{
 "title": "Assembly data for the constant c_M of the memory-region integrals (Sec. 7 and Table 7 of the paper): two implementations under the retarded routing, and the leading terms with Feynman propagators on the memory gravitons",
 "conventions": {
  "static_point": "x = 1",
  "gamma_stripping": "all values at gamma = sqrt(2), where the overall powers of (gamma^2 - 1) equal one",
  "measure": "Int_omega = Int d omega/(2 pi); omega_3 = omega_1 + omega_2",
  "routing": "retarded propagators on both memory gravitons, s_1 = s_2 = +1 in Eq. (57); the all-advanced routing gives identical values, so the average of the gamma-3 prescription is automatic",
  "identity": "I_2^(M) = r_1(eps) I_3^(M) + r_2(eps) I_1^(M), Eqs. (53)-(54)",
  "c_M": "I_2^(M) = -5 c_M/(6 (8 pi)^4 eps^2) + O(1/eps), Eq. (16)",
  "laurent_keys": "string k -> coefficient of eps^k (decimal strings as printed by the runs)",
  "equation_numbers": "as in the revised manuscript that this package accompanies (Sec. 7.1-7.2, Table 7)"
 },
 "retarded": {
  "implementation_A": {
   "method": "expansion of the integrands of I_1^(M), I_3^(M) in eps through second order; two-dimensional tanh-sinh quadrature of the resulting frequency integrals (cores), split into same-sign and opposite-sign frequency sectors; exact Laurent expansion of the prefactors N_1, N_3 and of the coefficients r_1, r_2 of the integration-by-parts identity I_2 = r_1 I_3 + r_2 I_1 (Eqs. (53)-(54)); I_2^(M) from that identity",
   "working_precision_digits": 35,
   "sector_integrals": {
    "definition": "A[sector,a,w]: raw two-dimensional integrals.  sector 'same': (u,v) in (0,inf)^2 with |omega_3| = u+v, integrand 2*sgn*(u v)^p K_0(u)K_0(v)K_0(u+v) * w; sector 'opp': (v,w) in (0,inf)^2 with u = v+w (the u > v half of one opposite-sign configuration), integrand 2*sgn*(u v)^p K_0(u)K_0(v)K_0(w) * w.  p = 1 for a = 1, p = 2 for a = 3; sgn = -1 except sgn(opp, a=1) = +1 (the retarded sign pattern of -omega_1^p omega_2^p summed over the two quadrants of each sector).  Weights w: '1'; 'b1' = -(3 ln u + 3 ln v + ln|omega_3|); 'b1sq' = b1^2; 'R' = sum_i kappa_2(|omega_i|)/K_0(|omega_i|) with kappa_2 = (1/2) d^2 K_nu/d nu^2 at nu = 0 (Eq. (56)).  Cores: j_a^(0) = (A[same,a,1] + 2 A[opp,a,1])/(2 pi)^2, j_a^(1) likewise with 'b1', j_a^(2) = ((A[same,a,b1sq]+2A[opp,a,b1sq])/2 + A[same,a,R] + 2A[opp,a,R] - 2 pi^2 A[same,a,1])/(2 pi)^2 (the -2 pi^2 term is the real part of the retarded phases, present in the same-sign sectors only).",
    "values_25_digits": {
     "same,1,1": "-0.228017582173806283607356",
     "same,1,b1": "-0.9363053870563734210357536",
     "same,1,b1sq": "-8.682056764705753152077277",
     "same,1,R": "-0.4102436439914343444220179",
     "same,3,1": "-0.113737589951934843818107",
     "same,3,b1": "0.05080750060809472258202251",
     "same,3,b1sq": "-1.378582178693526785358664",
     "same,3,R": "-0.121310185266184224067244",
     "opp,1,1": "0.771982417826193716392644",
     "opp,1,b1": "1.557195147473730675636082",
     "opp,1,b1sq": "19.08456911190713455726882",
     "opp,1,R": "1.807157456280905310286605",
     "opp,3,1": "-1.44707092328526817715144",
     "opp,3,b1": "3.250458608298017181504031",
     "opp,3,b1sq": "-24.77391953826925388689294",
     "opp,3,R": "-2.661178318962637097012074"
    }
   },
   "cores": {
    "definition": "I_a^(M)(eps) = N_a(eps) * sum_k j_a^(k) eps^k with N_a the prefactors of Eqs. (51), (52) (values at gamma = sqrt 2, i.e. overall powers of (gamma^2-1) stripped) and the frequency measure d omega/(2 pi) per frequency included in j",
    "values_25_digits": {
     "j_1^(0)": "0.03333333333333333333333333",
     "j_1^(1)": "0.05517153523525931749972088",
     "j_1^(2)": "0.5686271138877558463074322",
     "j_3^(0)": "-0.07619047619047619047619048",
     "j_3^(1)": "0.1659571258114707663498443",
     "j_3^(2)": "-0.7260115565251557098338957"
    },
    "exact_leading": {
     "j_1^(0)": "1/30",
     "j_3^(0)": "-8/105",
     "check_j_1^(0)_minus_1/30": "6.1713e-34"
    }
   },
   "prefactor_laurent": {
    "N_1": {
     "-1": "5.01268664779508554e-6",
     "0": "0.0000945100272444485154",
     "1": "0.00100139639567664462",
     "2": "0.00761650479258341618",
     "3": "0.0466948544883892074",
     "4": "0.245557010282393056",
     "5": "1.16065109679737139"
    },
    "N_3": {
     "-1": "7.83232288717982116e-8",
     "0": "1.51588079013040716e-6",
     "1": "0.000016483082306384574",
     "2": "0.000128921955785182168",
     "3": "0.000813896216075805031",
     "4": "0.00441235462308974134",
     "5": "0.0214924033738584983"
    },
    "note": "coefficients of eps^k, 18 significant digits printed"
   },
   "ibp_coefficient_laurent": {
    "r_2 (multiplies I_1)": {
     "-3": "-2.4",
     "-2": "42.6933333333333333",
     "-1": "-293.924",
     "0": "964.988266666666667",
     "1": "-1464.75210666666667",
     "2": "872.063296",
     "3": "123.367940266666667"
    },
    "r_1 (multiplies I_3)": {
     "-3": "-67.2",
     "-2": "971.413333333333333",
     "-1": "-4766.272",
     "0": "7794.49813333333333",
     "1": "1403.73034666666667",
     "2": "5324.101888",
     "3": "20709.7630208"
    },
    "note": "Laurent expansion of the rational functions r_1, r_2 of Eq. (54), printed to 18 digits"
   },
   "I1_laurent": {
    "-1": "1.67089554926502851e-7",
    "0": "3.42689185949375766e-6",
    "1": "0.0000414444926954499726"
   },
   "I3_laurent": {
    "-1": "-5.96748410451795898e-9",
    "0": "-1.02497381300193386e-7",
    "1": "-1.0611462403126798e-6"
   },
   "I2_laurent": {
    "-4": "-1.00498883783832961e-38",
    "-3": "-5.62492413967696171e-40",
    "-2": "-2.08861943658128564e-6"
   },
   "note_orders": "I_1, I_3 complete through eps^1 and I_2 through eps^-2 (the cores are expanded through second order); the eps^-4 and eps^-3 entries of I_2 are the numerical residuals of the pole cancellation",
   "c_M": {
    "value": "1.0",
    "c_M_minus_1": "-2.06836e-31",
    "digits_matching_1": "30.68",
    "I2_eps^-2": "-0.000002088619436581285642324378",
    "definition": "c_M = -(6 (8 pi)^4 / 5) * I_2^(M)[eps^-2]",
    "pole_cancellation_relative_to_eps^-2": {
     "eps^-4": "-4.812e-33",
     "eps^-3": "-2.693e-34"
    },
    "integer_relation_search": "basis [c_M, '1', 'pi^2', 'log2', 'zeta3', 'pi^2*log2'] -> [1, -1, 0, 0, 0, 0] (c_M = 1, no admixture)"
   }
  },
  "implementation_B": {
   "method": "independent implementation of the same expansion: sector decomposition of the two-frequency cores, tanh-sinh quadrature, kappa_2 by finite differences; prefactors in closed Gamma-function form",
   "working_precision_digits": 30,
   "reported_digits": 25,
   "I1_laurent": {
    "-1": "0.000000167089554926502851385950",
    "0": "0.00000342689185949375765535940",
    "1": "0.0000414444926954499725596585"
   },
   "I3_laurent": {
    "-1": "-5.96748410451795897806965e-9",
    "0": "-0.000000102497381300193386335937",
    "1": "-0.00000106114624031267979835865"
   },
   "I2_laurent": {
    "-4": "-2.35345638114070372102871e-35",
    "-3": "-2.45570368418670507767558e-35",
    "-2": "-0.00000208861943658128564232438",
    "-1": "0.0000139197867618983997584533"
   },
   "checks": {
    "I1 pole vs 1/(15 (8 pi)^4), digits": 28.2,
    "I1 eps^0, eps^1 vs implementation A (its 25-digit cores with exact prefactors), digits": [
     25.0,
     25.0
    ],
    "j_3^(0) vs -8/105, digits": 30.5,
    "c_M vs 1, digits": 25.4
   }
  },
  "agreement_A_vs_B_digits_from_printed_strings": {
   "I1[eps^-1]": 17.64,
   "I1[eps^0]": 17.87,
   "I1[eps^1]": 18.01,
   "I3[eps^-1]": 18.49,
   "I3[eps^0]": 17.48,
   "I3[eps^1]": 17.81
  },
  "note_agreement": "the comparison of the printed Laurent strings is limited by the 18 significant digits printed by implementation A; implementation B records agreement of I_1 at eps^0 and eps^1 with implementation A to 25 digits using the 25-digit cores"
 },
 "feynman_propagators": {
  "assignment": "Feynman propagators on the two memory gravitons (D_31 and D_24 of arXiv:2601.16256), applied to each frequency factor as a whole: (0^+ - i s omega_j)^(a - q eps) -> (0^+ - i|omega_j|)^(a - q eps), so ln(0^+ - i|omega|) = ln|omega| - i pi/2 for either sign of omega (Eq. (57) of the paper and the text following Table 7)",
  "consequence": "relative to the retarded assignment the contributions of the opposite-sign frequency sectors change sign at every order in eps; the leading cores and the eps^-1 pole of I_1^(M) change, and the eps^-4 pole of I_2^(M) in the identity (53) no longer cancels.  Beyond leading order the Feynman-propagator integrals are complex; only the leading (real) terms are given.",
  "quadrant_moments": {
   "definition": "S_same = Int_{omega_1, omega_2 > 0} omega_1 omega_2 K_0(omega_1) K_0(omega_2) K_0(omega_1+omega_2); S_opp = the same integral of |omega_1 omega_2| over one quadrant with omega_1 omega_2 < 0; T_same, T_opp likewise with omega_1^2 omega_2^2.  Retarded: Int_{R^2}(-omega_1 omega_2) prod K_0 = -2 S_same + 2 S_opp = 2 pi^2/15 and Int(-omega_1^2 omega_2^2) prod K_0 = -2 T_same - 2 T_opp = -32 pi^2/105 (Eq. (59)); Feynman: -2 S_same - 2 S_opp and -2 T_same + 2 T_opp",
   "values": {
    "S_same": {
     "numerical_25_digits": "0.114008791086903141803678",
     "closed_form": "1/3 - pi^2/45",
     "integer_relation": "[-45, 15, -1] . (x, 1, pi^2) = 0 at 24 and at 18 digits",
     "agreement_digits": 24.7
    },
    "S_opp": {
     "numerical_25_digits": "0.771982417826193716392644",
     "closed_form": "1/3 + 2 pi^2/45",
     "integer_relation": "[-45, 15, 2] . (x, 1, pi^2) = 0 at 24 and at 18 digits",
     "agreement_digits": 25.2
    },
    "T_same": {
     "numerical_25_digits": "0.0568687949759674219090535",
     "closed_form": "16 pi^2/315 - 4/9",
     "integer_relation": "[315, 140, -16] . (x, 1, pi^2) = 0 at 24 and at 18 digits",
     "agreement_digits": 24.7
    },
    "T_opp": {
     "numerical_25_digits": "1.44707092328526817715144",
     "closed_form": "4/9 + 32 pi^2/315",
     "integer_relation": "[-315, 140, 32] . (x, 1, pi^2) = 0 at 24 and at 18 digits",
     "agreement_digits": 24.6
    }
   },
   "independent_low_precision_check": {
    "S_same": "0.114008791087",
    "S_opp": "0.771982417826"
   }
  },
  "leading_cores": {
   "j_1F^(0)": {
    "closed_form": "-1/90 - 1/(3 pi^2)",
    "value": "-0.0448848389918903682590709",
    "from_implementation_B_quadratures": "-0.0448848389918903682590709",
    "recomputed_33_digits": "-0.0448848389918903682590709310554"
   },
   "j_3F^(0)": {
    "closed_form": "8/315 + 4/(9 pi^2)",
    "value": "0.0704284625711977396893433",
    "from_implementation_B_quadratures": "0.0704284625711977396893433",
    "recomputed_33_digits": "0.0704284625711977396893432360406"
   },
   "retarded_for_comparison": {
    "j_1^(0)": "1/30",
    "j_3^(0)": "-8/105"
   }
  },
  "I1_eps^-1": {
   "value": "-2.2499363310308107678e-7",
   "ratio_to_1/(15 (8 pi)^4)": "-1.3465451697567110478",
   "closed_form_of_ratio": "30 j_1F^(0) = -1/3 - 10/pi^2"
  },
  "I2_eps^-4": {
   "value_route_1": "1.6929711479424679298e-7",
   "value_route_3": "0.00000016929711479424679275258028228",
   "routes_agree_digits": 17.9,
   "closed_form": "1/(6144 pi^6)",
   "closed_form_value": "0.000000169297114794246792983372",
   "agreement_with_closed_form_digits": 17.9,
   "derivation": "I_2[eps^-4] = r_2[eps^-3] N_1[eps^-1] j_1F^(0) + r_1[eps^-3] N_3[eps^-1] j_3F^(0) with r_2[eps^-3] = -12/5, r_1[eps^-3] = -336/5, N_1[eps^-1] = 1/(2048 pi^4), N_3[eps^-1] = 1/(131072 pi^4); the rational parts of the two terms cancel as in the retarded case and the 1/pi^2 parts of the cores leave 1/(6144 pi^6)"
  },
  "I2_eps^-3_real_projection": {
   "value_route_1": "8.5370535203097680282e-7",
   "note": "real projection only; the full coefficient is complex"
  }
 },
 "source_sha256": {
  "implementation_A_log": {
   "description": "run log of implementation A (35-digit working precision, tanh-sinh quadrature of degree 5, kappa_2 by 4th-order finite differences)",
   "sha256": "d8a48c7e53d9dc584d46c89154a18ad832b8f60b8717dbebdf55d4091cdaa841"
  },
  "implementation_B_output": {
   "description": "output of implementation B (independent code, 30-digit working precision; c_M block at 25 digits)",
   "sha256": "a772bf0d7024f701f6827f42b255e2768f7e56dc438f0f7e01c97fdc24e9b88f"
  },
  "feynman_route_1": {
   "description": "Feynman-propagator assembly from the sector integrals of implementation A",
   "sha256": "c6854487da2af7f7e48988c155670f000ae11328fe017df87ee377bcf7f034e3"
  },
  "feynman_route_2": {
   "description": "Feynman-propagator assembly from the per-domain quadratures of implementation B; closed-form candidates",
   "sha256": "7c070c773a262c494b9ce63cbb1a7e9e3e5cb132b30e54c40000747accb4228d"
  },
  "feynman_route_3": {
   "description": "Feynman-propagator leading cores and eps^-4 coefficient recomputed at 33 digits",
   "sha256": "c8e50bfd6a34c4c91f53539350dff1f429b30a2988e974895e3e465999577b5c"
  },
  "quadrant_moment_relations": {
   "description": "integer-relation searches for the quadrant moments at two precisions",
   "sha256": "f7aa8d6c8b70585caf47ec08710c51427c4f23b416704e084a2cda040b667eec"
  },
  "quadrant_moment_check": {
   "description": "independent double-precision quadrature of the quadrant moments",
   "sha256": "5712960669412214b220497424404cba8c323aa560b534b2110c364680c0cd8e"
  }
 }
}