# Verification of the Euclidean realizability result

The accepted scope consists of two statements: the four-cycle cube has no Euclidean representative of its abstract type, and every strict metric type has a Euclidean representative after a common off-diagonal shift. The written proofs use the vertex criterion, the four-point collinearity trichotomy, the Gordon-Petrov criterion and an explicit positive-definite Gram construction. No known mathematical gap remains in that scope after the originating-researcher audit.

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The exact checker uses Fraction arithmetic and explicit runtime checks, not assertions. Its original normal and optimized runs agree. It independently enumerates the C4 hull, checks five named metrics, compares 72 strict four-point hulls under shifting, verifies 96 Gram and exposing-potential cases in sizes 2 through 9, and checks 14,688 balanced-cycle identities. A deliberate nonstrict control changes the number of vertices from 8 to 12. Portable replays are retained in the source package and bound by its manifest.

Finite computations supplement the general proof. They do not certify arbitrary metrics by enumeration. This is self-audit, not independent peer review or proof-assistant verification. The complete degenerate classification, the full five-part source and absolute novelty remain outside the accepted claim.