TOPOAUDIT3D: GEOMETRY IS NOT TOPOLOGY IN IMPLICIT SURFACE RECONSTRUCTION

Implicit surface reconstruction is usually ranked by overlap or surface distance, although a geometrically accurate level set can contain the wrong number of components, tunnels, or cavities. We present TopoAudit3D, a source-audited framework with two explicitly separated components: a model-agnostic Betti-stable output audit and TopoAudit3D-SDF, a controlled localized-residual SDF reconstruction probe. The audit maps heterogeneous outputs to one physical coordinate system, evaluates topology across reference-stable offsets, and binds aggregates to case identities, evaluator replay, and provenance. Cardiac surfaces are used only as a demanding topology stress test for closed implicit surfaces. The frozen archive contains 2,793 instances, five reconstruction tracks, and 13,965 method–instance records. We reserve the 1,920-instance, repair-free M&Ms track as primary evidence. NKSR obtains lower ASSD than TopoAudit3D-SDF (0.4479 vs. 0.5200 mm) but only 7.76% topology success versus 68.70%. Among 1,203 cases where TopoAudit3D-SDF passes the complete stable-offset criterion and NKSR fails, NKSR still has lower ASSD in 784. A controlled 1,080-run study shows why evaluation and causal attribution must remain separate: residual gating and hard-example sampling help within TopoAudit3D-SDF, yet a capacity-matched flat MLP is stronger on every reported endpoint of the 60-case subset, while removing gradient detachment slightly improves geometry. We therefore make neither an identical-input claim nor an architecture-superiority claim. Instead, we establish that geometric leadership does not certify topological reliability and derive sufficient conditions under which a bounded residual deformation preserves the isotopy class of a zero level set.