatial and orbital readouts. Its angular cross-term connects weak-field circular restoration to the stability boundary at six gravitational radii through exact finite differences. A fixed three-point optical update uses one scalar to determine direction and propagation time; at impact parameter six gravitational radii, the reported delay error is 0.033% for step 0.01. A radial-reading family distinguishes the consequences of the adopted spatial rule: the critical impact parameter is unchanged across the family, while bending and delay vary. A plane-restricted lattice-to-optics calculation further examines the source-to-readout connection. The principal contribution is a finite-state organization of the Newtonian and relativistic connections, with explicit dependencies between rules and observables. The native construction uses finite settlement, algebraic mappings and exact differences; continuum geometry and quadrature provide comparison benchmarks.

This record contains the author manuscript (v068) and its reproducibility materials. It is a preprint, not a peer-reviewed journal publication. High-resolution weak-field summaries and regeneration code are included; complete three-dimensional arrays are not stored.

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