{
 "row": 6,
 "name": "two-loop Sudakov vertex: the planar massless ladder I[1,0,1,1,1,1,1](s) of the seven-propagator vertex family (six lines plus one irreducible numerator)",
 "provenance": {
  "date": "2026-09-03",
  "family": "two loops k1,k2; p1^2 = p2^2 = 0, (p1+p2)^2 = s; d = 4-2eps; propagators D1 = k1^2, D2 = (k1+p1)^2, D3 = (k1+p1+p2)^2, D4 = k2^2, D5 = (k2+p1)^2, D6 = (k2+p1+p2)^2, D7 = (k1-k2)^2; the ladder drops D2",
  "normalisation": "I[a_1..a_7](s) = int prod_{l=1,2} d^d k_l/(i pi^{d/2}) prod_j 1/D_j^{a_j}, Minkowski propagators with +i0, no e^{eps gamma_E} or (4 pi)^eps factor; values of record at s = -1 (Euclidean), held-out point s = -2",
  "closed_form": "I[1,0,1,1,1,1,1](s) = (-s)^(-2-2eps) [ -3/eps G(1,2) T(1,1+eps,1) + (1-2eps)^2/eps^2 G(1,1)^2 + 3(1-3eps)/eps^2 G(1,2) G(1+eps,1) ], G(a,b) = Gamma(a+b-d/2) Gamma(d/2-a) Gamma(d/2-b)/(Gamma(a) Gamma(b) Gamma(d-a-b)), T(a,b,c) = Gamma(a+b+c-d/2) Gamma(d/2-a-b) Gamma(d/2-b-c)/(Gamma(a) Gamma(c) Gamma(d-a-b-c))",
  "closed_form_source": "the family's own reduction onto five Gamma-product masters (Kira 3.1, at s = -1): three genuine masters -- the sunrise, the bubble-inserted triangle and the square of the one-loop bubble -- plus two dotted companions that are exact rational multiples of them; reduction coefficients -6/(d-4), 4(d-3)^2/(d-4)^2, 6(3d-10)/(d-4)^2 of I[1,0,1,0,1,0,2], I[1,0,1,1,0,1,0], I[0,0,1,1,0,0,2]",
  "production_receipt_sha256": "f567a5b0a8d894dacc0036c8b95652db28c8614fd1f43ae467da7e268df00b04",
  "masters": "Gehrmann, Huber, Maitre, hep-ph/0507061 (A_2^2, A_3, A_4); the ladder itself is not in their paper, only the masters",
  "history": "the closed form shown for this row earlier in 2026 described a different object and was withdrawn; the row was re-derived from the family's own reduction and this data file carries only the re-derived form's reference inputs",
  "vendoring": "the three reference inputs below are carried in content (every number, order, index and integer relation as produced); machine paths and internal working notes of the originals are omitted and described in plain words; the sha256 of each ORIGINAL file is recorded beside it as provenance (the originals are not shipped, so those hashes document the source rather than gate this file). This file as a whole is sha256-pinned inside sudakov-evaluate.py, which refuses a mutated or truncated copy before any computation.",
  "digit_ceiling": "the numerical references are goal-70 auxiliary-mass-flow runs at working precision 100; their per-order ball radii grow from 1e-115 at eps^-4 to a few 1e-108 at eps^6, so agreement with the exact form is bounded near 78 digits at eps^6 by the numerical side, not by the closed form"
 },
 "sources": {
  "amflow_s1": {
   "description": "auxiliary-mass-flow evaluation of all ten integrals of the planar two-loop massless vertex family at s = -1 (values of record), goal 70 digits, working precision 100; each Laurent coefficient is an arb-style real ball '[value +/- radius]' with zero imaginary part; the ladder is the entry with indices [1,0,1,1,1,1,1], orders eps^-4 .. eps^6",
   "original_file": "g70_R0.json",
   "original_sha256": "d21e0731b99ec6c09048b478509ef9a02ab22001470b395297ed9a09d5763b0f",
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     "rationalize_pre": 100,
     "run_candidate": 10,
     "run_direction": "NegRe",
     "run_length": 1000,
     "run_radius": 2,
     "silent_mode": true,
     "test_x_order": 5,
     "working_pre": 100,
     "x_order": 100
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  },
  "zeta_tower": {
   "description": "integer-relation (PSLQ) search over weight-graded monomials in pi^2, zeta3, zeta5, zeta7, zeta9, zeta11 for the coefficients of the s = -1 ladder rescaled by e^{2 eps gamma_E}, taken from the amflow_s1 numbers; 'ladder' lists one entry per order eps^-4 .. eps^6 with the pool, height ladder, raw relation, re-verification residual and verdict as produced, and 'closed_form_reduced' (the raw relation reduced to lowest terms, added afterwards; raw vectors untouched). Order eps^6 has no closed form: the search returned NULL through its height ladder; since 2026-09-11 a second order-6 entry beside that NULL row carries the exact eps^6 coefficient from the closed form's own expansion (verdict 'CLOSED (exact route)', its receipt by sha256), which sudakov-evaluate.py's zeta-value control reads -- the search row stays as the record",
   "original_file": "PSLQ_LEG.json",
   "original_sha256": "11f0bb59e3f0d85beb41edd23ec5644b4a56781293b94796e60a90cd6d6a1c19",
   "content": {
    "stamp_utc": "2026-09-03T14:47:53Z",
    "protocol": {
     "normalisation": "AMFlow x exp(2 eps gamma_E)",
     "build_dps": 130,
     "dps_pair": [
      88,
      100
     ],
     "maxcoeff_ladder": [
      10000000,
      100000000,
      1000000000,
      10000000000
     ],
     "pool": "weight-graded monomials in pi^2, zeta3, zeta5, zeta7, zeta9, zeta11 of the coefficient's weight",
     "controls": "harness synthetic control planted in each pool (3 m0 - 7 m1)/5; degeneracy audit; capacity law; reverify at 100 d with a per-order tolerance matched to the measured AMFlow accuracy (cross-point scaling gate: >= 78 d at every order; heights up to 1e10 -> tol 1e-90 for w<=7, 1e-68 for w>=8)"
    },
    "results": {
     "ladder": [
      {
       "order": -4,
       "weight": 0,
       "target_rescaled": "0.25",
       "verdict": "RATIONAL: 0.25 -> (1/4)"
      },
      {
       "order": -3,
       "weight": 1,
       "target_rescaled": "4.903541240480252007244471816784658297419e-111",
       "verdict": "ZERO (|c|<1e-95)"
      },
      {
       "order": -2,
       "weight": 2,
       "target_rescaled": "2.056167583560283045590518958307531486524",
       "pool": [
        "pi2"
       ],
       "direct_ratio": "0.2083333333333333333333333333333333333333",
       "rational": "5/24",
       "two_precision_identical": true,
       "residual_95d": true,
       "closed_form": "(5/24)*pi2",
       "verdict": "CLOSED (direct ratio)",
       "closed_form_reduced": "(5/24)*pi2"
      },
      {
       "order": -1,
       "weight": 3,
       "target_rescaled": "5.809941698604705712765401113972008288697",
       "pool": [
        "z3"
       ],
       "direct_ratio": "4.833333333333333333333333333333333333333",
       "rational": "29/6",
       "two_precision_identical": true,
       "residual_95d": true,
       "closed_form": "(29/6)*z3",
       "verdict": "CLOSED (direct ratio)",
       "closed_form_reduced": "(29/6)*z3"
      },
      {
       "order": 0,
       "weight": 4,
       "target_rescaled": "9.132102284437728490916281189566104179662",
       "pool": [
        "pi2^2"
       ],
       "direct_ratio": "0.09375",
       "rational": "3/32",
       "two_precision_identical": true,
       "residual_95d": true,
       "closed_form": "(3/32)*pi2^2",
       "verdict": "CLOSED (direct ratio)",
       "closed_form_reduced": "(3/32)*pi2^2"
      },
      {
       "order": 1,
       "weight": 5,
       "target_rescaled": "-1.147004436084910198212765868714981906387",
       "pool": [
        "pi2*z3",
        "z5"
       ],
       "members_audit": "[]",
       "height_ladder": [
        {
         "maxcoeff": 10000000,
         "capacity": {
          "required": 41.0,
          "available": 88,
          "ok": true
         },
         "controls_ok": true,
         "result": "HIT"
        }
       ],
       "relation": [
        180,
        535,
        -5922
       ],
       "maxcoeff_used": 10000000,
       "closed_form": "(-535/180)*pi2*z3 + (5922/180)*z5",
       "reverify_100d": {
        "relresid": "7.091e-99",
        "ok": true
       },
       "reverify_tol": "1e-90",
       "verdict": "CLOSED",
       "closed_form_reduced": "(-107/36)*pi2*z3 + (329/10)*z5"
      },
      {
       "order": 2,
       "weight": 6,
       "target_rescaled": "-39.47986642509392591953152764481122030423",
       "pool": [
        "pi2^3",
        "z3^2"
       ],
       "members_audit": "[]",
       "height_ladder": [
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         "maxcoeff": 10000000,
         "capacity": {
          "required": 41.0,
          "available": 88,
          "ok": true
         },
         "controls_ok": true,
         "result": "HIT"
        }
       ],
       "relation": [
        60480,
        -1723,
        2798880
       ],
       "maxcoeff_used": 10000000,
       "closed_form": "(1723/60480)*pi2^3 + (-2798880/60480)*z3^2",
       "reverify_100d": {
        "relresid": "1.558e-98",
        "ok": true
       },
       "reverify_tol": "1e-90",
       "verdict": "CLOSED",
       "closed_form_reduced": "(1723/60480)*pi2^3 + (-833/18)*z3^2"
      },
      {
       "order": 3,
       "weight": 7,
       "target_rescaled": "-131.2187512347915245788664533604576566877",
       "pool": [
        "pi2*z5",
        "pi2^2*z3",
        "z7"
       ],
       "members_audit": "[]",
       "height_ladder": [
        {
         "maxcoeff": 10000000,
         "capacity": {
          "required": 48.0,
          "available": 88,
          "ok": true
         },
         "controls_ok": true,
         "result": "HIT"
        }
       ],
       "relation": [
        336,
        5908,
        511,
        -75576
       ],
       "maxcoeff_used": 10000000,
       "closed_form": "(-5908/336)*pi2*z5 + (-511/336)*pi2^2*z3 + (75576/336)*z7",
       "reverify_100d": {
        "relresid": "1.07e-98",
        "ok": true
       },
       "reverify_tol": "1e-90",
       "verdict": "CLOSED",
       "closed_form_reduced": "(-211/12)*pi2*z5 + (-73/48)*pi2^2*z3 + (3149/14)*z7"
      },
      {
       "order": 4,
       "weight": 8,
       "target_rescaled": "-307.2292518563273176362353766361937740937",
       "pool": [
        "pi2*z3^2",
        "pi2^4",
        "z3*z5"
       ],
       "members_audit": "[]",
       "height_ladder": [
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         "maxcoeff": 10000000,
         "capacity": {
          "required": 48.0,
          "available": 88,
          "ok": true
         },
         "controls_ok": true,
         "result": "NULL"
        },
        {
         "maxcoeff": 100000000,
         "capacity": {
          "required": 52.0,
          "available": 88,
          "ok": true
         },
         "controls_ok": true,
         "result": "NULL"
        },
        {
         "maxcoeff": 1000000000,
         "capacity": {
          "required": 56.0,
          "available": 88,
          "ok": true
         },
         "controls_ok": true,
         "result": "NULL"
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        {
         "maxcoeff": 10000000000,
         "capacity": {
          "required": 60.0,
          "available": 88,
          "ok": true
         },
         "controls_ok": true,
         "result": "HIT"
        }
       ],
       "relation": [
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        -41372800,
        -17727,
        1204600320
       ],
       "maxcoeff_used": 10000000000,
       "closed_form": "(41372800/2419200)*pi2*z3^2 + (17727/2419200)*pi2^4 + (-1204600320/2419200)*z3*z5",
       "reverify_100d": {
        "relresid": "7.358e-89",
        "ok": true
       },
       "reverify_tol": "1e-68",
       "verdict": "CLOSED",
       "closed_form_reduced": "(1847/108)*pi2*z3^2 + (5909/806400)*pi2^4 + (-7469/15)*z3*z5"
      },
      {
       "order": 5,
       "weight": 9,
       "target_rescaled": "-633.989347077190193285469010711238399608",
       "pool": [
        "pi2*z7",
        "pi2^2*z5",
        "pi2^3*z3",
        "z3^3",
        "z9"
       ],
       "members_audit": "[]",
       "height_ladder": [
        {
         "maxcoeff": 10000000,
         "capacity": {
          "required": 62.0,
          "available": 88,
          "ok": true
         },
         "controls_ok": true,
         "result": "NULL"
        },
        {
         "maxcoeff": 100000000,
         "capacity": {
          "required": 68.0,
          "available": 88,
          "ok": true
         },
         "controls_ok": true,
         "result": "NULL"
        },
        {
         "maxcoeff": 1000000000,
         "capacity": {
          "required": 74.0,
          "available": 88,
          "ok": true
         },
         "controls_ok": true,
         "result": "HIT"
        }
       ],
       "relation": [
        90720,
        10483560,
        731430,
        50293,
        -18606560,
        -146205360
       ],
       "maxcoeff_used": 1000000000,
       "closed_form": "(-10483560/90720)*pi2*z7 + (-731430/90720)*pi2^2*z5 + (-50293/90720)*pi2^3*z3 + (18606560/90720)*z3^3 + (146205360/90720)*z9",
       "reverify_100d": {
        "relresid": "7.321e-82",
        "ok": true
       },
       "reverify_tol": "1e-68",
       "verdict": "CLOSED",
       "closed_form_reduced": "(-9707/84)*pi2*z7 + (-129/16)*pi2^2*z5 + (-50293/90720)*pi2^3*z3 + (16613/81)*z3^3 + (29009/18)*z9"
      },
      {
       "order": 6,
       "weight": 10,
       "target_rescaled": "-1235.936250294814108200788319342058133199",
       "pool": [
        "pi2*z3*z5",
        "pi2^2*z3^2",
        "pi2^5",
        "z3*z7",
        "z5^2"
       ],
       "members_audit": "[]",
       "height_ladder": [
        {
         "maxcoeff": 10000000,
         "capacity": {
          "required": 62.0,
          "available": 88,
          "ok": true
         },
         "controls_ok": true,
         "result": "NULL"
        },
        {
         "maxcoeff": 100000000,
         "capacity": {
          "required": 68.0,
          "available": 88,
          "ok": true
         },
         "controls_ok": true,
         "result": "NULL"
        },
        {
         "maxcoeff": 1000000000,
         "capacity": {
          "required": 74.0,
          "available": 88,
          "ok": true
         },
         "controls_ok": true,
         "result": "NULL"
        },
        {
         "maxcoeff": 10000000000,
         "capacity": {
          "required": 80.0,
          "available": 88,
          "ok": true
         },
         "controls_ok": true,
         "result": "NULL"
        }
       ],
       "relation": null,
       "maxcoeff_used": null,
       "verdict": "NULL through the height ladder [10000000, 100000000, 1000000000, 10000000000] (two-precision)"
      },
      {
       "order": 6,
       "weight": 10,
       "route": "exact: route (A) of sudakov-evaluate.py carried out in exact arithmetic -- each ln Gamma(1 + x eps) = -gamma_E x eps + sum_{k>=2} (-1)^k zeta(k) (x eps)^k / k as a truncated series over Q[gamma_E, zeta(k)], the three exponentials and the reduction coefficients combined, times e^{2 eps gamma_E}; no gamma_E survives at any order and every order eps^-4 .. eps^5 of that exact tower equals the entry above it in this list identically",
       "closed_form_reduced": "(-64793/21)*z3*z7 + (-63209/50)*z5^2 + (46603/22809600)*pi2^5 + (6961/720)*pi2^2*z3^2 + (16907/90)*pi2*z3*z5",
       "closed_form_zeta_notation": "-64793*zeta(3)*zeta(7)/21 - 63209*zeta(5)**2/50 + 46603*pi**10/22809600 + 6961*pi**4*zeta(3)**2/720 + 16907*pi**2*zeta(3)*zeta(5)/90",
       "numeric_60d": "-1235.93625029481410820078831934205813319947005772511642087045",
       "verdict": "CLOSED (exact route)",
       "receipt": {
        "name": "ROW6_EPS6_RERUN_20260911T100039Z.json",
        "sha256": "e38970631db7bfdd128de8a42264a872b59d546810d61a29861ed0e609444723"
       },
       "note": "not an integer-relation result: the search at this order returned NULL through its whole height sequence (the entry above, kept as the record); sudakov-evaluate.py's zeta-value control reads this entry at eps^6"
      }
     ]
    }
   }
  }
 }
}