Middle-density triangle-free bipartization

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evidence/: Exact profile bounds, two-root switching traps and controls for hereditary and recursive comparisons, with bounded numerical discovery for the whole-band argument kept separate from evidence.

profile_envelope_obstructions: Exact root-count and linear-band envelope obstructions, a rooted repair for the Higman–Sims mixture, and an in-band mixture defeating constant-weight contradiction rows on at most four roots.

proof_sketch: Editable sketch of the attack on the open middle-density band of Problem 23, separating the held tail results, author-recorded partial results, tested routes with their exact obstructions, and the remaining gap.

two_root_switching_traps: A probabilistic construction with an exact finite gain table refutes the two-root stability implication for the middle band; an exhibited three-root move escapes, and the original bipartization bound holds throughout the construction.

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This folder holds the sustained attack on the case of Problem 23 that the known density tails leave open: the conjecture that every triangle-free graph on $N$ vertices whose edge density $p=2e/N^2$ satisfies $1243/5000<p<3197/10000$ can be made bipartite by deleting at most $N^2/25$ edges.

The proof sketch states the target, the held tail results, three author-recorded partial results, the routes tried and the gap that remains. The obstruction record records the exact Clebsch obstruction for at most five roots and the [[theory/extremal_graph_theory/L240_seven_root_linear_band_obstruction/_index|seven-root linear-band obstruction of L240]]. The latter uses a mixture of the Higman–Sims and balanced pentagon density vectors at interior density $1/4$, forcing slack at least $1/1200$ at every convex relaxation order. Its lower bound is checked by exact integer-Gram branching; stronger numerical optimum reports are kept separate. An explicit order-five rooted degree inequality removes this mixture in the contradiction formulation. Inside the band, a mixture of the $C_{13}(1,5)$ and Clebsch graphons defeats every certificate whose cut rows on at most four roots carry constant weights. The sketch records the localized rooted route, its floating-point linear-programming tests, and the unresolved full-positivity and exact cut-pricing test; the middle-band lead tracks that route.

The work here is by Plasma AI. It is author-recorded and has not been independently reviewed. The full problem remains open.