{
  "standardisation": {
    "id": "standardisation",
    "pdf_page": 55,
    "printed_page": 45,
    "latex": "Z_i = \\frac{x_i-\\bar{x}_i}{s_i}",
    "unicode_text": "Zᵢ = (xᵢ − x̄ᵢ) / sᵢ",
    "placement": "After For each metric…is given by; replace stray OCR s i.",
    "note": "Mean bar and all i subscripts visually confirmed."
  },
  "absolute-difference": {
    "id": "absolute-difference",
    "pdf_page": 56,
    "printed_page": 46,
    "section": "2.10.4.1",
    "latex": "d_{ij}=\\sum_{k=1}^{m}w_k\\left|c_{jk}-x_{ik}\\right|",
    "unicode_text": "dᵢⱼ = ∑ₖ₌₁ᵐ wₖ |cⱼₖ − xᵢₖ|",
    "placement": "After the first The distance is given by the following equation sentence, before Where m…",
    "note": "The original OCR omits the displayed equation."
  },
  "euclidean": {
    "id": "euclidean",
    "pdf_page": 56,
    "printed_page": 46,
    "section": "2.10.4.2",
    "latex": "d_{ij}=\\left(\\sum_{k=1}^{m}\\left(w_k\\left(c_{jk}-x_{ik}\\right)\\right)^2\\right)^{1/2}",
    "unicode_text": "dᵢⱼ = (∑ₖ₌₁ᵐ (wₖ(cⱼₖ − xᵢₖ))²)^(1/2)",
    "placement": "After Euclidean Distance explanatory paragraph; before section 2.10.4.3.",
    "note": "Weight wₖ is inside the squared quantity in the printed source. Do not silently move it outside the square."
  },
  "mahalanobis": {
    "id": "mahalanobis",
    "pdf_page": 56,
    "printed_page": 46,
    "section": "2.10.4.3",
    "latex": "d_{ij}=(c_j-x_i)'S^{-1}(c_j-x_i)",
    "unicode_text": "dᵢⱼ = (cⱼ − xᵢ)′ S⁻¹(cⱼ − xᵢ)",
    "placement": "After Mahalanobis Distance explanatory paragraph, bottom of page.",
    "note": "Remove added vector bold styling to follow the printed equation. Prime and absence of square root remain unchanged.",
    "source_evidence": {
      "image": "images/source-056.png",
      "region": "Bottom displayed equation: ordinary italic c and x, not added upright bold;400dpi image source-056-mahalanobis-detail.png",
      "dpi": 200
    }
  },
  "fuzzy-powerset": {
    "id": "fuzzy-powerset",
    "pdf_page": 59,
    "printed_page": 49,
    "latex": "2_1^D\\,x\\,2_2^D\\,x\\,\\ldots\\,x\\,2_k^D",
    "unicode_text": "2₁ᴰ x 2₂ᴰ x ... x 2ₖᴰ",
    "placement": "Inline in first paragraph after Cartesian product.",
    "note": "Literal printed notation retained. The source may intend2^{D_i}, but that interpretation is not silently substituted. Native inline rendering is required; do not leave caret ASCII in the reader edition.",
    "source_evidence": {
      "image": "images/source-059.png",
      "region": "First paragraph Cartesian product;400dpi source-059-powerset-0.png shows D superscript with1/2/k subscripts on2, italic x separators and baseline ellipsis.",
      "dpi": 200
    },
    "inline_old_text": "2^(D₁) × 2^(D₂) × ... × 2^(Dₖ)",
    "inline_text_anchor": "2_1^D x 2_2^D x ... x 2_k^D"
  },
  "similarity-map": {
    "id": "similarity-map",
    "pdf_page": 59,
    "printed_page": 49,
    "latex": "S:D\\times D\\to[0,1]",
    "unicode_text": "S: D × D → [0, 1]",
    "placement": "Inline after to the unit interval, i.e."
  },
  "reflexivity": {
    "id": "reflexivity",
    "pdf_page": 59,
    "printed_page": 49,
    "latex": "S_D(x,x)=1",
    "unicode_text": "S_D(x, x) = 1",
    "placement": "Numbered property 1. Reflexive."
  },
  "symmetry": {
    "id": "symmetry",
    "pdf_page": 59,
    "printed_page": 49,
    "latex": "S_D(x,y)=S_D(y,x)",
    "unicode_text": "S_D(x, y) = S_D(y, x)",
    "placement": "Numbered property 2. Symmetric."
  },
  "transitivity": {
    "id": "transitivity",
    "pdf_page": 59,
    "printed_page": 49,
    "latex": "S_D(x,z)\\geq\\max_{y\\in D}\\left(\\min\\left(S_D(x,y),S_D(y,z)\\right)\\right)",
    "unicode_text": "S_D(x, z) ≥ max_{y∈D}(min(S_D(x, y), S_D(y, z)))",
    "placement": "Numbered property 3. Transitive.",
    "note": "The OCR loses ≥, the y∈D condition, and S_D symbols. These are visible in the source."
  },
  "possibility-distribution": {
    "id": "possibility-distribution",
    "pdf_page": 60,
    "printed_page": 50,
    "latex": "\\Pi_{A(t)}",
    "unicode_text": "Π_{A(t)}",
    "placement": "Inline after possibility distribution.",
    "note": "Enlarged source crop math-page060-detail.png confirms the complete A(t) is subscript, not only A."
  },
  "exponential-feature-weight": {
    "id": "exponential-feature-weight",
    "pdf_page": 93,
    "printed_page": 83,
    "latex": "\\sqrt{\\frac{0.75}{2^i}}.",
    "unicode_text": "√(0.75 / 2ⁱ)",
    "placement": "After Exponential. For 0 <= i < n, exactly one feature has weight; OCR omitted this expression."
  },
  "linear-feature-weight": {
    "id": "linear-feature-weight",
    "pdf_page": 93,
    "printed_page": 83,
    "latex": "1-\\frac{1+i}{1+n}.",
    "unicode_text": "1 − (1+i)/(1+n)",
    "placement": "After Linear. For 0 <= i <= n-1, exactly one feature has weight; replace incomplete OCR fraction."
  },
  "interest-permutations": {
    "id": "interest-permutations",
    "pdf_page": 113,
    "printed_page": 103,
    "latex": "{}_5P_3\\ \\text{or}\\ \\frac{5!}{(5-3)!}=60",
    "unicode_text": "₅P₃ = 5!/(5−3)! = 60",
    "placement": "Inline after For INT…This is; original uses 'or' between the permutation symbol and factorial fraction."
  },
  "possibility-equation1": {
    "id": "possibility-equation1",
    "pdf_page": 124,
    "printed_page": 114,
    "latex": "\\operatorname{Poss}(\\mathrm{Poor}\\mid\\Pi_x)=\\sup_x[\\min(\\Pi_x,\\mu_{\\mathrm{poor}}(x))]",
    "unicode_text": "Poss(Poor | Πₓ) = supₓ[min(Πₓ, μ_poor(x))]",
    "placement": "Equation1 after Example introductory sentence.",
    "note": "Copied as printed, without validating the surrounding derivation."
  },
  "necessity-equation2-original": {
    "id": "necessity-equation2-original",
    "pdf_page": 124,
    "printed_page": 114,
    "latex": "\\operatorname{Nec}(\\mathrm{Poor}\\mid\\Pi_x)=\\inf_x[\\max(\\Pi_x,1-\\mu_{\\mathrm{poor}}(x))]",
    "unicode_text": "Nec(Poor | Πₓ) = infₓ[max(Πₓ, 1−μ_poor(x))]",
    "placement": "Equation2.",
    "note": "Preserve the original positions of Πₓ and the complement. The following worked example uses the complement of the possibility distribution instead. This is a source-level discrepancy, not an OCR repair."
  },
  "marks-possibility": {
    "id": "marks-possibility",
    "pdf_page": 125,
    "printed_page": 115,
    "latex": "\\Pi_{\\mathrm{marks}}(J)=0.2/30+0.5/40+1/50+0.8/60",
    "unicode_text": "Π_marks(J) = 0.2/30 + 0.5/40 + 1/50 + 0.8/60",
    "placement": "First displayed equation."
  },
  "poor-membership": {
    "id": "poor-membership",
    "pdf_page": 125,
    "printed_page": 115,
    "latex": "\\mathrm{Poor}=1/20+1/30+1/40+0.8/50+0.4/60+0.2/70",
    "unicode_text": "Poor = 1/20 + 1/30 + 1/40 + 0.8/50 + 0.4/60 + 0.2/70",
    "placement": "After definition of the linguistic term Poor."
  },
  "possibility-worked": {
    "id": "possibility-worked",
    "pdf_page": 125,
    "printed_page": 115,
    "latex": "\\begin{aligned}\\operatorname{Poss}(\\mathrm{Poor}\\mid\\Pi_{\\mathrm{marks}}J)&=\\max\\{0.2\\wedge1,0.5\\wedge1,1\\wedge0.8,0.8\\wedge0.4\\}\\\\&=\\max\\{0.2,0.5,0.8,0.4\\}\\\\&=0.8\\end{aligned}",
    "unicode_text": "Poss(Poor | Π_marks J) = max{0.2 ∧ 1, 0.5 ∧ 1, 1 ∧ 0.8, 0.8 ∧ 0.4} = max{0.2, 0.5, 0.8, 0.4} = 0.8",
    "placement": "Three lines following Using equation1; preserve line breaks for readable layout."
  },
  "marks-complement": {
    "id": "marks-complement",
    "pdf_page": 125,
    "printed_page": 115,
    "latex": "\\begin{aligned}1-\\Pi_{\\mathrm{marks}}(J)&=1/20+0.8/30+0.5/40+0/50+0.2/60\\\\&\\quad+1/70+1/80+1/90+1/95\\end{aligned}",
    "unicode_text": "1−Π_marks(J) = 1/20 + 0.8/30 + 0.5/40 + 0/50 + 0.2/60 + 1/70 + 1/80 + 1/90 + 1/95",
    "placement": "After complement introductory sentence."
  },
  "necessity-worked": {
    "id": "necessity-worked",
    "pdf_page": 125,
    "printed_page": 115,
    "latex": "\\begin{aligned}\\operatorname{Nec}(\\mathrm{Poor}\\mid\\Pi_{\\mathrm{marks}}J)&=\\min\\{1\\vee1,1\\vee0.8,1\\vee0.5,0.8\\vee0,0.4\\vee0.2,\\\\&\\qquad 0.2\\vee1,0\\vee1,0\\vee1,0\\vee1\\}\\\\&=\\min\\{1,1,1,0.8,0.4,1,1,1,1\\}\\\\&=0.4\\end{aligned}",
    "unicode_text": "Nec(Poor | Π_marks J) = min{1 ∨ 1, 1 ∨ 0.8, 1 ∨ 0.5, 0.8 ∨ 0, 0.4 ∨ 0.2, 0.2 ∨ 1, 0 ∨ 1, 0 ∨ 1, 0 ∨ 1} = min{1, 1, 1, 0.8, 0.4, 1, 1, 1, 1} = 0.4",
    "placement": "Three lines after The necessity measure is obtained using equation2.",
    "note": "This reproduces the worked example, which differs from the printed general equation2. Do not silently reconcile."
  },
  "interest-permutation-symbol": {
    "id": "interest-permutation-symbol",
    "pdf_page": 129,
    "printed_page": 119,
    "latex": "{}_5P_3",
    "unicode_text": "₅P₃",
    "placement": "Inline in the knowledge-base paragraph; source-collated by root in the merged68–134 review.",
    "source_evidence": "pages_068_134/corrected_pages.json, physical129, ROOT-P129-C001"
  }
}