ICS Equation-Manifold Specimen68: Recursive Geometry, Checkpoint Transport and Typed Operations

We formulate the executable mathematics of ICS v4 equation-manifold #68, identify its saved-state lineage, and reconcile the existing native ISA evidence. The baseline separates the declared formula, its normalized boundary carrier, the evolved density state and any registered operation that consumes it. This establishes a reference for indexing, reproduction and subsequent Lotus socket contracts. This preprint preserves the named finite computational results and their stated scope. The restricted accompanying package includes the manuscript, claim ledger, finding-specific data, source-result records and a portable archive/elementary-control verifier. Forge/ISA implementation is excluded. Copyright2026 David Glowalla; no additional reuse permission is granted.