{
  "schema": "kurpishev/formula-ledger/1.0",
  "version": "3.7.0",
  "edition": "C3.8U FORMULA WORKBENCH",
  "formulas": [
    {
      "id": "K-DEF",
      "family": "definition",
      "title": {
        "ru": "Ассоциатор",
        "en": "Associator",
        "zh": "结合子"
      },
      "tex": "K(x,y,z)=(x\\odot y)\\odot z-x\\odot(y\\odot z)=r_L-r_R",
      "condition": {
        "ru": "Объекты и область заданы в нормированной алгебре; это определение, а не экспериментальный результат.",
        "en": "Objects and domain are specified in a normed algebra; this is a definition, not an experimental result.",
        "zh": "对象与定义域在赋范代数中给出；这是定义，并非实验结果。"
      },
      "document": "publications/pn2/PN2_THEOREM_v1_0_FORMULA_EXCERPT_v1_0.pdf",
      "document_revision": "PN.2 v1.0 / source excerpt v1.0",
      "document_sha256": "5f686dd1f563d7cdb73f67f74ce674064a5f184c0a596df6af8a924f41a6dfc6",
      "section": "2; 3.1",
      "page": 2,
      "formula": "K",
      "anchor_status": "SOURCE_ANCHORED",
      "mathematical_status": "DEFINITION",
      "po": [
        "PO-1",
        "PO-3"
      ],
      "certificate": null,
      "truth_layer_promotion": 0,
      "evidence_id": "FORMULA-K-DEF"
    },
    {
      "id": "GAPS",
      "family": "definition",
      "title": {
        "ru": "Разрывы между ветвями",
        "en": "Branch separations",
        "zh": "分支间距"
      },
      "tex": "\\Delta_S=|S(r_L)-S(r_R)|,\\quad\\Delta_D=|\\mathfrak D(b_L)-\\mathfrak D(b_R)|,\\quad\\Delta_R=\\|K\\|",
      "condition": {
        "ru": "Объекты и область заданы в нормированной алгебре; это определение, а не экспериментальный результат.",
        "en": "Objects and domain are specified in a normed algebra; this is a definition, not an experimental result.",
        "zh": "对象与定义域在赋范代数中给出；这是定义，并非实验结果。"
      },
      "document": "publications/pn2/PN2_THEOREM_v1_0_FORMULA_EXCERPT_v1_0.pdf",
      "document_revision": "PN.2 v1.0 / source excerpt v1.0",
      "document_sha256": "5f686dd1f563d7cdb73f67f74ce674064a5f184c0a596df6af8a924f41a6dfc6",
      "section": "3.4",
      "page": 3,
      "formula": "Delta_S / Delta_D / Delta_R",
      "anchor_status": "SOURCE_ANCHORED",
      "mathematical_status": "DEFINITION",
      "po": [
        "PO-4"
      ],
      "certificate": null,
      "truth_layer_promotion": 0,
      "evidence_id": "FORMULA-GAPS"
    },
    {
      "id": "DET-ERROR",
      "family": "definition",
      "title": {
        "ru": "Детерминированная ошибка",
        "en": "Deterministic error",
        "zh": "确定性误差"
      },
      "tex": "e_X=\\max_{b\\in\\{L,R\\}}d_X(\\widehat x,x_b)",
      "condition": {
        "ru": "Теоремный пакет: A1–A4, один оцениватель для скрытых ветвей. Положительная граница требует ненулевых разрывов.",
        "en": "Theorem package: A1–A4, one estimator for hidden branches. A positive bound requires nonzero separations.",
        "zh": "定理文本的 A1–A4：两个隐藏分支共用同一估计。正下界需要非零间距。"
      },
      "document": "publications/pn2/PN2_THEOREM_v1_0_FORMULA_EXCERPT_v1_0.pdf",
      "document_revision": "PN.2 v1.0 / source excerpt v1.0",
      "document_sha256": "5f686dd1f563d7cdb73f67f74ce674064a5f184c0a596df6af8a924f41a6dfc6",
      "section": "3.3",
      "page": 3,
      "formula": "e_S / e_D / e_R",
      "anchor_status": "SOURCE_ANCHORED",
      "mathematical_status": "NOTATIONAL_UNIFICATION_OF_SOURCE_DEFINITIONS",
      "po": [
        "PO-6"
      ],
      "certificate": null,
      "truth_layer_promotion": 0,
      "evidence_id": "FORMULA-DET-ERROR"
    },
    {
      "id": "C-SEP",
      "family": "definition",
      "title": {
        "ru": "Условие c-разделения",
        "en": "c-separation condition",
        "zh": "c 分离条件"
      },
      "tex": "\\Delta_S(u)\\ge c\\|K(u)\\|,\\qquad c>0,\\quad u\\in\\Omega",
      "condition": {
        "ru": "Дополнительно c > 0 и Δ_S(u) ≥ c‖K(u)‖. Проверка одной числовой строки не доказывает разделение на всей Ω.",
        "en": "Additionally c > 0 and Δ_S(u) ≥ c‖K(u)‖. A numerical row does not prove separation over all of Ω.",
        "zh": "另需 c > 0 且 Δ_S(u) ≥ c‖K(u)‖。单行数值检验不能证明整个 Ω 上的分离。"
      },
      "document": "publications/pn2/PN2_THEOREM_v1_0_FORMULA_EXCERPT_v1_0.pdf",
      "document_revision": "PN.2 v1.0 / source excerpt v1.0",
      "document_sha256": "5f686dd1f563d7cdb73f67f74ce674064a5f184c0a596df6af8a924f41a6dfc6",
      "section": "3.5",
      "page": 3,
      "formula": "c-separation",
      "anchor_status": "SOURCE_ANCHORED",
      "mathematical_status": "ASSUMPTION",
      "po": [
        "PO-5"
      ],
      "certificate": null,
      "truth_layer_promotion": 0,
      "evidence_id": "FORMULA-C-SEP"
    },
    {
      "id": "PN2-S",
      "family": "deterministic",
      "title": {
        "ru": "ПН.2-S · скалярная форма",
        "en": "PN.2-S · scalar form",
        "zh": "PN.2-S · 标量形式"
      },
      "tex": "e_Se_D\\ge\\frac{\\Delta_S\\Delta_D}{4}",
      "condition": {
        "ru": "Теоремный пакет: A1–A4, один оцениватель для скрытых ветвей. Положительная граница требует ненулевых разрывов.",
        "en": "Theorem package: A1–A4, one estimator for hidden branches. A positive bound requires nonzero separations.",
        "zh": "定理文本的 A1–A4：两个隐藏分支共用同一估计。正下界需要非零间距。"
      },
      "document": "publications/pn2/PN2_THEOREM_v1_0_FORMULA_EXCERPT_v1_0.pdf",
      "document_revision": "PN.2 v1.0 / source excerpt v1.0",
      "document_sha256": "5f686dd1f563d7cdb73f67f74ce674064a5f184c0a596df6af8a924f41a6dfc6",
      "section": "7; 8",
      "page": 4,
      "formula": "PN.2-S",
      "anchor_status": "SOURCE_ANCHORED",
      "mathematical_status": "PROVED_IN_STATED_ABSTRACT_MODEL",
      "po": [
        "PO-5",
        "PO-6"
      ],
      "certificate": null,
      "truth_layer_promotion": 0,
      "evidence_id": "FORMULA-PN2-S"
    },
    {
      "id": "PN2-R",
      "family": "deterministic",
      "title": {
        "ru": "ПН.2-R · состояние",
        "en": "PN.2-R · full state",
        "zh": "PN.2-R · 完整状态"
      },
      "tex": "e_Re_D\\ge\\frac{\\|K\\|\\Delta_D}{4}",
      "condition": {
        "ru": "Теоремный пакет: A1–A4, один оцениватель для скрытых ветвей. Положительная граница требует ненулевых разрывов.",
        "en": "Theorem package: A1–A4, one estimator for hidden branches. A positive bound requires nonzero separations.",
        "zh": "定理文本的 A1–A4：两个隐藏分支共用同一估计。正下界需要非零间距。"
      },
      "document": "publications/pn2/PN2_THEOREM_v1_0_FORMULA_EXCERPT_v1_0.pdf",
      "document_revision": "PN.2 v1.0 / source excerpt v1.0",
      "document_sha256": "5f686dd1f563d7cdb73f67f74ce674064a5f184c0a596df6af8a924f41a6dfc6",
      "section": "7; 8",
      "page": 4,
      "formula": "PN.2-R",
      "anchor_status": "SOURCE_ANCHORED",
      "mathematical_status": "PROVED_IN_STATED_ABSTRACT_MODEL",
      "po": [
        "PO-3",
        "PO-6"
      ],
      "certificate": null,
      "truth_layer_promotion": 0,
      "evidence_id": "FORMULA-PN2-R"
    },
    {
      "id": "PN2-K",
      "family": "deterministic",
      "title": {
        "ru": "ПН.2-K · c-разделение",
        "en": "PN.2-K · c-separation",
        "zh": "PN.2-K · c 分离"
      },
      "tex": "e_Se_D\\ge\\frac{c\\|K\\|\\Delta_D}{4}",
      "condition": {
        "ru": "Дополнительно c > 0 и Δ_S(u) ≥ c‖K(u)‖. Проверка одной числовой строки не доказывает разделение на всей Ω.",
        "en": "Additionally c > 0 and Δ_S(u) ≥ c‖K(u)‖. A numerical row does not prove separation over all of Ω.",
        "zh": "另需 c > 0 且 Δ_S(u) ≥ c‖K(u)‖。单行数值检验不能证明整个 Ω 上的分离。"
      },
      "document": "publications/pn2/PN2_THEOREM_v1_0_FORMULA_EXCERPT_v1_0.pdf",
      "document_revision": "PN.2 v1.0 / source excerpt v1.0",
      "document_sha256": "5f686dd1f563d7cdb73f67f74ce674064a5f184c0a596df6af8a924f41a6dfc6",
      "section": "7; 8",
      "page": 5,
      "formula": "PN.2-K",
      "anchor_status": "SOURCE_ANCHORED",
      "mathematical_status": "PROVED_CONDITIONALLY_IN_MODEL",
      "po": [
        "PO-5",
        "PO-6"
      ],
      "certificate": null,
      "truth_layer_promotion": 0,
      "evidence_id": "FORMULA-PN2-K"
    },
    {
      "id": "TV",
      "family": "statistical",
      "title": {
        "ru": "Различимость данных",
        "en": "Data distinguishability",
        "zh": "数据可区分度"
      },
      "tex": "\\tau=\\operatorname{TV}(P_L,P_R)=\\frac12\\int|p-q|\\,d\\mu,\\quad\\mu=P_L+P_R",
      "condition": {
        "ru": "Два вероятностных закона на одном измеримом пространстве; τ использует весь доступный протокол данных.",
        "en": "Two probability laws on one measurable space; τ uses the entire accessible observation.",
        "zh": "同一可测空间上的两个概率分布；τ 必须基于全部可用观测。"
      },
      "document": "publications/pn2/PN2_STATISTICAL_AUDIT_C3_8T_v1_0_RU.pdf",
      "document_revision": "C3.8T v1.0 / 2026-09-06",
      "document_sha256": "9b1bcebaaa874bda300e09d98df318c414a8449204768f227810df1e9762ab9a",
      "section": "3",
      "page": 2,
      "formula": "(3)",
      "anchor_status": "SOURCE_ANCHORED",
      "mathematical_status": "DEFINITION",
      "po": [
        "PO-6"
      ],
      "certificate": null,
      "truth_layer_promotion": 0,
      "evidence_id": "FORMULA-TV"
    },
    {
      "id": "OVERLAP",
      "family": "statistical",
      "title": {
        "ru": "Общая часть распределений",
        "en": "Overlap of laws",
        "zh": "分布的重叠部分"
      },
      "tex": "\\nu(\\mathcal Y)=\\int\\min(p,q)\\,d\\mu=1-\\tau",
      "condition": {
        "ru": "Два вероятностных закона на одном измеримом пространстве; τ использует весь доступный протокол данных.",
        "en": "Two probability laws on one measurable space; τ uses the entire accessible observation.",
        "zh": "同一可测空间上的两个概率分布；τ 必须基于全部可用观测。"
      },
      "document": "publications/pn2/PN2_STATISTICAL_AUDIT_C3_8T_v1_0_RU.pdf",
      "document_revision": "C3.8T v1.0 / 2026-09-06",
      "document_sha256": "9b1bcebaaa874bda300e09d98df318c414a8449204768f227810df1e9762ab9a",
      "section": "3",
      "page": 3,
      "formula": "(4)",
      "anchor_status": "SOURCE_ANCHORED",
      "mathematical_status": "PROVED_UNDER_STATED_ASSUMPTIONS",
      "po": [],
      "certificate": null,
      "truth_layer_promotion": 0,
      "evidence_id": "FORMULA-OVERLAP"
    },
    {
      "id": "STAT-RISK",
      "family": "statistical",
      "title": {
        "ru": "Худший ожидаемый риск",
        "en": "Worst expected risk",
        "zh": "最坏期望风险"
      },
      "tex": "\\mathcal R_X=\\max_{b\\in\\{L,R\\}}\\mathbb E_b\\,d_X(\\widehat X,x_b)",
      "condition": {
        "ru": "Аудит C3.8T: A1–A6, общие законы полных данных и одно решающее ядро; риски конечны.",
        "en": "C3.8T audit: A1–A6, common complete-data laws and one decision kernel; finite risks.",
        "zh": "C3.8T 审计的 A1–A6：完整数据的共同分布与同一决策核；风险有限。"
      },
      "document": "publications/pn2/PN2_STATISTICAL_AUDIT_C3_8T_v1_0_RU.pdf",
      "document_revision": "C3.8T v1.0 / 2026-09-06",
      "document_sha256": "9b1bcebaaa874bda300e09d98df318c414a8449204768f227810df1e9762ab9a",
      "section": "2",
      "page": 2,
      "formula": "(1)",
      "anchor_status": "SOURCE_ANCHORED",
      "mathematical_status": "DEFINITION",
      "po": [
        "PO-6"
      ],
      "certificate": null,
      "truth_layer_promotion": 0,
      "evidence_id": "FORMULA-STAT-RISK"
    },
    {
      "id": "STAT-S",
      "family": "statistical",
      "title": {
        "ru": "Риск S",
        "en": "Risk S",
        "zh": "风险 S"
      },
      "tex": "\\mathcal R_S\\ge\\frac{\\Delta_S}{2}(1-\\tau)",
      "condition": {
        "ru": "Аудит C3.8T: A1–A6, общие законы полных данных и одно решающее ядро; риски конечны.",
        "en": "C3.8T audit: A1–A6, common complete-data laws and one decision kernel; finite risks.",
        "zh": "C3.8T 审计的 A1–A6：完整数据的共同分布与同一决策核；风险有限。"
      },
      "document": "publications/pn2/PN2_STATISTICAL_AUDIT_C3_8T_v1_0_RU.pdf",
      "document_revision": "C3.8T v1.0 / 2026-09-06",
      "document_sha256": "9b1bcebaaa874bda300e09d98df318c414a8449204768f227810df1e9762ab9a",
      "section": "4; 5",
      "page": 3,
      "formula": "(8)",
      "anchor_status": "SOURCE_ANCHORED",
      "mathematical_status": "PROVED_UNDER_A1_A6",
      "po": [],
      "certificate": null,
      "truth_layer_promotion": 0,
      "evidence_id": "FORMULA-STAT-S"
    },
    {
      "id": "STAT-D",
      "family": "statistical",
      "title": {
        "ru": "Риск D",
        "en": "Risk D",
        "zh": "风险 D"
      },
      "tex": "\\mathcal R_D\\ge\\frac{\\Delta_D}{2}(1-\\tau)",
      "condition": {
        "ru": "Аудит C3.8T: A1–A6, общие законы полных данных и одно решающее ядро; риски конечны.",
        "en": "C3.8T audit: A1–A6, common complete-data laws and one decision kernel; finite risks.",
        "zh": "C3.8T 审计的 A1–A6：完整数据的共同分布与同一决策核；风险有限。"
      },
      "document": "publications/pn2/PN2_STATISTICAL_AUDIT_C3_8T_v1_0_RU.pdf",
      "document_revision": "C3.8T v1.0 / 2026-09-06",
      "document_sha256": "9b1bcebaaa874bda300e09d98df318c414a8449204768f227810df1e9762ab9a",
      "section": "4; 5",
      "page": 3,
      "formula": "(8)",
      "anchor_status": "SOURCE_ANCHORED",
      "mathematical_status": "PROVED_UNDER_A1_A6",
      "po": [],
      "certificate": null,
      "truth_layer_promotion": 0,
      "evidence_id": "FORMULA-STAT-D"
    },
    {
      "id": "STAT-R",
      "family": "statistical",
      "title": {
        "ru": "Риск R",
        "en": "Risk R",
        "zh": "风险 R"
      },
      "tex": "\\mathcal R_R\\ge\\frac{\\|K\\|}{2}(1-\\tau)",
      "condition": {
        "ru": "Аудит C3.8T: A1–A6, общие законы полных данных и одно решающее ядро; риски конечны.",
        "en": "C3.8T audit: A1–A6, common complete-data laws and one decision kernel; finite risks.",
        "zh": "C3.8T 审计的 A1–A6：完整数据的共同分布与同一决策核；风险有限。"
      },
      "document": "publications/pn2/PN2_STATISTICAL_AUDIT_C3_8T_v1_0_RU.pdf",
      "document_revision": "C3.8T v1.0 / 2026-09-06",
      "document_sha256": "9b1bcebaaa874bda300e09d98df318c414a8449204768f227810df1e9762ab9a",
      "section": "4; 5",
      "page": 3,
      "formula": "(8)",
      "anchor_status": "SOURCE_ANCHORED",
      "mathematical_status": "PROVED_UNDER_A1_A6",
      "po": [],
      "certificate": null,
      "truth_layer_promotion": 0,
      "evidence_id": "FORMULA-STAT-R"
    },
    {
      "id": "STAT-SD",
      "family": "statistical",
      "title": {
        "ru": "Произведение рисков S/D",
        "en": "Product of S/D risks",
        "zh": "S/D 风险乘积"
      },
      "tex": "\\mathcal R_S\\mathcal R_D\\ge\\frac{\\Delta_S\\Delta_D}{4}(1-\\tau)^2",
      "condition": {
        "ru": "Аудит C3.8T: A1–A6, общие законы полных данных и одно решающее ядро; риски конечны.",
        "en": "C3.8T audit: A1–A6, common complete-data laws and one decision kernel; finite risks.",
        "zh": "C3.8T 审计的 A1–A6：完整数据的共同分布与同一决策核；风险有限。"
      },
      "document": "publications/pn2/PN2_STATISTICAL_AUDIT_C3_8T_v1_0_RU.pdf",
      "document_revision": "C3.8T v1.0 / 2026-09-06",
      "document_sha256": "9b1bcebaaa874bda300e09d98df318c414a8449204768f227810df1e9762ab9a",
      "section": "5",
      "page": 4,
      "formula": "T2-S",
      "anchor_status": "SOURCE_ANCHORED",
      "mathematical_status": "PROVED_UNDER_A1_A6",
      "po": [],
      "certificate": null,
      "truth_layer_promotion": 0,
      "evidence_id": "FORMULA-STAT-SD"
    },
    {
      "id": "STAT-RD",
      "family": "statistical",
      "title": {
        "ru": "Произведение рисков R/D",
        "en": "Product of R/D risks",
        "zh": "R/D 风险乘积"
      },
      "tex": "\\mathcal R_R\\mathcal R_D\\ge\\frac{\\|K\\|\\Delta_D}{4}(1-\\tau)^2",
      "condition": {
        "ru": "Аудит C3.8T: A1–A6, общие законы полных данных и одно решающее ядро; риски конечны.",
        "en": "C3.8T audit: A1–A6, common complete-data laws and one decision kernel; finite risks.",
        "zh": "C3.8T 审计的 A1–A6：完整数据的共同分布与同一决策核；风险有限。"
      },
      "document": "publications/pn2/PN2_STATISTICAL_AUDIT_C3_8T_v1_0_RU.pdf",
      "document_revision": "C3.8T v1.0 / 2026-09-06",
      "document_sha256": "9b1bcebaaa874bda300e09d98df318c414a8449204768f227810df1e9762ab9a",
      "section": "5",
      "page": 4,
      "formula": "T2-R",
      "anchor_status": "SOURCE_ANCHORED",
      "mathematical_status": "PROVED_UNDER_A1_A6",
      "po": [],
      "certificate": null,
      "truth_layer_promotion": 0,
      "evidence_id": "FORMULA-STAT-RD"
    },
    {
      "id": "STAT-cSD",
      "family": "statistical",
      "title": {
        "ru": "Риски при c-разделении",
        "en": "Risks with c-separation",
        "zh": "c 分离下的风险"
      },
      "tex": "\\mathcal R_S\\mathcal R_D\\ge\\frac{c\\|K\\|\\Delta_D}{4}(1-\\tau)^2",
      "condition": {
        "ru": "Дополнительно c > 0 и Δ_S(u) ≥ c‖K(u)‖. Проверка одной числовой строки не доказывает разделение на всей Ω.",
        "en": "Additionally c > 0 and Δ_S(u) ≥ c‖K(u)‖. A numerical row does not prove separation over all of Ω.",
        "zh": "另需 c > 0 且 Δ_S(u) ≥ c‖K(u)‖。单行数值检验不能证明整个 Ω 上的分离。"
      },
      "document": "publications/pn2/PN2_STATISTICAL_AUDIT_C3_8T_v1_0_RU.pdf",
      "document_revision": "C3.8T v1.0 / 2026-09-06",
      "document_sha256": "9b1bcebaaa874bda300e09d98df318c414a8449204768f227810df1e9762ab9a",
      "section": "5",
      "page": 4,
      "formula": "T2-c",
      "anchor_status": "SOURCE_ANCHORED",
      "mathematical_status": "PROVED_UNDER_A1_A6_AND_C_SEPARATION",
      "po": [
        "PO-5"
      ],
      "certificate": null,
      "truth_layer_promotion": 0,
      "evidence_id": "FORMULA-STAT-cSD"
    }
  ],
  "canonical_statistical_patch": "NOT_APPLIED",
  "independent_review": "NOT_PERFORMED",
  "physical_validation": "NOT_PERFORMED",
  "truth_layer_promotion": 0
}