{
 "what": "the Qiushi Engine's disproof of Bandeira's Open Problem 0.2(a) at n = 5 (Oxelra-AI/Bandeira-0.2a-open-source, 2026-06), audited: the finite core re-decided with our own certificate, the claimant's certificate decided and its convention named, the chain read against the sources",
 "preregistration": "corpus/bandeira/preregistration.json",
 "pins": "corpus/bandeira/PINS.json",
 "repositoryCommit": "c41373447abfadabc30afeebc3c4de74e1ccf8dc",
 "rows": [
  {
   "id": "bandeira-02a-sos-infeasible-120",
   "claim": "Lai--Lim's sum-of-squares SDP (Conjecture 1C, m = n = 5, Helton's form beta_a^T X_i beta_b) is infeasible at lambda = 120, hence lambda_2 > 120 = 5!",
   "verdict": "CERTIFIED",
   "how": "our own exact rational Farkas certificate (corpus/bandeira/certificate-cm.json; y0 = -1, denominators <= 10000, found by a float search and decided in exact rationals): m00 = -19/8568 < 0, the six 31 x 31 blocks positive definite by exact LDL^T, 120 y0 + b^T y = 1 > 0",
   "witness": {
    "certificate": "corpus/bandeira/certificate-cm.json",
    "minPivots": [
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    ],
    "convention": "leftT"
   },
   "sharesCode": "none with the claimant: the SDP built from the definition in Lai--Lim (2020) as read here; the certificate found here"
  },
  {
   "id": "bandeira-02a-claimant-certificate",
   "claim": "the claimant's certificate (y0 = -146/2061, y with denominators <= 10^4) is a Farkas certificate of infeasibility at lambda = 120",
   "verdict": "CERTIFIED",
   "scope": "for the SDP whose Gram form carries no transpose (beta_a X_i beta_b, the convention the claimant's export states and its Lean files encode); it is NOT a certificate for the sum-of-squares form under either transpose convention (the blocks lose positive definiteness)",
   "how": "decided here under the claimant's convention: m00 = -1060/20438937 < 0, six blocks positive definite with the smallest pivots equal to the claimant's published ones, scalar 1417280/1417281 > 0; under leftT and rightT the same y fails the cone condition",
   "witness": {
    "certificate": "corpus/bandeira/certificate-v2.json",
    "minPivots": [
     "90701022264822851934589061568971266809698915649048137526499278164633384308162623684349341653946243035594944854287045372574824434106513730606662063528396687885005319365607729422381589027656311077930899171784133686425359703457980234018252555808937094959804704590724988665161800462435363209980351557098325226824035645588/672455186246357533493931652216006034297722266862027033240686224551788872488111845123659423912349907508335800279962969734172899604336430229982503122653032899002725404927735009190842032945960322696846913754763199635156170112641930831894857593516605090955411801913736994662338757178459337212498663397870182345096709768943145",
     "90701022264822851934589061568971266809698915649048137526499278164633384308162623684349341653946243035594944854287045372574824434106513730606662063528396687885005319365607729422381589027656311077930899171784133686425359703457980234018252555808937094959804704590724988665161800462435363209980351557098325226824035645588/672455186246357533493931652216006034297722266862027033240686224551788872488111845123659423912349907508335800279962969734172899604336430229982503122653032899002725404927735009190842032945960322696846913754763199635156170112641930831894857593516605090955411801913736994662338757178459337212498663397870182345096709768943145",
     "90701022264822851934589061568971266809698915649048137526499278164633384308162623684349341653946243035594944854287045372574824434106513730606662063528396687885005319365607729422381589027656311077930899171784133686425359703457980234018252555808937094959804704590724988665161800462435363209980351557098325226824035645588/672455186246357533493931652216006034297722266862027033240686224551788872488111845123659423912349907508335800279962969734172899604336430229982503122653032899002725404927735009190842032945960322696846913754763199635156170112641930831894857593516605090955411801913736994662338757178459337212498663397870182345096709768943145",
     "90701022264822851934589061568971266809698915649048137526499278164633384308162623684349341653946243035594944854287045372574824434106513730606662063528396687885005319365607729422381589027656311077930899171784133686425359703457980234018252555808937094959804704590724988665161800462435363209980351557098325226824035645588/672455186246357533493931652216006034297722266862027033240686224551788872488111845123659423912349907508335800279962969734172899604336430229982503122653032899002725404927735009190842032945960322696846913754763199635156170112641930831894857593516605090955411801913736994662338757178459337212498663397870182345096709768943145",
     "12048452455146253821725160061200297721056739893559097568673233698798534821866774026495996150739235471792191001152451315384040134738543120784463003593333264286616547170541473700977087678617805439685301033327/88621307661265776452130203182564912777546332479291265081002419249193479576754187110079318735001595215750621904241846836052673846221200121202818258208494642591330734229781785278321278638219019740515438207987140",
     "119777578859029713849483952835033155574154232909/2242249067175716679139317172384087229756784705694200"
    ],
    "convention": "none",
    "underSOSForm": {
     "leftT": {
      "verdict": "REFUTED",
      "blocksPD": [
       false,
       false,
       false,
       false,
       false,
       false
      ]
     },
     "rightT": {
      "verdict": "REFUTED",
      "blocksPD": [
       false,
       false,
       false,
       false,
       false,
       false
      ]
     }
    }
   },
   "finding": "the claimant's paper identifies its SDP with Lai--Lim's by reproducing 'beta X_i beta^T' literally, without the transposes that make tr(beta X_i beta^T Y) a noncommutative sum of squares f^T X_i f; its certificate and its Lean files decide that SDP. In floating point the two SDPs have the same optimum on every case of Lai--Lim's Table 1 and at (5,5) (corpus/bandeira/sdp-screen.json), and the sum-of-squares SDP is infeasible at 120 by the certificate above, so the conclusion stands; whether the two SDPs are equivalent in general is not decided here, and the claimant's text does not argue it. This sits exactly at the boundary the paper declares external ('the identification of the concrete standard-form coefficient matrices C_i and right-hand-side vector b with Lai--Lim's monomial-matching SDP ... is not itself a Lean theorem')."
  },
  {
   "id": "bandeira-02a-equivalence",
   "claim": "Bandeira's Open Problem 0.2(a) for n matrices is the same inequality as Recht--Re's (RR_{n,n}) in spectral norm (the paper's Theorem 4.4)",
   "verdict": "CERTIFIED",
   "scope": "by reading: Bandeira (2015) writes (a) as ||(1/n!) sum_sigma prod A_sigma(j)|| <= ||(1/n^n) sum_k prod A_k_j||; the right side is ||(A_1+...+A_n)^n|| / n^n by the distributive law, which is (RR_{n,n})'s with-replacement side with Lai--Lim's normalisation, and the left side is its distinct-index side; the paper's Lemma 4.1 (||S^k|| = ||S||^k for S PSD) is only needed for the form ||mean||^n. The Lean files state this reduction over an abstract ordered type (BandeiraAMGM.lean) and the identities concretely (ConcreteIdentities.lean); the Lean lane says whether they build.",
   "how": "the two statements read side by side from the pinned sources (corpus/bandeira/sources/bandeira-open-problem-0.2_MIT18_S096F15.pdf, paper-main.tex, lai-lim-2020-pmlr-v119.pdf)"
  },
  {
   "id": "bandeira-02a-lean-reading",
   "claim": "the Lean 4 development (Formalization/, the default Lake target, Lean v4.29.1 and Mathlib at the manifest's pin) states and proves what the paper's 'Lean formalization overview' lists: the equivalence 0.2(a) <=> (RR)_{n,n} over an abstract ordered type, the PSD spectral-norm power identity, the abstract scalar Farkas contradiction, the standard-form entry equalities Q = -M, six 31 x 31 blocks Q_b = L D L^T entrywise with positive pivots hence PosSemidef, and the 187 x 187 block-diagonal -M PosSemidef",
   "verdict": "CERTIFIED",
   "scope": "by reading the theorem statements and the import graph at the pins (whether they compile is the 'lean' entry below, PENDING until lake build finishes); for the claimant's matrices, i.e. the SDP without the basis transposes (row bandeira-02a-claimant-certificate)",
   "how": "every theorem statement of Core/, Farkas/, Bridges/, Certificate/FarkasStandard/ and the six Blocks/B*/B*FinalAsm.lean read against paper-main.tex; the import graph followed: AllBlocksFarkasChain imports the six FinalAsm and nothing imports AllBlocksFarkasChain; the NativePSD native_decide probes and the integer bound 144 > 120 (ConcreteIdentities.lean, native_decide) sit on the default target but on no theorem of the -M chain",
   "finding": "the Lean contains no definition of Lai--Lim's SDP and no theorem that any SDP is infeasible. Its two IsEmpty theorems (Farkas/SDPFarkasSemantics.lean no_lailim_feasible_from_certificate_semantics, Bridges/NativeFarkasAdapter.lean no_native_farkas_feasible) are about structures whose FIELDS assume what the certificate must deliver \u2014 the standard-form pairing pairM = y0 c0 + lin, the cone pairing or (-M).PosSemidef, the bound c0 <= 120 and the scalar inequality \u2014 and the proved theorem negM_full_posSemidef is never supplied to either; GeneratedFarkasTheoremShape takes three arbitrary Props as hypotheses. The files say so in their own docstrings ('the remaining work is to fill GeneratedCertificateAdapter from the actual generated theorem families rather than by assumptions'). So what Lean checks is the certificate's ARITHMETIC \u2014 5,766 entries, the LDL rows, the pivots, the block-diagonal assembly \u2014 for the claimant's C_i; the step 'these matrices are Lai--Lim's SDP, and a feasible point pairs with M to a positive number' is outside Lean, as the paper's boundary section says of the identification of C_i and b, while its abstract and overview say 'the entire certificate' and 'an SDP infeasibility semantic bridge' are machine-checked. Also at the pin: Tests/UnformalizedDependencies.lean still lists the spectral-norm power identity as unformalized while Core/SpectralNormPSD.lean proves it, and calls the 144 > 120 bound 'floating-point and NOT a rigorous bound' \u2014 it is not load-bearing; the bound that matters is the exact certificate's."
  },
  {
   "id": "bandeira-02a-headline",
   "claim": "Bandeira's Open Problem 0.2(a) is false for n = 5",
   "verdict": "CERTIFIED",
   "scope": "the finite core \u2014 the sum-of-squares SDP infeasible at lambda = 120 \u2014 is decided here by our own exact certificate; the step from that infeasibility to a PSD tuple violating the inequality is the noncommutative Positivstellensatz for bounded linear matrix-inequality domains (Helton--Klep--McCullough 2012, as Lai--Lim's Theorem 3 states it), a published theorem CITED and not re-derived, which the claimant's paper also declares external; the reduction from the inequality to the Loewner form and the equivalence with Bandeira's statement are read above; the Lean development certifies the claimant's certificate's arithmetic and not the SDP's infeasibility (row bandeira-02a-lean-reading), so nothing here rests on it. A claim whose analytic step rests on a cited theorem is certified with that step named, as the register does for Mathlib's theorems.",
   "how": "composition of the rows above; the kinds of evidence are not merged"
  }
 ],
 "decisions": {
  "claimant": {
   "leftT": {
    "convention": "leftT",
    "conventionMeaning": "beta_j^T * X_i * beta_k  (the j-th basis monomial reversed; Helton's SOS form f^T l f)",
    "verdict": "REFUTED",
    "what": "the Lai--Lim (5,5) SDP is infeasible at lambda = 120: -M is PSD and 120 y0 + b^T y > 0 for the published (y0, y), the constraints built here from the definition",
    "monomials": 3906,
    "basis": 31,
    "distinctWords": 120,
    "y0": "-146/2061",
    "yConstant": "702/9917",
    "m00": "-1060/20438937",
    "m00Negative": true,
    "scalar": "1417280/1417281",
    "scalarPositive": true,
    "maxDenominator": 9999,
    "seconds": 0.2,
    "blocks": [
     {
      "block": 1,
      "constraint": "X_1",
      "positiveDefinite": false,
      "minPivot": "-1869842540982186513329307809191673857936889597392539811061305442973683370327054..."
     },
     {
      "block": 2,
      "constraint": "X_2",
      "positiveDefinite": false,
      "minPivot": "-3592004156751867280467935034091952097448978254420198690791752039970918823870752..."
     },
     {
      "block": 3,
      "constraint": "X_3",
      "positiveDefinite": false,
      "minPivot": "-2933519483819810035311347158800606481853803678684153453779543222508884148730151..."
     },
     {
      "block": 4,
      "constraint": "X_4",
      "positiveDefinite": false,
      "minPivot": "-2933519483819810035311347158800606481853803678684153453779543222508884148730151..."
     },
     {
      "block": 5,
      "constraint": "X_5",
      "positiveDefinite": false,
      "minPivot": "-2933519483819810035311347158800606481853803678684153453779543222508884148730151..."
     },
     {
      "block": 6,
      "constraint": "5 - X_1 - ... - X_5",
      "positiveDefinite": false,
      "minPivot": "-7351300341460322285160903708549160622327998089067777575561022576350008002199995..."
     }
    ]
   },
   "rightT": {
    "convention": "rightT",
    "conventionMeaning": "beta_j * X_i * beta_k^T  (the k-th basis monomial reversed; Helton's SOS form f^T l f written on the right factor)",
    "verdict": "REFUTED",
    "what": "the Lai--Lim (5,5) SDP is infeasible at lambda = 120: -M is PSD and 120 y0 + b^T y > 0 for the published (y0, y), the constraints built here from the definition",
    "monomials": 3906,
    "basis": 31,
    "distinctWords": 120,
    "y0": "-146/2061",
    "yConstant": "702/9917",
    "m00": "-1060/20438937",
    "m00Negative": true,
    "scalar": "1417280/1417281",
    "scalarPositive": true,
    "maxDenominator": 9999,
    "seconds": 0.2,
    "blocks": [
     {
      "block": 1,
      "constraint": "X_1",
      "positiveDefinite": false,
      "minPivot": "-1620420712020605994873153772455555997913599094713762970215774606788467712344794..."
     },
     {
      "block": 2,
      "constraint": "X_2",
      "positiveDefinite": false,
      "minPivot": "-1609694608541501273099076463335304999031113652372508806621984560987300633965603..."
     },
     {
      "block": 3,
      "constraint": "X_3",
      "positiveDefinite": false,
      "minPivot": "-2850323798014303013091765354316580833716532812086544728922564687966478346954755..."
     },
     {
      "block": 4,
      "constraint": "X_4",
      "positiveDefinite": false,
      "minPivot": "-2850323798014303013091765354316580833716532812086544728922564687966478346954755..."
     },
     {
      "block": 5,
      "constraint": "X_5",
      "positiveDefinite": false,
      "minPivot": "-2850323798014303013091765354316580833716532812086544728922564687966478346954755..."
     },
     {
      "block": 6,
      "constraint": "5 - X_1 - ... - X_5",
      "positiveDefinite": false,
      "minPivot": "-1994095539129093726796431063239314452498481252902070811295349252632026648673735..."
     }
    ]
   },
   "none": {
    "convention": "none",
    "conventionMeaning": "beta_j * X_i * beta_k  (no reversal; the convention the claimant's export states)",
    "verdict": "CERTIFIED",
    "what": "the Lai--Lim (5,5) SDP is infeasible at lambda = 120: -M is PSD and 120 y0 + b^T y > 0 for the published (y0, y), the constraints built here from the definition",
    "monomials": 3906,
    "basis": 31,
    "distinctWords": 120,
    "y0": "-146/2061",
    "yConstant": "702/9917",
    "m00": "-1060/20438937",
    "m00Negative": true,
    "scalar": "1417280/1417281",
    "scalarPositive": true,
    "maxDenominator": 9999,
    "seconds": 1.2,
    "blocks": [
     {
      "block": 1,
      "constraint": "X_1",
      "positiveDefinite": true,
      "minPivot": "90701022264822851934589061568971266809698915649048137526499278164633384308162623..."
     },
     {
      "block": 2,
      "constraint": "X_2",
      "positiveDefinite": true,
      "minPivot": "90701022264822851934589061568971266809698915649048137526499278164633384308162623..."
     },
     {
      "block": 3,
      "constraint": "X_3",
      "positiveDefinite": true,
      "minPivot": "90701022264822851934589061568971266809698915649048137526499278164633384308162623..."
     },
     {
      "block": 4,
      "constraint": "X_4",
      "positiveDefinite": true,
      "minPivot": "90701022264822851934589061568971266809698915649048137526499278164633384308162623..."
     },
     {
      "block": 5,
      "constraint": "X_5",
      "positiveDefinite": true,
      "minPivot": "12048452455146253821725160061200297721056739893559097568673233698798534821866774..."
     },
     {
      "block": 6,
      "constraint": "5 - X_1 - ... - X_5",
      "positiveDefinite": true,
      "minPivot": "119777578859029713849483952835033155574154232909/2242249067175716679139317172384..."
     }
    ]
   }
  },
  "ours": {
   "leftT": {
    "convention": "leftT",
    "conventionMeaning": "beta_j^T * X_i * beta_k  (the j-th basis monomial reversed; Helton's SOS form f^T l f)",
    "verdict": "CERTIFIED",
    "what": "the Lai--Lim (5,5) SDP is infeasible at lambda = 120: -M is PSD and 120 y0 + b^T y > 0 for the published (y0, y), the constraints built here from the definition",
    "monomials": 3906,
    "basis": 31,
    "distinctWords": 120,
    "y0": "-1",
    "yConstant": "8549/8568",
    "m00": "-19/8568",
    "m00Negative": true,
    "scalar": "1",
    "scalarPositive": true,
    "maxDenominator": 9970,
    "seconds": 3.9,
    "blocks": [
     {
      "block": 1,
      "constraint": "X_1",
      "positiveDefinite": true,
      "minPivot": "78702720238728098463848405296165469807343399620786268111239228434449522999863783..."
     },
     {
      "block": 2,
      "constraint": "X_2",
      "positiveDefinite": true,
      "minPivot": "17251335276807175082795777966912979455718530047949947199168740883422145824336046..."
     },
     {
      "block": 3,
      "constraint": "X_3",
      "positiveDefinite": true,
      "minPivot": "27232472137104447844576322434208879795554704932124178498808105134291449201771527..."
     },
     {
      "block": 4,
      "constraint": "X_4",
      "positiveDefinite": true,
      "minPivot": "12795423362520228842048408108375192466142777563006197141475445300549482136726917..."
     },
     {
      "block": 5,
      "constraint": "X_5",
      "positiveDefinite": true,
      "minPivot": "45089211760382586733907062424908964873881154796970786586113733394390276288939793..."
     },
     {
      "block": 6,
      "constraint": "5 - X_1 - ... - X_5",
      "positiveDefinite": true,
      "minPivot": "59514429869917963878911522435327207131552929199333413152200291773225739636428923..."
     }
    ]
   },
   "rightT": {
    "convention": "rightT",
    "conventionMeaning": "beta_j * X_i * beta_k^T  (the k-th basis monomial reversed; Helton's SOS form f^T l f written on the right factor)",
    "verdict": "CERTIFIED",
    "what": "the Lai--Lim (5,5) SDP is infeasible at lambda = 120: -M is PSD and 120 y0 + b^T y > 0 for the published (y0, y), the constraints built here from the definition",
    "monomials": 3906,
    "basis": 31,
    "distinctWords": 120,
    "y0": "-1",
    "yConstant": "8549/8568",
    "m00": "-19/8568",
    "m00Negative": true,
    "scalar": "1",
    "scalarPositive": true,
    "maxDenominator": 9970,
    "seconds": 3.8,
    "blocks": [
     {
      "block": 1,
      "constraint": "X_1",
      "positiveDefinite": true,
      "minPivot": "78702720238728098463848405296165469807343399620786268111239228434449522999863783..."
     },
     {
      "block": 2,
      "constraint": "X_2",
      "positiveDefinite": true,
      "minPivot": "17251335276807175082795777966912979455718530047949947199168740883422145824336046..."
     },
     {
      "block": 3,
      "constraint": "X_3",
      "positiveDefinite": true,
      "minPivot": "74804405506909874961098548857220636468984884990361248381930462120136638112647895..."
     },
     {
      "block": 4,
      "constraint": "X_4",
      "positiveDefinite": true,
      "minPivot": "34546023033325913752027568353664002437944557836423434878973143663843367065439278..."
     },
     {
      "block": 5,
      "constraint": "X_5",
      "positiveDefinite": true,
      "minPivot": "12243108503386251463280427121490751578004503839748008412966035554169417445760094..."
     },
     {
      "block": 6,
      "constraint": "5 - X_1 - ... - X_5",
      "positiveDefinite": true,
      "minPivot": "63680439960812221350435329005800111630761634243286752072854312197351541410978948..."
     }
    ]
   },
   "none": {
    "convention": "none",
    "conventionMeaning": "beta_j * X_i * beta_k  (no reversal; the convention the claimant's export states)",
    "verdict": "REFUTED",
    "what": "the Lai--Lim (5,5) SDP is infeasible at lambda = 120: -M is PSD and 120 y0 + b^T y > 0 for the published (y0, y), the constraints built here from the definition",
    "monomials": 3906,
    "basis": 31,
    "distinctWords": 120,
    "y0": "-1",
    "yConstant": "8549/8568",
    "m00": "-19/8568",
    "m00Negative": true,
    "scalar": "1",
    "scalarPositive": true,
    "maxDenominator": 9970,
    "seconds": 0.3,
    "blocks": [
     {
      "block": 1,
      "constraint": "X_1",
      "positiveDefinite": false,
      "minPivot": "-5132675695414234971414818144922261558908765572315658416290622705440194162982183..."
     },
     {
      "block": 2,
      "constraint": "X_2",
      "positiveDefinite": false,
      "minPivot": "-1223744635914464779855714104342842199632319998921665661297576120715975188153968..."
     },
     {
      "block": 3,
      "constraint": "X_3",
      "positiveDefinite": false,
      "minPivot": "-8088736993626896633557319644031708251634536685532872571417846464598120150664959..."
     },
     {
      "block": 4,
      "constraint": "X_4",
      "positiveDefinite": false,
      "minPivot": "-3041107209494378820633769718398561457563481320981142659061546461678740279087325..."
     },
     {
      "block": 5,
      "constraint": "X_5",
      "positiveDefinite": false,
      "minPivot": "-7002049964135092552196687991121473889614328122004019364706410730419829574716907..."
     },
     {
      "block": 6,
      "constraint": "5 - X_1 - ... - X_5",
      "positiveDefinite": false,
      "minPivot": "-2490903987396910911589871958822083518120395212126753420671684778207635910252742..."
     }
    ]
   }
  }
 },
 "floatScreen": {
  "file": "corpus/bandeira/sdp-screen.json",
  "lambda2At55": {
   "none": 144.64875348561657,
   "leftT": 144.64875346800687
  },
  "laiLim": 144.6488,
  "isAVerdict": false
 },
 "lean": {
  "status": "BUILT",
  "what": "the default Lake target of Oxelra-AI/Bandeira-0.2a-open-source at c4137344 built at its pins (lean-toolchain v4.29.1, Mathlib from lake-manifest.json via lake exe cache get); then the axioms of its main theorems asked of the kernel with #print axioms from a scratch file imported against the built oleans",
  "jobs": 3759,
  "start": "2026-10-15T03:37:44Z",
  "cacheDone": "2026-10-15T03:42:01Z",
  "end": "2026-10-15T20:38:04Z",
  "note": "the three NativePSD native_decide modules dominated the wall time (one SIGSTOP pause of ~40 min on battery is included); the Blocks compiled in minutes once they ran",
  "slowestModules": [
   [
    "Formalization.Certificate.NativePSD.NativeSplitB1Entry30_30",
    53734
   ],
   [
    "Formalization.Certificate.NativePSD.NativeSplitB1Entry00_00",
    53381
   ],
   [
    "Formalization.Tests.SplitTestAssembly",
    53381
   ],
   [
    "Formalization.Certificate.NativePSD.NativePSDProbeB1K31",
    27224
   ]
  ],
  "axioms": {
   "QiushiFarkasChain.negM_full_posSemidef": [
    "propext",
    "Classical.choice",
    "Quot.sound"
   ],
   "QiushiFarkasNativeAdapter.no_native_farkas_feasible": [
    "propext",
    "Classical.choice",
    "Quot.sound"
   ],
   "QiushiFarkas.no_lailim_feasible_from_certificate_semantics": [
    "propext",
    "Classical.choice",
    "Quot.sound"
   ],
   "QiushiFarkas.no_generated_certificate_adapter": [
    "propext",
    "Classical.choice",
    "Quot.sound"
   ],
   "QiushiB1.block1_Q_posSemidef": [
    "propext",
    "Classical.choice",
    "Quot.sound"
   ],
   "Qiushi.BandeiraAMGM.rrnn_iff_bandeira02a": [],
   "Qiushi.BandeiraAMGM.exists_not_bandeira02a_of_exists_not_rrnn": [],
   "spectralNormPSD_pow_full": [
    "propext",
    "Classical.choice",
    "Quot.sound"
   ],
   "QiushiFarkasStandardFull.m00_reconstructed": [
    "propext",
    "Classical.choice",
    "Quot.sound"
   ],
   "Qiushi.BandeiraAMGM.qiushi_lailim_int_bound": [
    "Qiushi.BandeiraAMGM.qiushi_lailim_int_bound._native.native_decide.ax_1_1"
   ],
   "Qiushi.BandeiraAMGM.qiushi_rrnn_iff_bandeira": []
  },
  "axiomsSummary": "every theorem of the -M chain and the Farkas files uses only propext, Classical.choice and Quot.sound; the two Core equivalence theorems use no axiom at all; the one native_decide on the default target is the integer bound 144 > 5! (ConcreteIdentities.lean), which is not load-bearing",
  "structuresPrinted": {
   "NativeFarkasFeasible": {
    "M": "Matrix n n \u211d",
    "Y": "Matrix n n \u211d",
    "c0": "\u211d",
    "lin": "\u211d",
    "pairM": "\u211d",
    "hY": "self.Y.PosSemidef",
    "hNSD": "(-self.M).PosSemidef",
    "hTrace": "self.pairM = (self.M * self.Y).trace",
    "hBound": "self.c0 \u2264 120",
    "hStandard": "self.pairM = -146 / 2061 * self.c0 + self.lin",
    "hScalar": "120 * (-146 / 2061) + self.lin > 0"
   }
  },
  "reading": "the kernel confirms the statements read in row bandeira-02a-lean-reading: NativeFarkasFeasible carries hNSD, hTrace, hStandard, hBound, hScalar as FIELDS; no SDP is defined and negM_full_posSemidef is never supplied to the contradiction",
  "summary": "BUILT: 3759 jobs, every main theorem on the standard three axioms (the Core equivalence on none; the integer bound 144 > 120 on native_decide). The structures printed by the kernel carry the pairing as fields, as read.",
  "logs": {
   "build": "e4aaf4bc82dbe7f4406d02c6d3bdeca681d5b078664515194ea3305c851d4c3a",
   "axioms": "3db45b3cdf2f51c55f2b51fb518ea2dbd8a2099202335175d88ac4a7c6635d4d"
  }
 },
 "secondImplementation": {
  "what": "instruments/bandeira/verify.js, BigInt rationals, written from the same definition and sharing no code with verify.py (the same author: an N-version check); every verdict, m00, scalar and smallest pivot must be equal, exactly",
  "checks": {
   "ours": {
    "convention": "leftT",
    "jsVerdict": "CERTIFIED",
    "pythonVerdict": "CERTIFIED",
    "agree": true,
    "jsMs": 79470
   },
   "claimant": {
    "convention": "none",
    "jsVerdict": "CERTIFIED",
    "pythonVerdict": "CERTIFIED",
    "agree": true,
    "jsMs": 9573
   },
   "claimantUnderSOS": {
    "convention": "leftT",
    "jsVerdict": "REFUTED",
    "pythonVerdict": "REFUTED",
    "agree": true,
    "jsMs": 655
   }
  }
 },
 "decider": {
  "verify.py": "fd78a61cb6b9ef550ccd467be5a36eed8b58d6b96b16cb4f4d7bb0b99a5a6a39",
  "verify.js": "6e2970efca85e093834dfbbe8ea94275be4265d13ab17219a0c13af5727692fa",
  "certify.py": "d6da73eb62f4a15707f71220f34dbc5fc03538dce8b7cfc85f3ef50c56d4e674",
  "battery.py": "d700242ee093e3f65f75ac35c649384a9b56ca90f7cd057529fcf7c01bd29e50",
  "python": "3.9.6"
 },
 "seconds": 99.6,
 "generated": "2026-10-15T20:51:18Z"
}