Discrete Gravitational Field in the ZEBTS Framework: Canonical Construction, Convergence, and Reproducible Examples

This paper develops a discrete gravitational framework within the ZEBTS (Zero‑Entropy Background Topological Structure) axiomatic system and rigorously connects it to the modular ZEBTS‑SUPER program. We construct gravity as an emergent phenomenon of a discrete topological background, spectral objects (flytrons), and controlled topological invariants, thereby addressing the long‑standing questions of “how” and “why” gravity operates at a fundamental level. Key results include: (i) a discrete Poisson equation on graph Laplacians with explicit control over gauge choices and solution uniqueness; (ii) a quantitative stability and error analysis grounded in the condition number of the Laplacian on the subspace orthogonal to constants, linking numerical robustness directly to graph connectivity and topological properties; (iii) explicit computational checks on a four‑vertex chain graph that verify convergence rates, perturbation bounds, and force computations; (iv) a systematic closure of gap F3 of the ZEBTS‑SUPER program by demonstrating geometric consistency, including the reproduction of continuum limits and the structural role of flytron modes; (v) the embedding of spectral constructions (F1) and topological diagnostics (F5) as foundational elements of the discrete model. We further outline the immediate next step: the derivation of a discrete Weitzenböck identity and the establishment of equivalence with a Yang–Mills formulation via the adjoint‑valued field χχχ, which will solidify the unification of gravity and gauge interactions in the ZEBTS framework. The approach yields not only a numerically stable discrete theory but also a self‑diagnosing model where topological invariants serve as explicit control parameters for convergence and stability. The results are supported by linear‑format formulas suitable for direct implementation and by a minimal Python‑style verification script, ensuring reproducibility and integration into the ZEBTS‑SUPER codebase.