Sparse halves in triangle-free graphs of minimum degree greater than one third

We prove that every triangle-free graph on n vertices with minimum degree strictly greater than n/3 contains a set of ⌊n/2⌋ vertices spanning at most n²/50 edges. A theorem of Łuczak, Polcyn and Reiher reduces the problem to proper blow-ups of Andrásfai and Vega graphs. For all four Vega families we establish the requisite weighted sparse-half inequality. A thirteen-coordinate aggregate description of the original weighted graphs yields the uniform index reduction, and the required cost bounds are certified in smaller coordinate systems: a seven-variable certificate for the two families retaining both gadget vertices x, y, a nine-variable certificate for the other two families, and small-index certificates. A separate certificate in the thirteen aggregate coordinates gives a second proof of the bound 1/50 for all indices i ≥ 3. A finite rounding argument transfers the weighted conclusion to induced vertex subsets. Files: - sparse_halves_one_third.pdf: the paper. - sparse_halves_one_third_supplement.zip: the computer-assisted part of the proof. It contains the paper source, the seven-variable, nine-variable and index-two certificates with their verifiers and generators, the thirteen-coordinate certificates for i ≥ 3, and the record of a complete verification run. Instructions are in README.md at the top level of the archive; Appendix F of the paper describes the verification. Verification: the certificates used in the proof of the main theorem (seven-variable, nine-variable and index-two) are replayed by "python3 verifiers/run.py --mode full" (Python 3 with NumPy and SciPy, and Node.js); the recorded run took 3046.640 seconds (about 51 minutes) with fourteen worker processes. The thirteen-coordinate certificates in audit/d3_complete/ give a second proof for i ≥ 3 and are not needed for the main theorem; checking one type loads about 1.2 GB of leaf files and needs about 4.6 GB of RAM. The ZIP expands to 2.36 GB. Licenses: the paper, all rights reserved; programs (.py, .mjs, .cpp, requirements.txt), MIT License; certificates, data, logs and documentation, CC BY 4.0.