[
  {
    "claim_id": "A001",
    "statement": "附件研究來源共85 MD與89 HTML；加規則及manifest共176檔",
    "status": "VERIFIED",
    "status_type": "inventory",
    "internal_source": "results/archive_inventory.csv",
    "external_source": [],
    "hypotheses": "對實際上傳ZIP解壓後計數",
    "normalization": "byte-level SHA256",
    "quantifiers": "此附件",
    "depends_on": [],
    "reproduced_here": true,
    "notes": "逐檔hash與閱讀分工已保存；176不是176份數學證明。"
  },
  {
    "claim_id": "A002",
    "statement": "原包沒有執行碼、JSON/CSV計算證書；47筆literal缺失檔引用",
    "status": "VERIFIED",
    "status_type": "inventory",
    "internal_source": "results/missing_asset_references.csv",
    "external_source": [],
    "hypotheses": "以隨附檔案為全集",
    "normalization": "literal filename scan",
    "quantifiers": "此附件",
    "depends_on": [
      "A001"
    ],
    "reproduced_here": true,
    "notes": "不推論被引用資產從未在其他地方存在。"
  },
  {
    "claim_id": "Z001",
    "statement": "rank equality、Sha finiteness、leading term是不同義務",
    "status": "VERIFIED",
    "status_type": "logical_decomposition",
    "internal_source": "phase0/files/01_BSD_Statement_and_Quantifier_Audit.md",
    "external_source": [],
    "hypotheses": "明示各命題之定義",
    "normalization": "Néron period、實高度、factorial須固定",
    "quantifiers": "所有E的目標；不聲稱已成立",
    "depends_on": [],
    "reproduced_here": false,
    "notes": "保留問題分解；不是BSD證明。"
  },
  {
    "claim_id": "Z002",
    "statement": "忠實非負全域和的零性編碼不提供可判定的BSD oracle",
    "status": "VERIFIED",
    "status_type": "logical_nogo",
    "internal_source": "phase0/files/07_BSD_Certificate_Globalizer.md",
    "external_source": [],
    "hypotheses": "每項非負並忠實表示一個義務",
    "normalization": "精確零而非浮點近零",
    "quantifiers": "所有被編碼義務",
    "depends_on": [
      "Z001"
    ],
    "reproduced_here": false,
    "notes": "形式等價保留，oracle建構未供應。"
  },
  {
    "claim_id": "Z003",
    "statement": "僅靠解析函數收斂不能穩定讀取零點重數",
    "status": "VERIFIED",
    "status_type": "counterexample",
    "internal_source": "phase0/files/05_Internal_Grid_Rank_Audit.md",
    "external_source": [],
    "hypotheses": "以z(z+a)、a→0為例",
    "normalization": "零點z=0處重數",
    "quantifiers": "所稱一般收斂推論",
    "depends_on": [],
    "reproduced_here": false,
    "notes": "重讀反例的代數內容；無新數值實驗。"
  },
  {
    "claim_id": "P101",
    "statement": "BH族定理可轉為Algorithm1/2的條件接口",
    "status": "CONDITIONAL",
    "status_type": "conditional_reduction",
    "internal_source": "phase1/files/01_Theorem_2_18_Condition_Map.md",
    "external_source": [
      "https://arxiv.org/html/2601.16044v3"
    ],
    "hypotheses": "最適、rank0、semistable、2-BSD、無指定同源、a3與twist各條件",
    "normalization": "基本判別式；d≡1 mod4；期間與2部固定",
    "quantifiers": "每個通過全部假設的base及twist",
    "depends_on": [],
    "reproduced_here": false,
    "notes": "原始定理與程式實現分開；未重跑repo。"
  },
  {
    "claim_id": "P102",
    "statement": "46a1有7、106d1有21個twist fixture重播",
    "status": "UNVERIFIED",
    "status_type": "project_assertion",
    "internal_source": "phase1/files/03_Algorithm2_Independent_Reproduction.md",
    "external_source": [],
    "hypotheses": "原稿外部curve data與mirror正確",
    "normalization": "原稿Algorithm2語義",
    "quantifiers": "兩個fixture",
    "depends_on": [
      "P101",
      "A002"
    ],
    "reproduced_here": false,
    "notes": "沒有隨附原程式或結果。"
  },
  {
    "claim_id": "P103",
    "statement": "Algorithm1一個commit更動了3/5/7排除與獨立a3條件",
    "status": "UNVERIFIED",
    "status_type": "project_assertion",
    "internal_source": "phase1/files/07_One_Commit_Semantic_Autopsy.md",
    "external_source": [],
    "hypotheses": "兩個原稿所列commit內容",
    "normalization": "舊動態排除；新一律排除",
    "quantifiers": "指定commit pair",
    "depends_on": [],
    "reproduced_here": false,
    "notes": "記錄語義差異；本輪未抓取repo。"
  },
  {
    "claim_id": "P104",
    "statement": "小fixture25→12，removed13已分類",
    "status": "UNVERIFIED",
    "status_type": "project_assertion",
    "internal_source": "phase1/files/08_Removed_13_First_Failure_Closure.md",
    "external_source": [],
    "hypotheses": "原稿predicate trace與曲線metadata",
    "normalization": "first-failure順序不可換",
    "quantifiers": "25個base",
    "depends_on": [
      "P103"
    ],
    "reproduced_here": false,
    "notes": "取代舊unexplained狀態；不是本輪重播。"
  },
  {
    "claim_id": "P105",
    "statement": "500K base40749→36687，刪4062",
    "status": "UNVERIFIED",
    "status_type": "project_assertion",
    "internal_source": "phase1/files/11_500K_One_Commit_Global_Impact.md",
    "external_source": [],
    "hypotheses": "來源資料完整；同一conductor母集合",
    "normalization": "1353 isogeny only、2707 a3 only、2 both",
    "quantifiers": "conductor<500000指定清單",
    "depends_on": [
      "P103"
    ],
    "reproduced_here": false,
    "notes": "不可推成全部橢圓曲線。"
  },
  {
    "claim_id": "P106",
    "statement": "最終stable-base twists268697→247391，刪21306、增0",
    "status": "UNVERIFIED",
    "status_type": "project_assertion",
    "internal_source": "phase1/files/V0_5_EXACT_CENSUS_REPORT.md",
    "external_source": [],
    "hypotheses": "匹配base keys與原twist map",
    "normalization": "與upstream24785刪除項分開",
    "quantifiers": "stable-base清單",
    "depends_on": [
      "P105"
    ],
    "reproduced_here": false,
    "notes": "取代v0.4可能新增的推測；母集合不同不可混算。"
  },
  {
    "claim_id": "P107",
    "statement": "Phase1 Reproduction COMPLETE只是條件式semantic replay",
    "status": "CONDITIONAL",
    "status_type": "conditional_reduction",
    "internal_source": "phase1/files/16_Phase1_Closure_and_Phase2_Interface.md",
    "external_source": [],
    "hypotheses": "繼承base、torsion及2-descent輸入可靠",
    "normalization": "timeout=UNKNOWN；禁test-only flags",
    "quantifiers": "指定歷史輸入",
    "depends_on": [
      "P101",
      "P102",
      "P106"
    ],
    "reproduced_here": false,
    "notes": "不等於此輪完成Sage Algorithm1。"
  },
  {
    "claim_id": "T201",
    "statement": "FW ordinary H2排除a_p²≡1，故需要hybrid router",
    "status": "CONDITIONAL",
    "status_type": "proved_lemma",
    "internal_source": "phase2/files/10_FW_H2_and_Ordinary_Obstruction.md",
    "external_source": [
      "https://arxiv.org/pdf/2107.13726v3"
    ],
    "hypotheses": "ordinary局部表示與FW forbidden ratio設定",
    "normalization": "E[p]與FW慣例須scalar twist對齊",
    "quantifiers": "每個符合設定的ordinary p",
    "depends_on": [],
    "reproduced_here": false,
    "notes": "比全odd FW-only路線嚴格。"
  },
  {
    "claim_id": "T202",
    "statement": "gcd witness可縮小例外素數，但p本身必須leave-one-out",
    "status": "VERIFIED",
    "status_type": "proved_lemma",
    "internal_source": "phase2/files/00_GCD_Witness_Lemmas.md",
    "external_source": [],
    "hypotheses": "給定有限multiplicative primes及正整數valuations",
    "normalization": "gcd of nonempty witness set；排除ell=p",
    "quantifiers": "每個p",
    "depends_on": [],
    "reproduced_here": false,
    "notes": "空witness集合不能當成有可用素點。"
  },
  {
    "claim_id": "T203",
    "statement": "HTML09/索引的g_-=2^a推出所有odd p有H3 witness",
    "status": "INVALIDATED",
    "status_type": "counterexample",
    "internal_source": "phase2/p/09-fw-h3-exact-compiler/index.html；phase2/index.html",
    "external_source": [
      "https://arxiv.org/pdf/2107.13726v3"
    ],
    "hypotheses": "原例W_-={29}, n29=1",
    "normalization": "H3須ell≠p",
    "quantifiers": "原HTML的所有odd p",
    "depends_on": [
      "T202"
    ],
    "reproduced_here": false,
    "notes": "p=29反例；MD restricted branch及final supersingular route未被否定。"
  },
  {
    "claim_id": "T204",
    "statement": "FW nonsplit weight2公式需用E[p](-1)對接",
    "status": "VERIFIED",
    "status_type": "normalization_repair",
    "internal_source": "phase2/files/08_FW_Weight2_Exact_Translation.md",
    "external_source": [
      "https://arxiv.org/pdf/2107.13726v3"
    ],
    "hypotheses": "nonsplit multiplicative且residual ramified",
    "normalization": "0→μω→E[p]→μ→0；FW為0→μ→ρ→μω^-1→0",
    "quantifiers": "所用FW weight2實例",
    "depends_on": [],
    "reproduced_here": false,
    "notes": "scalar twist保留H1/H2；不是曲線族反例。"
  },
  {
    "claim_id": "T205",
    "statement": "696e1的rank0、singleton、全odd maximal image、exact Lalg等metadata",
    "status": "UNVERIFIED",
    "status_type": "project_assertion",
    "internal_source": "phase2/files/15_696e1_Base_Certificate.md",
    "external_source": [],
    "hypotheses": "原稿資料庫及數學證書可靠",
    "normalization": "LMFDB696e1=Cremona696b1",
    "quantifiers": "這一base",
    "depends_on": [
      "A002"
    ],
    "reproduced_here": false,
    "notes": "本輪未重認證全部算術實例；analytic Sha≠證明。"
  },
  {
    "claim_id": "T206",
    "statement": "Creutz–Miller修復696e1 base BSD anchor",
    "status": "CONDITIONAL",
    "status_type": "external_theorem_application",
    "internal_source": "phase2/files/21_Base_BSD_Anchor_Repair.md",
    "external_source": [
      "https://arxiv.org/pdf/1105.4018v2"
    ],
    "hypotheses": "N<5000且analytic rank≤1",
    "normalization": "fullBSD的標準Néron正規化",
    "quantifiers": "所有满足條件E；具體base待核",
    "depends_on": [
      "T205"
    ],
    "reproduced_here": false,
    "notes": "Theorem1.1已精確核對；舊過廣Miller引證被取代。"
  },
  {
    "claim_id": "T207",
    "statement": "696e1支持素數集合密度1/24",
    "status": "CONDITIONAL",
    "status_type": "conditional_reduction",
    "internal_source": "phase2/files/16_696e1_Chebotarev_Support.md",
    "external_source": [],
    "hypotheses": "S3 splitting field、quadratic resolvent及intersection如稿",
    "normalization": "支持素數，非所有squarefree integers",
    "quantifiers": "prime density",
    "depends_on": [
      "T205"
    ],
    "reproduced_here": false,
    "notes": "群階48/共軛元素2的推導可保留；27667掃描未重播。"
  },
  {
    "claim_id": "T208",
    "statement": "696e1 twist族的hybrid odd-prime routing",
    "status": "CONDITIONAL",
    "status_type": "conditional_reduction",
    "internal_source": "phase2/files/29_Theorem_Note_v1.0.md",
    "external_source": [
      "https://arxiv.org/pdf/2107.13726v3",
      "https://arxiv.org/pdf/1407.1093v1",
      "https://arxiv.org/pdf/2409.01350v2",
      "https://arxiv.org/html/2601.16044v3"
    ],
    "hypotheses": "base arithmetic、2-BSD、bad-prime splitting、ordinarity、residual及period條件",
    "normalization": "3與29互為distinct witnesses；supersing使用29",
    "quantifiers": "每個所定義twist及每個素數",
    "depends_on": [
      "T204",
      "T205",
      "T206",
      "T207"
    ],
    "reproduced_here": false,
    "notes": "引用條件吻合不等於所有base certificates已提供。"
  },
  {
    "claim_id": "T209",
    "statement": "Two-witness criterion推出rank0族的strongBSD",
    "status": "CONDITIONAL",
    "status_type": "conditional_reduction",
    "internal_source": "phase2/files/30_Two_Witness_Criterion_v0.1.md",
    "external_source": [
      "https://arxiv.org/pdf/2107.13726v3",
      "https://arxiv.org/pdf/1407.1093v1",
      "https://arxiv.org/pdf/2409.01350v2",
      "https://arxiv.org/html/2601.16044v3"
    ],
    "hypotheses": "T1–T7全部成立，尤其全odd residual hypothesis",
    "normalization": "twist discriminants、period及Tamagawa固定",
    "quantifiers": "所有符合T1–T7之base與其指定twists",
    "depends_on": [
      "T208"
    ],
    "reproduced_here": false,
    "notes": "非有限資料庫closure；T6不是一份執行證書。"
  },
  {
    "claim_id": "T210",
    "statement": "Witness network把大部分p歸於generic routes，留有限例外",
    "status": "CONDITIONAL",
    "status_type": "conditional_reduction",
    "internal_source": "phase2/files/31_Witness_Network_Criterion_v0.2.md",
    "external_source": [
      "https://arxiv.org/pdf/2107.13726v3",
      "https://arxiv.org/pdf/1407.1093v1",
      "https://arxiv.org/pdf/2409.01350v2"
    ],
    "hypotheses": "所有global hypotheses及每個exception的twist-uniform local/period證书",
    "normalization": "gcd與LOO及空集處理",
    "quantifiers": "每個家族成員與每個exception prime",
    "depends_on": [
      "T202",
      "T209"
    ],
    "reproduced_here": false,
    "notes": "有限prime集合不自動供應uniform in twists證明。"
  },
  {
    "claim_id": "T211",
    "statement": "H2 forbidden ratio等價某Jordan–Hölder constituent平方為1",
    "status": "VERIFIED",
    "status_type": "proved_lemma",
    "internal_source": "phase2/files/03_FW_H2_Jordan_Holder_Lemma.md",
    "external_source": [
      "https://arxiv.org/pdf/2107.13726v3"
    ],
    "hypotheses": "二維表示det=ω；可約情形",
    "normalization": "λμ=ω；平方條件與dual一起检查",
    "quantifiers": "所有符合設定的局部表示",
    "depends_on": [],
    "reproduced_here": false,
    "notes": "純角色代數，不是backend完成。"
  },
  {
    "claim_id": "T212",
    "statement": "局部p-isogeny與dual kernel的線性因子提供H2精確判準",
    "status": "CONDITIONAL",
    "status_type": "proved_lemma",
    "internal_source": "phase2/files/04_Local_p_Isogeny_Kernel_Criterion.md",
    "external_source": [],
    "hypotheses": "給定定義於Qp的cyclic p-isogeny及dual",
    "normalization": "x(P)有理⇔kernel character取±1",
    "quantifiers": "每個此類局部isogeny",
    "depends_on": [
      "T211"
    ],
    "reproduced_here": false,
    "notes": "不可約shortcut有效；無local p-torsion不充分。"
  },
  {
    "claim_id": "T213",
    "statement": "H2 local-agent實作與fixtures尚待提供",
    "status": "OPEN",
    "status_type": "open_obligation",
    "internal_source": "phase2/files/07_Local_Agent_Implementation_Spec.md",
    "external_source": [],
    "hypotheses": "須實際Sage/Magma local backend",
    "normalization": "kernel polynomials與Hensel證書",
    "quantifiers": "指定fixtures A–F",
    "depends_on": [
      "T212",
      "A002"
    ],
    "reproduced_here": false,
    "notes": "原稿是實作規格，未宣稱已執行；本輪明確保留為待辦。"
  },
  {
    "claim_id": "R301",
    "statement": "RUGZPB R1–R8 ⇒ fullBSD",
    "status": "CONDITIONAL",
    "status_type": "conditional_reduction",
    "internal_source": "p5/files/BSD_Rank_Uniform_Zeta_Primitivity_Reduction_v0.1.md",
    "external_source": [],
    "hypotheses": "R1–R8全部成立，含global finiteness與arch comparison",
    "normalization": "rank-uniform zeta、integral lattice、real comparison",
    "quantifiers": "所有欲涵蓋之rank/E",
    "depends_on": [
      "Z001"
    ],
    "reproduced_here": false,
    "notes": "接口未構造；不能把BSD相近要求藏入已知輸入。"
  },
  {
    "claim_id": "R302",
    "statement": "389a1的最小不變量及a2,a11",
    "status": "VERIFIED",
    "status_type": "exact_computation",
    "internal_source": "05_NEW_MATHEMATICS.md §4、7；results/exact_results.json",
    "external_source": [],
    "hypotheses": "給定Weierstrass方程",
    "normalization": "最小模型；小特徵直接枚舉",
    "quantifiers": "E=389a1，q=2,5,11",
    "depends_on": [],
    "reproduced_here": true,
    "notes": "Δ389,c4=112；#E(F2)=5、#E(F5)=9、#E(F11)=16。"
  },
  {
    "claim_id": "R303",
    "statement": "389a1模11殘餘像為GL2(F11)",
    "status": "VERIFIED",
    "status_type": "proved_lemma",
    "internal_source": "05_NEW_MATHEMATICS.md §7",
    "external_source": [],
    "hypotheses": "Frob2不可約；I1慣性transvection；Weil配對",
    "normalization": "算術Frob t²+2t+2；質數域root subgroups",
    "quantifiers": "這一E,p",
    "depends_on": [
      "R302"
    ],
    "reproduced_here": true,
    "notes": "有限部分重播；滿射結論來自文中群論與Tate表示證明，非枚舉整個Galois群。"
  },
  {
    "claim_id": "R304",
    "statement": "390 Manin系統rank325、E plus eigenspace dimension1",
    "status": "UNVERIFIED",
    "status_type": "project_assertion",
    "internal_source": "p5/files/BSD_RUGZPB_P2_P4_389a1_p11_v0.2.md",
    "external_source": [],
    "hypotheses": "原稿Manin/Hecke矩陣與消元正確",
    "normalization": "F11、relative lattice、plus involution",
    "quantifiers": "N389,p11",
    "depends_on": [
      "A002"
    ],
    "reproduced_here": false,
    "notes": "本輪未重算；不得以主定理局部群計算替代。"
  },
  {
    "claim_id": "R305",
    "statement": "指定delta^lambda_397*991=6",
    "status": "UNVERIFIED",
    "status_type": "project_assertion",
    "internal_source": "p5/files/BSD_RUGZPB_P2_P4_389a1_p11_v0.2.md",
    "external_source": [],
    "hypotheses": "R304及path decomposition、logs求和正確",
    "normalization": "原稿primitive roots5、6與deterministic eigenvector",
    "quantifiers": "僅n393427",
    "depends_on": [
      "R304"
    ],
    "reproduced_here": false,
    "notes": "非零在unit換基下不變；數字6非canonical invariant。"
  },
  {
    "claim_id": "R306",
    "statement": "僅Cor1.6整族非零足以推出canonical plus模11非零",
    "status": "INVALIDATED",
    "status_type": "counterexample",
    "internal_source": "p5/files/BSD_RUGZPB_P2_P4_389a1_p11_v0.2.md §4",
    "external_source": [
      "https://arxiv.org/html/2203.12159v6"
    ],
    "hypotheses": "只使用該Corollary給出的非零性",
    "normalization": "Z11/I_n不可直接當F11",
    "quantifiers": "原聲稱之推論",
    "depends_on": [],
    "reproduced_here": true,
    "notes": "11 mod121≠0但mod11=0；只否定引證邊，非否定Sha結論。"
  },
  {
    "claim_id": "R307",
    "statement": "canonical plus modular-symbol向量模11 primitive",
    "status": "VERIFIED",
    "status_type": "external_theorem_application",
    "internal_source": "05_NEW_MATHEMATICS.md §7",
    "external_source": [
      "https://arxiv.org/pdf/2601.14504v1",
      "https://arxiv.org/html/2203.12159v6"
    ],
    "hypotheses": "non-CM、goodordinary、surj、Manin11unit、Tam11unit；文中逐項補上",
    "normalization": "optimal quotient經prime-to11 isogeny傳回Néron period",
    "quantifiers": "存在某一canonical mod11 witness",
    "depends_on": [
      "R302",
      "R303"
    ],
    "reproduced_here": false,
    "notes": "修復R306所缺前提；不認證R305的指定n。"
  },
  {
    "claim_id": "R308",
    "statement": "指定canonical witness若驗證，Kim可給Sel corank2及Sha11∞=0",
    "status": "CONDITIONAL",
    "status_type": "conditional_reduction",
    "internal_source": "p5/files/BSD_RUGZPB_P2_P4_389a1_p11_v0.2.md；05§7",
    "external_source": [
      "https://arxiv.org/html/2203.12159v6"
    ],
    "hypotheses": "R304,R305,R307；rank下界2；Kim1.8 Selmer部分全部條件",
    "normalization": "先使用Sel/div結構，避免預設Sha finite",
    "quantifiers": "這一E,p=11",
    "depends_on": [
      "R303",
      "R304",
      "R305",
      "R307",
      "R313"
    ],
    "reproduced_here": false,
    "notes": "不能繼承原CLOSED；尚未重認證。"
  },
  {
    "claim_id": "R309",
    "statement": "本例的Kato IMC formulation由CS A+B可得",
    "status": "VERIFIED",
    "status_type": "external_theorem_application",
    "internal_source": "05_NEW_MATHEMATICS.md §7；08_SOURCE_AUDIT.md",
    "external_source": [
      "https://arxiv.org/pdf/2601.14504v1"
    ],
    "hypotheses": "同R307且采用CS Conj2.2.2的精確formulation",
    "normalization": "integral determinant formulation；與BCS不同表示先另核",
    "quantifiers": "E389a1,p11",
    "depends_on": [
      "R307"
    ],
    "reproduced_here": false,
    "notes": "已知定理推導；不推出rank2複數BSD。"
  },
  {
    "claim_id": "R310",
    "statement": "已出版11-adic regulator的非零數值",
    "status": "UNVERIFIED",
    "status_type": "numerical_evidence",
    "internal_source": "p5/files/BSD_P5_uGPR_Minimal_Gate_389a1_p11_v0.6.md",
    "external_source": [],
    "hypotheses": "原稿MST引文與normalization正確",
    "normalization": "11-adic height与指定點基底",
    "quantifiers": "此數值引用",
    "depends_on": [],
    "reproduced_here": false,
    "notes": "未重算或核對原MST全文，不作主定理輸入。"
  },
  {
    "claim_id": "R311",
    "statement": "v0.8 Euler/log局部抵消是11-unit且模11為4",
    "status": "UNVERIFIED",
    "status_type": "project_assertion",
    "internal_source": "p5/files/BSD_P5_Explicit_Local_Unit_Cancellation_389a1_p11_v0.8.md",
    "external_source": [],
    "hypotheses": "16P exact rational計算與formal log truncation證書",
    "normalization": "omega_short=omega_min/6；S={∞,11,389}",
    "quantifiers": "原指定局部scalar",
    "depends_on": [
      "A002"
    ],
    "reproduced_here": false,
    "notes": "本輪不重算；與本轮有限域norm矩陣不同。"
  },
  {
    "claim_id": "R312",
    "statement": "good ell≠p、totallyramified degreep下norm商等於E(k)/p",
    "status": "VERIFIED",
    "status_type": "proved_lemma",
    "internal_source": "05_NEW_MATHEMATICS.md §2",
    "external_source": [],
    "hypotheses": "良好約化、完全分歧、循環degreep",
    "normalization": "NE(L)=pE(K)，經約化及Kummer",
    "quantifiers": "所有符合假設的local fields",
    "depends_on": [],
    "reproduced_here": false,
    "notes": "用[p]在形式群可逆取代過強log同構。"
  },
  {
    "claim_id": "R313",
    "statement": "397/991局部列(1,2),(1,4)與det2",
    "status": "VERIFIED",
    "status_type": "exact_computation",
    "internal_source": "05_NEW_MATHEMATICS.md §4；results/exact_results.json",
    "external_source": [],
    "hypotheses": "給定E,P,Q；各Lell為單一degree11子域完備化",
    "normalization": "局部norm商以P為基底；短模型scale6為unit",
    "quantifiers": "兩個指定素點",
    "depends_on": [
      "R302",
      "R312"
    ],
    "reproduced_here": true,
    "notes": "#群374/1045；兩種點數法與軌道核對。"
  },
  {
    "claim_id": "R314",
    "statement": "原v1.2 full Selmer norm cube維度2,1,1,0",
    "status": "CONDITIONAL",
    "status_type": "conditional_reduction",
    "internal_source": "p5/files/BSD_P5_Norm_Selmer_Core_Vertex_389a1_p11_v1.2.md",
    "external_source": [],
    "hypotheses": "先證U=Sel_11，尤其Sha11=0及rank upper bound",
    "normalization": "norm-local conditions；不是只看U",
    "quantifiers": "整個Sel_11",
    "depends_on": [
      "R308",
      "R313"
    ],
    "reproduced_here": false,
    "notes": "U上的維度可直接得；full-Selmer不能悄悄換入。"
  },
  {
    "claim_id": "R315",
    "statement": "Mazur–Tate initial form等於6XY且augmentation order2",
    "status": "UNVERIFIED",
    "status_type": "project_assertion",
    "internal_source": "p5/files/BSD_P5_Determinantal_Kurihara_Semilocal_389a1_p11_v1.3.md",
    "external_source": [],
    "hypotheses": "缺失v1.0計算正確；低次及pure-square係數處理",
    "normalization": "F11[G]，prime-labelled generators",
    "quantifiers": "n397*991",
    "depends_on": [
      "R304",
      "R305",
      "A002"
    ],
    "reproduced_here": false,
    "notes": "單一mixed coefficient非零不足以證全部initial-form形状。"
  },
  {
    "claim_id": "R316",
    "statement": "手定義B_N之det等於2(e397∧e991)XY",
    "status": "VERIFIED",
    "status_type": "exact_computation",
    "internal_source": "05_NEW_MATHEMATICS.md §4–5",
    "external_source": [],
    "hypotheses": "局部列已認證",
    "normalization": "外積W與J²/J³乘法；不是A/J²內乘法",
    "quantifiers": "本例二維U",
    "depends_on": [
      "R313"
    ],
    "reproduced_here": true,
    "notes": "數字本身是重播；新內容另列N402。"
  },
  {
    "claim_id": "R317",
    "statement": "modular initial form與finite determinant生成同mixed line",
    "status": "CONDITIONAL",
    "status_type": "conditional_reduction",
    "internal_source": "p5/files/BSD_P5_Determinantal_Kurihara_Semilocal_389a1_p11_v1.3.md",
    "external_source": [],
    "hypotheses": "R315成立",
    "normalization": "比例3取決於基底；只論線",
    "quantifiers": "指定mixed line",
    "depends_on": [
      "R315",
      "R316"
    ],
    "reproduced_here": false,
    "notes": "不能推canonical scalar或regulator equality。"
  },
  {
    "claim_id": "R318",
    "statement": "P5-CANON-BocID完整Selmer複形／regulator comparison",
    "status": "OPEN",
    "status_type": "open_obligation",
    "internal_source": "03_CANONICAL_FRONTIER.md；05§8",
    "external_source": [
      "https://arxiv.org/html/2603.23978v1",
      "https://arxiv.org/html/1910.07404v2"
    ],
    "hypotheses": "需filtered Selmer morphism、torsion與格修正",
    "normalization": "E[p] vs TpE；增廣degree2 vs r−1；local cone",
    "quantifiers": "此E,p與實際C11²擴張",
    "depends_on": [
      "N402",
      "R308",
      "R317"
    ],
    "reproduced_here": false,
    "notes": "N402只關閉局部來源子義務。"
  },
  {
    "claim_id": "R319",
    "statement": "P5-CPLX-GPR的rank2複數主項比較",
    "status": "OPEN",
    "status_type": "open_obligation",
    "internal_source": "05_NEW_MATHEMATICS.md §8",
    "external_source": [
      "https://arxiv.org/html/1910.07404v2"
    ],
    "hypotheses": "需真正的複數／算術比較定理",
    "normalization": "L''/2!、Néron實週期、實NT regulator",
    "quantifiers": "E389a1,p11；非所有rank",
    "depends_on": [
      "R318"
    ],
    "reproduced_here": false,
    "notes": "不先假設複數比值屬Q或有canonical Q11 realization。"
  },
  {
    "claim_id": "R320",
    "statement": "beta=L''/(2!ΩReg)有理且v11(beta)=0",
    "status": "OPEN",
    "status_type": "open_obligation",
    "internal_source": "p5/files/BSD_P5_Rank2_Scalar_Collapse_389a1_p11_v0.3.md",
    "external_source": [],
    "hypotheses": "有理性、分母界及嚴格誤差須獨立提供",
    "normalization": "複數實數與p-adic估值不可混合",
    "quantifiers": "此scalar",
    "depends_on": [
      "R319"
    ],
    "reproduced_here": false,
    "notes": "數值近1與VAL目標不等於已證估值。"
  },
  {
    "claim_id": "N401",
    "statement": "tame字符Bockstein在H1_unr上為同構，cross-unram字符作用零",
    "status": "VERIFIED",
    "status_type": "proved_lemma",
    "internal_source": "05_NEW_MATHEMATICS.md §3",
    "external_source": [
      "https://www.jmilne.org/math/Books/ADTnot.pdf"
    ],
    "hypotheses": "ell≠p且ell≡1 modp；V unram；own tame character非零",
    "normalization": "算術Frob；H1I=V(-1)；direct-action cup sign",
    "quantifiers": "任意符合設定之V",
    "depends_on": [],
    "reproduced_here": false,
    "notes": "容許nonsemisimple F；character尺度以unit追蹤。"
  },
  {
    "claim_id": "N402",
    "statement": "B_N是實際全域環境Bockstein的兩地局部投影",
    "status": "VERIFIED",
    "status_type": "proved_lemma",
    "internal_source": "05_NEW_MATHEMATICS.md §1、5",
    "external_source": [
      "https://www.jmilne.org/math/Books/ADTnot.pdf"
    ],
    "hypotheses": "實際cyclotomic F與其tautological exact sequence；已認證局部資料",
    "normalization": "eell=βψell(locP)；J²/J³；只投影397/991",
    "quantifiers": "U=spanKummer(P,Q)，不是整個Selmer",
    "depends_on": [
      "N401",
      "R312",
      "R313",
      "R316"
    ],
    "reproduced_here": true,
    "notes": "LOCAL-BOC-REALIZATION閉合；非全global target determinant。"
  },
  {
    "claim_id": "N403",
    "statement": "若λ|U同構則kerλ≅S/U；U=MW/p時等於Sha[p]",
    "status": "VERIFIED",
    "status_type": "proved_lemma",
    "internal_source": "05_NEW_MATHEMATICS.md §6",
    "external_source": [],
    "hypotheses": "S=Sel_p；U含於MW/p；指定λ|U為同構",
    "normalization": "Kummer quotient與unique subtraction",
    "quantifiers": "所有满足假設的Selmer情形",
    "depends_on": [],
    "reproduced_here": true,
    "notes": "一般證明不依赖Sha finite；抽象k4反模型另有測試。"
  },
  {
    "claim_id": "N404",
    "statement": "相同殘餘複形不決定一階Bockstein",
    "status": "VERIFIED",
    "status_type": "logical_nogo",
    "internal_source": "05_NEW_MATHEMATICS.md §8",
    "external_source": [],
    "hypotheses": "R=k[X,Y]/(X,Y)²；兩項自由複形",
    "normalization": "微分0 vs [[X,2X],[Y,4Y]]",
    "quantifiers": "對residual-data-only推論",
    "depends_on": [],
    "reproduced_here": true,
    "notes": "抽象代數反例，不主張任何任意複形皆來自E。"
  },
  {
    "claim_id": "G001",
    "statement": "一般BSD與本例rank2複數主項的全部依賴閉合",
    "status": "OPEN",
    "status_type": "open_obligation",
    "internal_source": "00_EXECUTIVE_STATE.md",
    "external_source": [],
    "hypotheses": "須閉合全部全域與複數依賴",
    "normalization": "所有prime、rank、period及Sha類型",
    "quantifiers": "一般BSD所有E；本例leading term亦未閉合",
    "depends_on": [
      "R319",
      "Z001"
    ],
    "reproduced_here": false,
    "notes": "本輪VERIFIED_ADVANCE僅指明示子義務，不能換成BSD解決。"
  }
]