{
 "description": "single-valuedness certificates for the 26 deep Laurent coefficients c_{3-w} of the two-vertex four-edge modular graph functions of weights w = 11 (transcendental weight 19, eleven functions) and w = 12 (transcendental weight 21, fifteen functions): each written exactly as a rational combination of single-valued multiple zeta values zsv(a,b,c) of the same weight and of odd-zeta monomials; with the data that make the check a substitution (the zsv values in the reduced basis of laurent_C4_w7-12.json and the double-shuffle reductions of the monomials of zsv_depth3_w11-21.json), and the f-alphabet images of the depth-two and depth-three basis generators through weight 21",
 "convention": {
  "mzv": "ascending: zeta(n1,...,nr) = sum_{0<k1<...<kr} prod k_i^(-n_i)",
  "even_zeta": "powers of zeta(2)",
  "coefficient": "c_k = (sum_i num_i * monomial_i) / den, integers as computed (not reduced)",
  "monomial_syntax": "zeta(n), zeta(2)^j, zeta(n1,n2), zeta(n1,n2,n3) joined by *; '1' = rational",
  "zsv": "zsv(a,b,c) = zeta_sv(a,b,c), single-valued multiple zeta value, indices in the same ascending convention (that of Brown, arXiv:1309.5309); 'zsv(a,b,c)' here is the entry 'a,b,c' of zsv_depth3_w11-21.json",
  "basis": "the reduced basis of the Laurent file at a given weight: monomials in zeta(2), odd zeta values and the depth-two and depth-three generators listed under basis_generators (products written with '*'); c_{3-w} in laurent_C4_w7-12.json, and zsv_reduced and reductions here, are written in it",
  "f_alphabet": "U = Q<f3,f5,f7,...> with the shuffle product and deconcatenation, H = U tensor Q[f2]; a key 'f3 f5 f11' is the word f3f5f11 read left to right, a leading 'f2^j' is the j-th power of the commuting generator f2; phi is multiplicative (shuffle product on words, powers of f2 multiply), phi(zeta(n)) = f_n for n odd, phi(zeta(2)) = f2, and the images of the generators are listed under phi. Word orientation: these words are the REVERSALS of Brown's (arXiv:1102.1310, arXiv:1309.5309): the coefficient of a word u f_m in phi(x) is the coefficient of u in phi(D_m x), D_m the weight-m derivation, so the last letter is the one a derivation strips (Eq. (67) of the paper). In this orientation Brown's single-valued map (Eq. (10) of the paper; his eq. (7.3)) reads sv(f2) = 0, sv(w) = sum over deconcatenations w = uv of (u reversed) shuffle v, and zsv(a,b,c) = phi^{-1}(sv(phi(zeta(a,b,c)))). Normalization: the isomorphism H = U tensor Q[f2] is not canonical; the one choice made beyond phi(zeta(2)) = f2 is that the image of each even-weight (depth-two) generator has no pure power-of-f2 word. None of the 83 formulas of zsv_depth3_w11-21.json and none of the 26 certificates depends on that choice (sv annihilates f2; re-deriving the formulas with that part changed returns them unchanged)."
 },
 "generated": "2026-09-28T16:39:13Z",
 "counts": {
  "certificates": 26,
  "certificates_tw19": 11,
  "certificates_tw21": 15,
  "terms_per_certificate_tw19": "11 (11 zsv + 0 odd-zeta monomials)",
  "terms_per_certificate_tw21": "14 (11 zsv + 3 odd-zeta monomials)",
  "zsv_reduced": 49,
  "reductions": 42,
  "phi_generators": 32,
  "phi_depth2": 12,
  "phi_depth3": 20
 },
 "how_to_verify": [
  "(1) certificate -> Laurent coefficient: in certificates.<a,b,c,d>.decomposition replace each zsv(a,b,c) by zsv_reduced.<a,b,c>.value and add the odd-zeta monomial terms; the result equals certificates.<a,b,c,d>.certifies, which is the entry functions.<a,b,c,d>.coefficients.<k> of laurent_C4_w7-12.json (k = 3-w), monomial by monomial with exact rational arithmetic.",
  "(2) formula -> reduced value: in the entry <a,b,c> of zsv_depth3_w11-21.json (2*zeta(a,b,c) + correction) replace every monomial that is a key of reductions by reductions.<monomial>.value and keep the others; the result equals zsv_reduced.<a,b,c>.value exactly. The reductions are identities among multiple zeta values that follow from the regularized double-shuffle relations in depth at most three; they are the one ingredient a reader must take from this file or re-derive (e.g. with the MZV data mine), everything else is substitution.",
  "(3) phi, sv and the values are consistent: extend phi multiplicatively from the table (phi(zeta(n)) = f_n, phi(zeta(2)) = f2, shuffle product); then for each of the 49 triples phi(zsv_reduced.<a,b,c>) = sv(phi(zeta(a,b,c))) word by word, where zeta(a,b,c) is a basis generator or is first replaced by reductions.<zeta(a,b,c)>.",
  "The script check_svcert_anc.py in the same directory performs (1)-(3), the term counts, and the comparison of the C_{8,1,1,1} certificate with Eq. (69) of the paper (given the paper source), using only these three files and exact rational arithmetic."
 ],
 "basis_generators": {
  "depth2": [
   "zeta(3,5)",
   "zeta(3,7)",
   "zeta(3,9)",
   "zeta(5,9)",
   "zeta(3,11)",
   "zeta(5,11)",
   "zeta(3,13)",
   "zeta(5,13)",
   "zeta(3,15)",
   "zeta(7,13)",
   "zeta(5,15)",
   "zeta(3,17)"
  ],
  "depth3": [
   "zeta(3,3,5)",
   "zeta(3,5,5)",
   "zeta(3,3,7)",
   "zeta(3,5,7)",
   "zeta(3,3,9)",
   "zeta(3,7,7)",
   "zeta(5,3,9)",
   "zeta(3,5,9)",
   "zeta(3,3,11)",
   "zeta(5,5,9)",
   "zeta(3,7,9)",
   "zeta(5,3,11)",
   "zeta(3,5,11)",
   "zeta(3,3,13)",
   "zeta(7,3,11)",
   "zeta(5,5,11)",
   "zeta(3,7,11)",
   "zeta(5,3,13)",
   "zeta(3,5,13)",
   "zeta(3,3,15)"
  ]
 },
 "certificates": {
  "8,1,1,1": {
   "weight": 11,
   "k": -8,
   "transcendental_weight": 19,
   "n_terms": 11,
   "n_zsv_terms": 11,
   "n_odd_zeta_monomial_terms": 0,
   "decomposition": {
    "zsv(11,3,5)": "1035353002567/2904224526773504",
    "zsv(13,3,3)": "15632744128772707/3717407394270085120",
    "zsv(3,11,5)": "332095881374857/1858703697135042560",
    "zsv(3,13,3)": "3105372750663279/675892253503651840",
    "zsv(3,3,13)": "9191194867742227/1858703697135042560",
    "zsv(3,5,11)": "-3242382217911/47966547022839808",
    "zsv(3,7,9)": "22488039/119057776640",
    "zsv(5,11,3)": "-1831621015863949/7434814788540170240",
    "zsv(5,3,11)": "-62658272742847/135178450700730368",
    "zsv(5,5,9)": "1022409/5411717120",
    "zsv(7,3,9)": "3546837/23811555328"
   },
   "certifies": {
    "den": 5365303623680000,
    "num": {
     "zeta(19)": -34338640474763625,
     "zeta(2)*zeta(17)": 26976074447322000,
     "zeta(2)^2*zeta(15)": 1658086803172800,
     "zeta(2)^3*zeta(13)": -50338486598400,
     "zeta(2)^6*zeta(7)": -606699314688,
     "zeta(2)^4*zeta(11)": -40654705291200,
     "zeta(2)^5*zeta(9)": -5937658272000,
     "zeta(3)^2*zeta(13)": 1394048921265000,
     "zeta(3)*zeta(5)*zeta(11)": 501032381850000,
     "zeta(3)*zeta(7)*zeta(9)": 56980153230000,
     "zeta(5)^2*zeta(9)": 70475782377000,
     "zeta(5)*zeta(7)^2": 41261490270000,
     "zeta(5)*zeta(5,9)": 204324120000,
     "zeta(5)*zeta(3,11)": 241405164000,
     "zeta(5,3,11)": -2263204944000,
     "zeta(3,5,11)": 5171166000,
     "zeta(3,7,9)": 2297258964000,
     "zeta(5,5,9)": 2299655358000,
     "zeta(3,3,13)": -2228141916000,
     "zeta(7)*zeta(3,9)": 1964832870000
    }
   }
  },
  "7,2,1,1": {
   "weight": 11,
   "k": -8,
   "transcendental_weight": 19,
   "n_terms": 11,
   "n_zsv_terms": 11,
   "n_odd_zeta_monomial_terms": 0,
   "decomposition": {
    "zsv(11,3,5)": "5416975452289655/11709833291950768128",
    "zsv(13,3,3)": "-7504066181200669699/234196665839015362560",
    "zsv(3,11,5)": "36119291669959511/117098332919507681280",
    "zsv(3,13,3)": "-64395758137312183/2027676760510955520",
    "zsv(3,3,13)": "-1881067251391983437/58549166459753840640",
    "zsv(3,5,11)": "97375842248159/335765829159878656",
    "zsv(3,7,9)": "308908013/833404436480",
    "zsv(5,11,3)": "-24304333056127847/468393331678030725120",
    "zsv(5,3,11)": "-6266401640648549/8516242394146013184",
    "zsv(5,5,9)": "1903909/5411717120",
    "zsv(7,3,9)": "-38564927/166680887296"
   },
   "certifies": {
    "den": 112671376097280000,
    "num": {
     "zeta(19)": -1796819806656499625,
     "zeta(2)*zeta(17)": 973651543770906000,
     "zeta(2)^2*zeta(15)": 60224171682074400,
     "zeta(2)^3*zeta(13)": -1792973445463200,
     "zeta(2)^6*zeta(7)": -95058508616704,
     "zeta(2)^4*zeta(11)": -1483315093260000,
     "zeta(2)^5*zeta(9)": -238437763200000,
     "zeta(3)^2*zeta(13)": -27459521774685000,
     "zeta(3)*zeta(5)*zeta(11)": 7880944641570000,
     "zeta(3)*zeta(7)*zeta(9)": 4882609681950000,
     "zeta(5)^2*zeta(9)": 4319746992561000,
     "zeta(5)*zeta(7)^2": 4442487119070000,
     "zeta(5)*zeta(3,11)": -127693493928000,
     "zeta(5)*zeta(5,9)": 92259907740000,
     "zeta(7)*zeta(3,9)": 169630571110000,
     "zeta(9)*zeta(3,7)": 70733983320000,
     "zeta(11)*zeta(3,5)": 907752785940000,
     "zeta(5,5,9)": 70393737414000,
     "zeta(3,7,9)": 74704051422000,
     "zeta(5,3,11)": -59327400132000,
     "zeta(3,5,11)": 7399875483000,
     "zeta(3,3,13)": -88009524603000
    }
   }
  },
  "6,3,1,1": {
   "weight": 11,
   "k": -8,
   "transcendental_weight": 19,
   "n_terms": 11,
   "n_zsv_terms": 11,
   "n_odd_zeta_monomial_terms": 0,
   "decomposition": {
    "zsv(11,3,5)": "-2099963920988343923/234196665839015362560",
    "zsv(13,3,3)": "-177289200441333507449/468393331678030725120",
    "zsv(3,11,5)": "-2669350259932330697/234196665839015362560",
    "zsv(3,13,3)": "-1563386040872163083/4055353521021911040",
    "zsv(3,3,13)": "-89247221700916056179/234196665839015362560",
    "zsv(3,5,11)": "-41184158527313413/3357658291598786560",
    "zsv(3,7,9)": "-438545103/416702218240",
    "zsv(5,11,3)": "167988217807705613/936786663356061450240",
    "zsv(5,3,11)": "99827378582489339/17032484788292026368",
    "zsv(5,5,9)": "-5516533/5411717120",
    "zsv(7,3,9)": "1367968887/166680887296"
   },
   "certifies": {
    "den": 225342752194560000,
    "num": {
     "zeta(19)": 5858315293801845875,
     "zeta(2)*zeta(17)": -3004659179770248000,
     "zeta(2)^2*zeta(15)": -144159260280235200,
     "zeta(2)^3*zeta(13)": 28514976936045600,
     "zeta(2)^6*zeta(7)": 3345306691274752,
     "zeta(2)^4*zeta(11)": 11158627234154400,
     "zeta(2)^5*zeta(9)": 2343721030560000,
     "zeta(3)*zeta(5)*zeta(11)": -50008926207240000,
     "zeta(3)^2*zeta(13)": 35402358651660000,
     "zeta(3)*zeta(7)*zeta(9)": 29873318955480000,
     "zeta(5)^2*zeta(9)": 28166206950882000,
     "zeta(5)*zeta(7)^2": 8637405296520000,
     "zeta(5)*zeta(3,11)": 6620286136464000,
     "zeta(5)*zeta(5,9)": -2715959446200000,
     "zeta(9)*zeta(3,7)": -1343945683080000,
     "zeta(7)*zeta(3,9)": -348430362280000,
     "zeta(11)*zeta(3,5)": -29955841936020000,
     "zeta(3,7,9)": 151487289954000,
     "zeta(5,3,11)": -604147197654000,
     "zeta(3,5,11)": -462205943199000,
     "zeta(3,3,13)": 961439768289000,
     "zeta(5,5,9)": 170893751028000
    }
   }
  },
  "6,2,2,1": {
   "weight": 11,
   "k": -8,
   "transcendental_weight": 19,
   "n_terms": 11,
   "n_zsv_terms": 11,
   "n_odd_zeta_monomial_terms": 0,
   "decomposition": {
    "zsv(11,3,5)": "793823099519191571/117098332919507681280",
    "zsv(13,3,3)": "116536248342310181411/234196665839015362560",
    "zsv(3,11,5)": "1072402647173292887/117098332919507681280",
    "zsv(3,13,3)": "1020819905334837317/2027676760510955520",
    "zsv(3,3,13)": "58495924011599470031/117098332919507681280",
    "zsv(3,5,11)": "17333023123982131/1678829145799393280",
    "zsv(3,7,9)": "243415919/416702218240",
    "zsv(5,11,3)": "128610495529980493/468393331678030725120",
    "zsv(5,3,11)": "-36960969606686429/8516242394146013184",
    "zsv(5,5,9)": "392523/676464640",
    "zsv(7,3,9)": "-308496025/41670221824"
   },
   "certifies": {
    "den": 112671376097280000,
    "num": {
     "zeta(19)": -879537525685944125,
     "zeta(2)*zeta(17)": 426182087078748000,
     "zeta(2)^2*zeta(15)": 15021502571275200,
     "zeta(2)^3*zeta(13)": -8093927447205600,
     "zeta(2)^6*zeta(7)": -1464718501199872,
     "zeta(2)^4*zeta(11)": -2989003101084000,
     "zeta(2)^5*zeta(9)": -747565037856000,
     "zeta(5)^2*zeta(9)": -4891515480852000,
     "zeta(5)*zeta(7)^2": 8362328694720000,
     "zeta(5)*zeta(3,11)": -2713783015944000,
     "zeta(5)*zeta(5,9)": 1170141532560000,
     "zeta(9)*zeta(3,7)": 1202477716440000,
     "zeta(7)*zeta(3,9)": 623506964080000,
     "zeta(11)*zeta(3,5)": 12708539003160000,
     "zeta(3,7,9)": -150618281814000,
     "zeta(5,3,11)": 345641870574000,
     "zeta(3,5,11)": 196559992629000,
     "zeta(3,3,13)": -303215291379000,
     "zeta(5,5,9)": -153529984608000
    }
   }
  },
  "5,4,1,1": {
   "weight": 11,
   "k": -8,
   "transcendental_weight": 19,
   "n_terms": 11,
   "n_zsv_terms": 11,
   "n_odd_zeta_monomial_terms": 0,
   "decomposition": {
    "zsv(11,3,5)": "93086429485515107/2365622887262781440",
    "zsv(13,3,3)": "58193150559793665/946249154905112576",
    "zsv(3,11,5)": "94162189560479157/2365622887262781440",
    "zsv(3,13,3)": "16436075882603355/270356901401460736",
    "zsv(3,3,13)": "23688125800093767/473124577452556288",
    "zsv(3,5,11)": "12335113232322273/305241662872616960",
    "zsv(3,7,9)": "4401941/3788201984",
    "zsv(5,11,3)": "473383093958595/1892498309810225152",
    "zsv(5,3,11)": "-14192881125223837/1892498309810225152",
    "zsv(5,5,9)": "1037179/1082343424",
    "zsv(7,3,9)": "-116183365/15152807936"
   },
   "certifies": {
    "den": 15022850146304000,
    "num": {
     "zeta(19)": 101809123250159325,
     "zeta(2)*zeta(17)": -26077612480920000,
     "zeta(2)^2*zeta(15)": -6647565676752000,
     "zeta(2)^3*zeta(13)": -2180291237124000,
     "zeta(2)^6*zeta(7)": -67957393017856,
     "zeta(2)^4*zeta(11)": -482456536642080,
     "zeta(2)^5*zeta(9)": -90343164038400,
     "zeta(3)*zeta(5)*zeta(11)": 1815505571880000,
     "zeta(3)*zeta(7)*zeta(9)": -2695750697640000,
     "zeta(5)*zeta(3,11)": -291673438056000,
     "zeta(7)*zeta(3,9)": -311753482040000,
     "zeta(5)^2*zeta(9)": -2848367459938000,
     "zeta(5)*zeta(7)^2": -2695750697640000,
     "zeta(5)*zeta(5,9)": 88720055424000,
     "zeta(9)*zeta(3,7)": -518715877680000,
     "zeta(3,7,9)": -4062980922000,
     "zeta(3,5,11)": 10103260167000,
     "zeta(3,3,13)": -39865843017000,
     "zeta(5,5,9)": -10465515060000,
     "zeta(5,3,11)": 18849656826000,
     "zeta(11)*zeta(3,5)": 544651671564000
    }
   }
  },
  "5,3,2,1": {
   "weight": 11,
   "k": -8,
   "transcendental_weight": 19,
   "n_terms": 11,
   "n_zsv_terms": 11,
   "n_odd_zeta_monomial_terms": 0,
   "decomposition": {
    "zsv(11,3,5)": "-4211255849429779/522760414819230720",
    "zsv(13,3,3)": "17145862102595233501/33456666548430766080",
    "zsv(3,11,5)": "-50828598206571617/16728333274215383040",
    "zsv(3,13,3)": "1071899417271336979/2027676760510955520",
    "zsv(3,3,13)": "4431140512771865753/8364166637107691520",
    "zsv(3,5,11)": "-607248632347273/239832735114199040",
    "zsv(3,7,9)": "61313643/119057776640",
    "zsv(5,11,3)": "-83847208635074587/66913333096861532160",
    "zsv(5,3,11)": "-7821569223135985/1216606056306573312",
    "zsv(5,5,9)": "4561363/5411717120",
    "zsv(7,3,9)": "-267738711/23811555328"
   },
   "certifies": {
    "den": 16095910871040000,
    "num": {
     "zeta(19)": -314816194862294125,
     "zeta(2)*zeta(17)": 156596566178052000,
     "zeta(2)^2*zeta(15)": 8451818640064800,
     "zeta(2)^3*zeta(13)": -1504884935954400,
     "zeta(2)^6*zeta(7)": -411642420889088,
     "zeta(2)^4*zeta(11)": -757166298688800,
     "zeta(2)^5*zeta(9)": -183409349760000,
     "zeta(5)*zeta(3,11)": -635380661916000,
     "zeta(7)*zeta(3,9)": 339261142220000,
     "zeta(5)^2*zeta(9)": -2755932202023000,
     "zeta(5)*zeta(7)^2": 2448181756020000,
     "zeta(5)*zeta(5,9)": 264792708180000,
     "zeta(9)*zeta(3,7)": 654289345710000,
     "zeta(3,7,9)": -44661973356000,
     "zeta(3,5,11)": 55691376741000,
     "zeta(3,3,13)": -72794062341000,
     "zeta(5,5,9)": -34548475962000,
     "zeta(5,3,11)": 88794091386000,
     "zeta(11)*zeta(3,5)": 2788097842530000
    }
   }
  },
  "5,2,2,2": {
   "weight": 11,
   "k": -8,
   "transcendental_weight": 19,
   "n_terms": 11,
   "n_zsv_terms": 11,
   "n_odd_zeta_monomial_terms": 0,
   "decomposition": {
    "zsv(11,3,5)": "-692860812169711169/13010925879945297920",
    "zsv(13,3,3)": "-79287464841957633521/26021851759890595840",
    "zsv(3,11,5)": "-200618130412882393/2602185175989059584",
    "zsv(3,13,3)": "-2109296608801386261/675892253503651840",
    "zsv(3,3,13)": "-40329078165465735461/13010925879945297920",
    "zsv(3,5,11)": "-138886154378273961/1678829145799393280",
    "zsv(3,7,9)": "-1946649021/416702218240",
    "zsv(5,11,3)": "212317407721516577/52043703519781191680",
    "zsv(5,3,11)": "44110600509824207/946249154905112576",
    "zsv(5,5,9)": "-4059807/676464640",
    "zsv(7,3,9)": "2958640491/41670221824"
   },
   "certifies": {
    "den": 37557125365760000,
    "num": {
     "zeta(19)": 3440870310202450125,
     "zeta(2)*zeta(17)": -1818893707987356000,
     "zeta(2)^2*zeta(15)": -71302881504254400,
     "zeta(2)^3*zeta(13)": 29605915121983200,
     "zeta(2)^6*zeta(7)": 5325898864041984,
     "zeta(2)^4*zeta(11)": 11843357345820000,
     "zeta(2)^5*zeta(9)": 2700502590240000,
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    "f5 f5 f11": "1260",
    "f5 f7 f9": "1050",
    "f5 f9 f7": "420",
    "f5 f11 f5": "270",
    "f7 f5 f9": "420",
    "f7 f7 f7": "225",
    "f7 f9 f5": "168",
    "f9 f5 f7": "90",
    "f9 f7 f5": "90",
    "f11 f5 f5": "36",
    "f2 f19": "-50217/2",
    "f2^2 f17": "-2628",
    "f2^3 f15": "-13728/35",
    "f2^4 f13": "-14652/175",
    "f2^5 f11": "-912/77",
    "f2^6 f9": "-44224/35035"
   }
  },
  "zeta(3,3,15)": {
   "weight": 21,
   "depth": 3,
   "image": {
    "f3 f3 f15": "-90",
    "f3 f5 f13": "330",
    "f3 f7 f11": "630",
    "f3 f9 f9": "756",
    "f3 f11 f7": "660",
    "f3 f13 f5": "390",
    "f5 f5 f11": "270",
    "f5 f7 f9": "420",
    "f5 f9 f7": "420",
    "f5 f11 f5": "270",
    "f7 f5 f9": "168",
    "f7 f7 f7": "225",
    "f7 f9 f5": "168",
    "f9 f5 f7": "90",
    "f9 f7 f5": "90",
    "f11 f5 f5": "36",
    "f2 f19": "-11457/2",
    "f2^2 f17": "-4368/5",
    "f2^3 f15": "-7648/35",
    "f2^4 f13": "-9504/175",
    "f2^5 f11": "-576/55",
    "f2^6 f9": "-176896/125125",
    "f2^7 f7": "-2304/25025"
   }
  }
 }
}