{
 "file": "frobenius_coefficients_rows31_32_20260909T042148Z.json",
 "stamp_utc": "2026-09-09T04:21:48Z",
 "purpose": "Table 2 rows 31 and 32: the Frobenius coefficients of the eps^0 layer of the physical master in the IBP operator's z = 0 Frobenius basis under the stated particular-solution convention -- the full-precision strings the paper's footnote does not print (it prints 9 significant digits with an ellipsis and the floored digit claim)",
 "source_receipt": {
  "path": "runs/amplitudes/strengthen_2026-09-09/R1_frobenius_convention/RECEIPTS/R1_FROBENIUS_COEFFICIENTS_v2_20260909T041725Z.json",
  "sha256": "f367d80f571f5b4f68fda3da49d871ccb043399746904ae98452f40305466154"
 },
 "row31_K3_1119": {
  "convention": "f_0 = a_0 Pi_0 + a_1 Pi_1 + a_2 Pi_2 + a_3 Pi_3 + f_p with Pi_0 = z varpi_0 (varpi_0 = 1 - 12 z + ..., the operator's holomorphic solution in the Feynman normalisation), Pi_1 = Pi_0 log z + z h_1, Pi_2 = (1/2) Pi_0 log^2 z + z h_2 log z + z h_3 (h_i(0) = 0, no z^2 term in h_1, h_3), Pi_3 = z^2 (1 + O(z)), and f_p the particular solution whose z = 0 expansion has no z, z log z, z log^2 z or z^2 term",
  "variable": "z = -1/p^2 (the IBP gauge of leg (I): L_IBP = L_record(theta_z -> theta_z - 1), exponents {1,1,1,2} at z = 0)",
  "basis_first_coefficients": {
   "Pi_0/z": [
    "1",
    "-12",
    "204",
    "-4224",
    "99324",
    "-2546352",
    "69359424",
    "-1973611008"
   ],
   "Pi_1: p_1(u)": [
    "0",
    "-12"
   ],
   "Pi_2: p_1(u)": [
    "0",
    "-18",
    "-6"
   ],
   "Pi_3/z^2 head": [
    "1",
    "-23",
    "1594/3",
    "-13103",
    "1716506/5"
   ]
  },
  "served_31_digit_strings": {
   "a_0": "19.23291045055350856639581058418",
   "a_2": "-13.18334746401731629674294284307",
   "a_3": "-38.53322028786146962605645725544"
  },
  "a_1": {
   "value": "0 at the floor",
   "bound": "|a_1| <= 10^-31",
   "abs_over_a0_by_leg": {
    "goal60_dps40": "2.1e-33",
    "goal80_dps50": "2.73e-44"
   }
  },
  "digits": {
   "served": 31,
   "rule": "int(min) of the two-precision agreement over {a_0, a_2, a_3}; the two members coincide at the served length",
   "members_two_precision_digits": {
    "a_0": 32.2,
    "a_2": 32.6,
    "a_3": 31.7
   }
  },
  "members_full": {
   "a_0": {
    "goal60_dps40": "19.23291045055350856639581058418308637789",
    "goal80_dps50": "19.232910450553508566395810584183199852239777578731"
   },
   "a_2": {
    "goal60_dps40": "-13.1833474640173162967429428430702776343",
    "goal80_dps50": "-13.183347464017316296742942843070308455769885107796"
   },
   "a_3": {
    "goal60_dps40": "-38.53322028786146962605645725543677500609",
    "goal80_dps50": "-38.533220287861469626056457255437532353840895982169"
   }
  },
  "legs": {
   "goal60_dps40": "the goal-60 AMFlow point at pB (pp = -80) + the goal-60 completed masters, read at dps 40 / N 175",
   "goal80_dps50": "the goal-80 point + the goal-80 completion, read at dps 50 / N 204"
  },
  "floor_source": {
   "reading": "the completed masters at pB (from the LEG_IIIab receipt) enter the matching, so the goal-60 completion floor bounds the pair",
   "key": "LEG_IIIab_RECEIPT IIIa.completion.<goalkey>.residual_floor_digits_int",
   "values": {
    "goal60_dps40": 33,
    "goal80_dps50": 43
   }
  }
 },
 "row32_CY3_111116": {
  "convention": "f_0 = sum_{k=0}^{5} a_k Pi_k + f_p with Pi_0 = z varpi_0 (varpi_0 = 1 - 20 z + ..., the operator's holomorphic solution in the Feynman normalisation), Pi_k = Pi_0 log^k z / k! + lower log powers (k = 1, 2, 3; the correction series start at O(z^2) relative, no z^2 or z^2 log z term), Pi_4 = z^2 (1 + O(z)), Pi_5 = z^2 log z + ... (no z^2 term), and f_p the particular solution whose z = 0 expansion has no z log^j z (j = 0..3), z^2 or z^2 log z term",
  "variable": "z = -1/p^2 (the IBP gauge of leg (I): L_IBP = L_record(theta_z -> theta_z - 1), exponents {1,1,1,1,2,2} at z = 0)",
  "basis_first_coefficients": {
   "Pi_0/z": [
    "1",
    "-20",
    "540",
    "-17600",
    "653980",
    "-26742480",
    "1175054400",
    "-54549619200"
   ],
   "Pi_k: p_1(u)": [
    [
     "-20"
    ],
    [
     "0"
    ],
    [
     "0",
     "0",
     "-10"
    ],
    [
     "0",
     "0",
     "-16",
     "-10/3"
    ],
    [
     "1"
    ],
    [
     "0",
     "1"
    ]
   ]
  },
  "served_31_digit_strings": {
   "a_0": "53.32495061105228758052880189834",
   "a_1": "96.16455225276754283197905292092",
   "a_3": "-66.54212933375474970405428365998",
   "a_4": "-3210.349930756070870031438260789",
   "a_5": "-1818.616975715274861374712519859"
  },
  "a_2": {
   "value": "0 at the floor",
   "bound": "|a_2| <= 10^-31",
   "abs_over_a0_by_leg": {
    "dps40": "1.0e-33",
    "dps50": "1.06e-43"
   }
  },
  "digits": {
   "served": 31,
   "rule": "int(min) of the two-precision agreement over the five non-vanishing coefficients; the two members coincide at the served length",
   "members_two_precision_digits": {
    "a_0": 32.0,
    "a_1": 33.7,
    "a_3": 33.3,
    "a_4": 33.2,
    "a_5": 32.7
   }
  },
  "members_full": {
   "a_0": {
    "dps40": "53.32495061105228758052880189834039575275",
    "dps50": "53.32495061105228758052880189833984664923733354435"
   },
   "a_1": {
    "dps40": "96.16455225276754283197905292091598117875",
    "dps50": "96.164552252767542831979052920915999261198894112372"
   },
   "a_3": {
    "dps40": "-66.54212933375474970405428365998498255731",
    "dps50": "-66.542129333754749704054283659984950535248010166656"
   },
   "a_4": {
    "dps40": "-3210.349930756070870031438260789280045811",
    "dps50": "-3210.3499307560708700314382607892818774288607217068"
   },
   "a_5": {
    "dps40": "-1818.616975715274861374712519858557391939",
    "dps50": "-1818.6169757152748613747125198585607766600100852727"
   }
  },
  "pair_note": "ONE landed point inside the disc (pB at goal 60): the pair is the read at two dps on the same object; the data's own two-goal agreement is the pA pair (LEG_III_ROW32_SECOND_POINTS two_precision_pair_pA.min_digits_int_floor 62), not repeated here",
  "legs": {
   "dps40": "the goal-60 AMFlow point at pB (pp = -80), all nine masters, read at dps 40 / N 472",
   "dps50": "the same point read at dps 50 / N 576"
  }
 },
 "how_to_check": "the reader scripts frob_convention_k3_20260909T034956Z.py and frob_convention_cy3_20260909T035622Z.py (runs/amplitudes/strengthen_2026-09-09/R1_frobenius_convention/scripts/) match the exact eps-layered local expansion at z = 0 to the landed AMFlow point at pB inside the disc; python-flint arb; the exact Frobenius basis is checked against the BFKNS period series of the record",
 "PRODUCER": {
  "script": "emit_r1_datafile.py",
  "script_sha256": "b628339b7611fac7357ab7f81a92ddb1976fec5d5c8517ca1b05047dd430fbb5",
  "stamp_source": "date -u"
 }
}