{
 "row": 2,
 "name": "massless planar double box (Smirnov hep-ph/9905323), calibration row",
 "provenance": {
  "closed_form_source": "the arXiv TeX source of hep-ph/9905323: eq. (2box) definition, eq. (K0t), the final K_{1t} block (right-contour closure), S_{1,2}=Li_{2,1}, S_{2,2}=Li_{3,1}; converted to G-words by exact Fraction shuffle algebra in a derivation script, verified to <1e-100 at 3 points vs a direct mpmath evaluation.",
  "paper_row": "the results paper, Sec. 2.3 (massless double box)",
  "date": "2026-07-03"
 },
 "normalization": {
  "g": "g(x,eps) = -K(x,eps)/4 with Smirnov's K; g = 1/eps^4 + ... (c_empty = 1)",
  "I_row": "I_row = LS * g, LS = 1/(-2 s^2 t), x = t/s",
  "raw_relation": "DBox_raw = -8 (i pi^{d/2} e^{-gammaE eps})^2 (-s)^{-2eps} * LS * g (Euclidean s<0, t<0, x=t/s>0)",
  "grading": "order eps^{w-4} is pure weight w, w=0..4"
 },
 "alphabet": [
  "x",
  "1+x"
 ],
 "letters": [
  0,
  -1
 ],
 "orders": {
  "-4": {
   "weight": 0,
   "words": [
    [
     "1",
     0,
     ""
    ]
   ],
   "reduced": [
    [
     "1",
     0,
     0,
     ""
    ]
   ]
  },
  "-3": {
   "weight": 1,
   "words": [
    [
     "-5/4",
     0,
     "0"
    ]
   ],
   "reduced": [
    [
     "-5/4",
     0,
     1,
     ""
    ]
   ]
  },
  "-2": {
   "weight": 2,
   "words": [
    [
     "1",
     0,
     "0,0"
    ],
    [
     "-15/4",
     2,
     ""
    ]
   ],
   "reduced": [
    [
     "1/2",
     0,
     2,
     ""
    ],
    [
     "-15/4",
     2,
     0,
     ""
    ]
   ]
  },
  "-1": {
   "weight": 3,
   "words": [
    [
     "1",
     0,
     "0,0,0"
    ],
    [
     "-1",
     0,
     "-1,0,0"
    ],
    [
     "33/4",
     2,
     "0"
    ],
    [
     "-3",
     2,
     "-1"
    ],
    [
     "-65/12",
     3,
     ""
    ]
   ],
   "reduced": [
    [
     "-1",
     0,
     0,
     "0,0,-1"
    ],
    [
     "1",
     0,
     1,
     "0,-1"
    ],
    [
     "-1/2",
     0,
     2,
     "-1"
    ],
    [
     "1/6",
     0,
     3,
     ""
    ],
    [
     "-3",
     2,
     0,
     "-1"
    ],
    [
     "33/4",
     2,
     1,
     ""
    ],
    [
     "-65/12",
     3,
     0,
     ""
    ]
   ]
  },
  "0": {
   "weight": 4,
   "words": [
    [
     "-8",
     0,
     "0,0,0,0"
    ],
    [
     "5",
     0,
     "0,-1,0,0"
    ],
    [
     "4",
     0,
     "-1,0,0,0"
    ],
    [
     "-1",
     0,
     "-1,-1,0,0"
    ],
    [
     "-18",
     2,
     "0,0"
    ],
    [
     "15",
     2,
     "0,-1"
    ],
    [
     "5",
     2,
     "-1,0"
    ],
    [
     "-3",
     2,
     "-1,-1"
    ],
    [
     "22/3",
     3,
     "0"
    ],
    [
     "-1",
     3,
     "-1"
    ],
    [
     "-87/4",
     4,
     ""
    ]
   ],
   "reduced": [
    [
     "11",
     0,
     0,
     "0,0,0,-1"
    ],
    [
     "-1",
     0,
     0,
     "0,0,-1,-1"
    ],
    [
     "-1",
     0,
     0,
     "0,-1,0,-1"
    ],
    [
     "-1",
     0,
     0,
     "-1,0,0,-1"
    ],
    [
     "-6",
     0,
     1,
     "0,0,-1"
    ],
    [
     "1",
     0,
     1,
     "0,-1,-1"
    ],
    [
     "1",
     0,
     1,
     "-1,0,-1"
    ],
    [
     "1/2",
     0,
     2,
     "0,-1"
    ],
    [
     "-1/2",
     0,
     2,
     "-1,-1"
    ],
    [
     "2/3",
     0,
     3,
     "-1"
    ],
    [
     "-1/3",
     0,
     4,
     ""
    ],
    [
     "10",
     2,
     0,
     "0,-1"
    ],
    [
     "-3",
     2,
     0,
     "-1,-1"
    ],
    [
     "5",
     2,
     1,
     "-1"
    ],
    [
     "-9",
     2,
     2,
     ""
    ],
    [
     "-1",
     3,
     0,
     "-1"
    ],
    [
     "22/3",
     3,
     1,
     ""
    ],
    [
     "-87/4",
     4,
     0,
     ""
    ]
   ]
  }
 },
 "references": {
  "amflow_heldout": {
   "what": "the auxiliary-mass-flow (AMFlow) values of this integral at the two Euclidean points held out of the fit that produced the served tower, x = 2/5 and x = 3/4, as the raw goal-40 and goal-70 output strings of the runs of record; the evaluator's AMFLOW HELD-OUT GATE compares the served tower with them",
   "normalisation": {
    "formula": "J = e^{-2 gamma_E eps} K(x,eps) / ((-s)^{2+2 eps} (-t)) (the AMFlow measure d^d l/(i pi^{d/2})); at s = -1, t = -x: J_j = (1/x) [e^{-2 gamma_E eps} K]_j = (1/x) sum_{m>=0} (-2 gamma_E)^m/m! K_{j-m}; K = -4 g (the served g = -K/4, normalization.g above)",
    "K_from_g_factor": "-4",
    "exponent_gamma_coefficient": "-2",
    "source": {
     "record_relative": "scripts/dbox_common.jl",
     "sha256": "9956a26df0f422113bab9e3ef11534e2cacdc53bdda41b9c8af8e9d1941e4627",
     "lines": "9-12",
     "quote": [
      "# amflow-cpp measure is d^d\u2113/(i\u03c0^{d/2}), so",
      "#   J_amflow = e^{-2\u03b3_E \u03b5} K(x,\u03b5) / ((-s)^{2+2\u03b5} (-t)),",
      "#   i.e.  K(x,\u03b5) = e^{2\u03b3_E \u03b5} (-s)^{2+2\u03b5} (-t) \u00b7 J.",
      "# Euclidean sampling at s = -1, t = -x (x>0):    K_j = x \u00b7 [e^{2\u03b3\u03b5} J]_j."
     ]
    },
    "served_g": "normalization.g of this file"
   },
   "run_config": {
    "common": {
     "amf_options": {
      "blackbox": {
       "ibp_dot": 1,
       "ibp_rank": 0,
       "integral_order": 5,
       "n_thread": 8
      },
      "ending_schemes": [
       "Tradition",
       "SingleMass"
      ],
      "max_recursion_depth": 16
     },
     "family": {
      "conservation": {
       "p4": "-p1 - p2 - p3"
      },
      "legs": [
       "p1",
       "p2",
       "p3",
       "p4"
      ],
      "loops": [
       "l1",
       "l2"
      ],
      "name": "doublebox",
      "propagators": [
       "l1^2",
       "(l1 + p1)^2",
       "(l1 + p1 + p2)^2",
       "(l2 + p1 + p2)^2",
       "(l2 + p1 + p2 + p4)^2",
       "l2^2",
       "(l1 - l2)^2",
       "(l1 + p3)^2",
       "(l2 + p1)^2"
      ],
      "replacement": {
       "(p1 + p2)^2": "s",
       "(p1 + p3)^2": "t",
       "p1^2": "0",
       "p2^2": "0",
       "p3^2": "0",
       "p4^2": "0"
      }
     },
     "integrals": [
      {
       "indices": [
        1,
        1,
        1,
        1,
        1,
        1,
        1,
        0,
        0
       ]
      }
     ],
     "mode": "solve_integrals",
     "options": {
      "d0": "4",
      "silent_mode": true
     }
    },
    "per_run": {
     "goal40": {
      "goal_digits": 40,
      "eps_order": 6
     },
     "goal70": {
      "goal_digits": 70,
      "eps_order": 8
     }
    },
    "note": "the four config.json files (shas per point below) are identical except goal_digits / eps_order, the point's numeric_values (s, t: stated per point) and the run's working directory (a machine path: not copied)"
   },
   "output_form": "each out.json lists result[0].coefficients = [{order, value: {re, im}}] from leading_order -4; re is an arb-style ball string '[midpoint +/- radius]' printed under working_pre 100 (about 110 significant digits; the radius is the engine's own and is not used as a certification), im is the plain string '0'; the strings below are copied verbatim for orders -4..0 (the runs also carry higher orders, beyond the served tower)",
   "output_options": {
    "chop_pre": 20,
    "d0": "4",
    "extra_x_order": 20,
    "learn_x_order": -1,
    "rationalize_pre": 100,
    "run_candidate": 10,
    "run_direction": "NegIm",
    "run_length": 1000,
    "run_radius": 2,
    "silent_mode": true,
    "test_x_order": 5,
    "working_pre": 100,
    "x_order": 100
   },
   "points": {
    "2/5": {
     "s": "-1",
     "t": "-2/5",
     "g40": {
      "source": {
       "record_relative": "work/dbox_eucl_x2_5/g40/out.json",
       "sha256": "dbede5242aff2b7f9c70390ce8b3f8bacc1b3e542f9dce44bda64c334d0e658c"
      },
      "config": {
       "record_relative": "work/dbox_eucl_x2_5/g40/config.json",
       "sha256": "2a0db24f7de5fa9200a63a7a73c0b055eea67db7299af1d1da2cb9d56b6612f4",
       "goal_digits": 40,
       "eps_order": 6
      },
      "leading_order": -4,
      "orders_in_file": [
       -4,
       -3,
       -2,
       -1,
       0,
       1,
       2
      ],
      "orders": {
       "-4": {
        "im": "0",
        "re": "[-10.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 +/- 1e-113]"
       },
       "-3": {
        "im": "0",
        "re": "[0.090679149603718897336151654547910227716921469945191128715746172937537463828755305935559377125423449568068669986 +/- 4.00e-112]"
       },
       "-2": {
        "im": "0",
        "re": "[64.045959605108058861749005759729470501675442485979677599765087472034126367212121396888355131934418591201866572 +/- 1.24e-109]"
       },
       "-1": {
        "im": "0",
        "re": "[144.49702811285982869088581478699412585848051979899061159825379936239304333890831292751816069575135202762284363 +/- 3.17e-108]"
       },
       "0": {
        "im": "0",
        "re": "[144.46856025804766393539036946708942833505034295468300202532366839749135268366297050013646820982480378261756758 +/- 4.52e-108]"
       }
      }
     },
     "g70": {
      "source": {
       "record_relative": "work/dbox_eucl_x2_5/g70/out.json",
       "sha256": "443fbf50ecb5c08fdc36778417a6439e1e6ad5e3568fca6e0ff3d196331b2a3d"
      },
      "config": {
       "record_relative": "work/dbox_eucl_x2_5/g70/config.json",
       "sha256": "4e67959218afd6474979be14f3af84092f404ad1b9b4990ba0dd1bd43a6008df",
       "goal_digits": 70,
       "eps_order": 8
      },
      "leading_order": -4,
      "orders_in_file": [
       -4,
       -3,
       -2,
       -1,
       0,
       1,
       2,
       3,
       4
      ],
      "orders": {
       "-4": {
        "im": "0",
        "re": "[-10.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 +/- 3e-113]"
       },
       "-3": {
        "im": "0",
        "re": "[0.090679149603718897336151654547910227716921469945191128715746172937537463828755305935559377125423449568068669982 +/- 1.70e-112]"
       },
       "-2": {
        "im": "0",
        "re": "[64.045959605108058861749005759729470501675442485979677599765087472034126367212121396888355131934418591202026719 +/- 3.22e-109]"
       },
       "-1": {
        "im": "0",
        "re": "[144.49702811285982869088581478699412585848051979899061159825379936239304333890831292751816069575134774509749190 +/- 1.12e-108]"
       },
       "0": {
        "im": "0",
        "re": "[144.46856025804766393539036946708942833505034295468300202532366839749135268366297050013646829065110773950078279 +/- 1.62e-108]"
       }
      }
     },
     "stored_ball_40d": {
      "source": {
       "record_relative": "dbox_eucl_x2_5.json",
       "sha256": "93105b3e0d8bc6988bb1ea659263d738c1c6c877c1c676f924519caf5c96949d"
      },
      "form": "the goal-40 ball of record: '[midpoint +/- radius]' for re, a zero-centred ball for im; midpoint printed to about 40 digits",
      "orders": {
       "-4": {
        "re": "[-10.000000000000000000000000000000000000000 +/- 1.01e-40]",
        "im": "[+/- 1.01e-40]"
       },
       "-3": {
        "re": "[0.090679149603718897336151654547910227717 +/- 1.79e-40]",
        "im": "[+/- 1.01e-40]"
       },
       "-2": {
        "re": "[64.045959605108058861749005759729470501675 +/- 5.43e-40]",
        "im": "[+/- 1.01e-40]"
       },
       "-1": {
        "re": "[144.497028112859828690885814786994125858481 +/- 5.81e-40]",
        "im": "[+/- 1.01e-40]"
       },
       "0": {
        "re": "[144.468560258047663935390369467089428335050 +/- 4.43e-40]",
        "im": "[+/- 1.01e-40]"
       }
      },
      "meta": {
       "phys": false,
       "x": "2/5",
       "goal": 40,
       "wall_s": 1440.6,
       "staggered": "40+70"
      }
     },
     "wall_s_pair": 1440.6,
     "pair_certified_digits_per_order": {
      "-4": 110,
      "-3": 109.36,
      "-2": 104.6,
      "-1": 97.53,
      "0": 90.25
     },
     "pair_certified_digits_min_over_orders": 90.25,
     "digit_count_of_record": {
      "served_tower_dps": 110,
      "two_precision_partner_dps": 140,
      "vs_goal40_digits_per_order": {
       "-4": 110,
       "-3": 109.34,
       "-2": 104.6,
       "-1": 97.53,
       "0": 90.25
      },
      "vs_goal70_digits_per_order": {
       "-4": 110,
       "-3": 110,
       "-2": 110,
       "-1": 110,
       "0": 109.95
      },
      "vs_goal40_min_over_orders": 90.25,
      "vs_goal70_min_over_orders": 109.95,
      "tower_two_precision_min_over_orders": 117.51,
      "certified_transport_bound": "1.0e-137",
      "inside_stored_ball_every_order": true
     }
    },
    "3/4": {
     "s": "-1",
     "t": "-3/4",
     "g40": {
      "source": {
       "record_relative": "work/dbox_eucl_x3_4/g40/out.json",
       "sha256": "07b960c2d00c49042dccceaa3e12c956d42edc38fc7f028a43cd998f54d2829d"
      },
      "config": {
       "record_relative": "work/dbox_eucl_x3_4/g40/config.json",
       "sha256": "387f973359faf2191cce2164333c86fed704febb12dfd6600f0c78dbd4517b2b",
       "goal_digits": 40,
       "eps_order": 6
      },
      "leading_order": -4,
      "orders_in_file": [
       -4,
       -3,
       -2,
       -1,
       0,
       1,
       2
      ],
      "orders": {
       "-4": {
        "im": "0",
        "re": "[-5.3333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333 +/- 3.34e-110]"
       },
       "-3": {
        "im": "0",
        "re": "[4.2390866092711443302080022542534430544263015107074446772170792697766360808232200610733626805845967528491825508 +/- 3.14e-110]"
       },
       "-2": {
        "im": "0",
        "re": "[31.338148866595200624424142844782477266451401354624917569173258429706694745007608331863406559047929666182779048 +/- 1.83e-109]"
       },
       "-1": {
        "im": "0",
        "re": "[37.461220975964120589791867866378865672244706643298023217155558412102361899814364088824561898251311797024540522 +/- 6.05e-110]"
       },
       "0": {
        "im": "0",
        "re": "[-7.2447981825593492878480131517564628323647155150205021186349034935510609682875856668521017907649846400296088375 +/- 4.16e-110]"
       }
      }
     },
     "g70": {
      "source": {
       "record_relative": "work/dbox_eucl_x3_4/g70/out.json",
       "sha256": "8d104e03f4f5e9bfb70c25aff5ad83d781d24deee204ab9d089a1d9010b9d39b"
      },
      "config": {
       "record_relative": "work/dbox_eucl_x3_4/g70/config.json",
       "sha256": "b44d5e74189a50904273ee51497c20db4198d666eaeb76723402a3f77729011d",
       "goal_digits": 70,
       "eps_order": 8
      },
      "leading_order": -4,
      "orders_in_file": [
       -4,
       -3,
       -2,
       -1,
       0,
       1,
       2,
       3,
       4
      ],
      "orders": {
       "-4": {
        "im": "0",
        "re": "[-5.3333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333 +/- 3.34e-110]"
       },
       "-3": {
        "im": "0",
        "re": "[4.2390866092711443302080022542534430544263015107074446772170792697766360808232200610733626805845967528491825508 +/- 3.01e-110]"
       },
       "-2": {
        "im": "0",
        "re": "[31.338148866595200624424142844782477266451401354624917569173258429706694745007608331863406559047929666182833403 +/- 1.69e-109]"
       },
       "-1": {
        "im": "0",
        "re": "[37.461220975964120589791867866378865672244706643298023217155558412102361899814364088824561898251310343507225843 +/- 2.72e-109]"
       },
       "0": {
        "im": "0",
        "re": "[-7.2447981825593492878480131517564628323647155150205021186349034935510609682875856668521017633320046490490228203 +/- 2.46e-110]"
       }
      }
     },
     "stored_ball_40d": {
      "source": {
       "record_relative": "dbox_eucl_x3_4.json",
       "sha256": "78b8ec99b78c780cb4eb71bfc720cc931dc785aa3bf39bdaffd3d871f7d6fbc9"
      },
      "form": "the goal-40 ball of record: '[midpoint +/- radius]' for re, a zero-centred ball for im; midpoint printed to about 40 digits",
      "orders": {
       "-4": {
        "re": "[-5.333333333333333333333333333333333333333 +/- 4.34e-40]",
        "im": "[+/- 1.01e-40]"
       },
       "-3": {
        "re": "[4.239086609271144330208002254253443054426 +/- 4.02e-40]",
        "im": "[+/- 1.01e-40]"
       },
       "-2": {
        "re": "[31.338148866595200624424142844782477266451 +/- 5.02e-40]",
        "im": "[+/- 1.01e-40]"
       },
       "-1": {
        "re": "[37.461220975964120589791867866378865672245 +/- 3.94e-40]",
        "im": "[+/- 1.01e-40]"
       },
       "0": {
        "re": "[-7.244798182559349287848013151756462832365 +/- 3.85e-40]",
        "im": "[+/- 1.01e-40]"
       }
      },
      "meta": {
       "phys": false,
       "x": "3/4",
       "goal": 40,
       "wall_s": 1429.4,
       "staggered": "40+70"
      }
     },
     "wall_s_pair": 1429.4,
     "pair_certified_digits_per_order": {
      "-4": 110,
      "-3": 110,
      "-2": 104.76,
      "-1": 97.41,
      "0": 89.42
     },
     "pair_certified_digits_min_over_orders": 89.42,
     "digit_count_of_record": {
      "served_tower_dps": 110,
      "two_precision_partner_dps": 140,
      "vs_goal40_digits_per_order": {
       "-4": 110,
       "-3": 110,
       "-2": 104.76,
       "-1": 97.41,
       "0": 89.42
      },
      "vs_goal70_digits_per_order": {
       "-4": 110,
       "-3": 110,
       "-2": 110,
       "-1": 110,
       "0": 110
      },
      "vs_goal40_min_over_orders": 89.42,
      "vs_goal70_min_over_orders": 110,
      "tower_two_precision_min_over_orders": 118.05,
      "certified_transport_bound": "7.83e-138",
      "inside_stored_ball_every_order": true
     }
    }
   },
   "held_out_of_fit": {
    "statement": "true (dbox_common.jl 50-53 'Last two = held-out pair', N_HOLDOUT = 2; oracle_dbox_massless.jl 192, 209-211: the fit runs over columns 1:ptr, the hold-out residual over ptr+1:p)",
    "sources": [
     {
      "record_relative": "scripts/dbox_common.jl",
      "sha256": "9956a26df0f422113bab9e3ef11534e2cacdc53bdda41b9c8af8e9d1941e4627",
      "lines": "50-53",
      "quote": [
       "# Euclidean: s=-1, t=-x, x>0 (K real there).  Last two = held-out pair.",
       "const XS_EUCLID = [1//7, 1//5, 1//4, 1//3, 1//2, 4//7, 3//5, 2//3, 5//6, 8//9,",
       "                   2//5, 3//4]",
       "const N_HOLDOUT = 2"
      ]
     },
     {
      "record_relative": "scripts/oracle_dbox_massless.jl",
      "sha256": "516ac6e6c6602398872bb17d0803b3fb5c7fc4a982457357c965a5f85f9dfbec",
      "lines": "192, 209-211",
      "quote": [
       "    ptr = p - N_HOLDOUT     # XS_EUCLID ordered so the held-out pair is last",
       "        fr = lattice_fit(fv[1:ptr], B[:, 1:ptr]; digits = 30)",
       "        res_fit = fit_residual(fr, fv[1:ptr], B[:, 1:ptr])",
       "        res_cv  = fit_residual(fr, fv[ptr+1:p], B[:, ptr+1:p])"
      ]
     }
    ]
   },
   "fit_points": {
    "x": [
     "1/7",
     "1/5",
     "1/4",
     "1/3",
     "1/2",
     "4/7",
     "3/5",
     "2/3",
     "5/6",
     "8/9"
    ],
    "statement": "x in {1/7, 1/5, 1/4, 1/3, 1/2, 4/7, 3/5, 2/3, 5/6, 8/9} (Euclidean, s = -1)"
   },
   "never_fit_physical_points": {
    "x": [
     "1/5",
     "1/3",
     "1/2",
     "2/3"
    ],
    "statement": "x in {1/5, 1/3, 1/2, 2/3} at s = +1 + i0 (dbox_phys_x*.json; listed for the record, NOT gated: the served transport runs on x > 0 only)"
   },
   "sample_set": "sixteen staggered goal-40/70 pairs on this family: twelve Euclidean points at s = -1 (ten fit, two held out) and four physical points at s = +1 + i0",
   "digit_count_record": {
    "sha256": "0796a5bdd10427e8a6634ba97f37048e13ecc71bf87e970559487b65d1e32270",
    "date": "2026-09-05",
    "law": "pair-certified: min over points and orders of the served tower's agreement with the goal-40 and goal-70 strings and of the goal-40 vs goal-70 agreement; capped by string length",
    "certified_floor_digits": 89,
    "raw_min_over_points_and_orders": 89.42
   },
   "gate": {
    "stage": "5. AMFLOW HELD-OUT GATE",
    "rule": "PASS iff, at every order eps^-4..eps^0 and at both points, the relative agreement of the served tower (mapped g -> J by the normalisation above) with BOTH the goal-40 and the goal-70 string is >= min(pair-certified digits of that order, working dps - 5); the pair-certified digits are the whole digits on which the goal-40 and goal-70 strings agree (computed from the strings at run time and compared with the per-order figures above; capped by the strings' own length); under --check the rerun's working dps is dps + 60 higher",
    "record_relative_names": "sample files relative to LandauBootstrap/results/samples; sources relative to LandauBootstrap"
   }
  }
 }
}