files / problems / caccetta-haggkvist-k3 / compute / q4 / certs / keep / f4_or_new_certificate.json
{
"c": 0.3464,
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"stored_f4": 0.34645,
"personal_communication_3388": 0.3388,
"css_beta": 0.8616,
"css_source": "Chen-Karson-Liu-Shen, Electron. J. Linear Algebra 28 (2015), Theorem 2.5; arXiv:0909.2468. Every 3-free digraph satisfies beta(G)<0.8616 gamma(G). Hamburger-Haxell-Kostochka, Electron. J. Combin. 14 (2007), Lemma 5, with the feedback-arc budget 0.8616 k instead of k, gives a vertex of out-degree < sqrt(2*0.8616 k). The HKN Lemmas 3.5-4.7 chain then yields Psi(kappa)>=3(3c-1)^2/0.8616. The 4 Psi(kappa) type coefficients are rebuilt from flags4.enumerate_labeled and match HKN (4.7) exactly; only the uniform penalty changes.",
"method": "F4 Farkas certificate: Q\u227d0 on the 8x8 beta-flag Cauchy-Schwarz slices (SDP warm start, Cholesky-refined), out-regularity, order-3 IndT/IndV, and the CKLS-tightened CSS-fork form. Linear multipliers fit by HiGHS.",
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"all_negative": true,
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