files / problems / simon-lieb-thirring / compute / q3 / certs / covering_local.json

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{
  "analytic_constant_alpha3": {
    "alpha": 3.0,
    "L_local": 0.5773502691896258,
    "ratio_over_classical": 2.720699046351327
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  "best_sample": {
    "E0": -3.00000000000074,
    "abs_E0": 3.00000000000074,
    "integral": 5.196152422706632,
    "L_upper_local": 0.5773502691897682,
    "ratio_over_classical": 2.7206990463519976,
    "alpha": 3.0,
    "n_neg": 1,
    "v_min": 3.0,
    "v_max": 3.0,
    "v_std": 0.0,
    "tag": "constant depth=0.5",
    "analytic_1_over_sqrt_alpha": 0.5773502691896258
  },
  "best_constant": {
    "E0": -3.00000000000074,
    "abs_E0": 3.00000000000074,
    "integral": 5.196152422706632,
    "L_upper_local": 0.5773502691897682,
    "ratio_over_classical": 2.7206990463519976,
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    "v_min": 3.0,
    "v_max": 3.0,
    "v_std": 0.0,
    "tag": "constant depth=0.5",
    "analytic_1_over_sqrt_alpha": 0.5773502691896258
  },
  "random_best": {
    "E0": -3.008134343920681,
    "abs_E0": 3.008134343920681,
    "integral": 5.416198790124355,
    "L_upper_local": 0.5553958524206263,
    "ratio_over_classical": 2.617241294698321,
    "alpha": 3.0,
    "n_neg": 1,
    "v_min": 1.207039335917259,
    "v_max": 4.703441128491818,
    "v_std": 0.9993084699640877,
    "tag": "random-hist"
  },
  "beats_CCR": false,
  "note": "A covering bound is an *upper* bound on L equal to the worst local ratio. The constant well already gives ~2.72 > 1.44655, and random histograms did not produce a larger local ratio. This handle cannot dent CCR at gamma=1 with the Poincar\u00e9 partition \u03b1\u22643.",
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}