Toward a Formal No-Tear Theorem in a Continuous Substrate Framework

This paper develops a continuity‑layer result within a continuous‑substrate framework in which spacetime geometry is treated as an induced structure arising from admissible substrate configurations. Physical states are represented as admissible pairs (S,D) on a differentiable manifold, and geometry is introduced through a continuous induced‑geometry map gμν=GS,D. The No‑Tear Theorem shows that geometric discontinuities cannot arise within the admissible class once continuity of the substrate and continuity of the geometry map are taken as structural conditions. The result is intentionally narrow: it does not impose curvature bounds, exclude collapse, or replace classical singularity theorems. Instead, it provides a continuity‑layer closure statement within the broader structural‑admissibility program. The framework serves as a foundational layer beneath later analyses involving bounded deformation, finite‑capacity response, saturation behavior, admissible continuation, and collapse constraints. The paper addresses a core structural question: whether geometric rupture is mathematically admissible when continuity of the underlying support structure is taken as primitive. Version 2.0 updates the manuscript to align with the later structural‑admissibility architecture developed in the displacement, continuity, finite‑capacity, and invariant‑substrate papers. The theorem itself is unchanged, while the interpretive framing, admissibility language, and induced‑geometry formulation have been modernized and normalized across the broader program.