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    "summary": "For a C^q map from a region of R^m to R^n, Sard proves measure-zero results for the images of points where the derivative has less than maximal rank. If m>n, q≥m−n+1 gives zero n-dimensional Lebesgue measure; if m≤n, C^1 suffices for the rank-deficient image to have zero m-dimensional Hausdorff measure.",
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    "area": "Differential analysis and geometry",
    "topic": "Mathematics",
    "year": 1942,
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        "title": "The measure of the critical values of differentiable maps (original1942)",
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            "citation": "The measure of the critical values of differentiable maps (original1942)",
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        "description": "The original defines critical rank as less than min(m,n); for m<n this is not the modern convention rank<n for every point. Retain the distinct measure dimensions and integer differentiability threshold. The proof uses A. P. Morse's decomposition theorem and Whitney's earlier critical-set example; these are inherited, not newly proved here as Sard's sole inventions. No claim that the critical-point set is null or empty, that arbitrary C^1 maps with m>n suffice, or that the result covers arbitrary Sobolev or infinite-dimensional maps.",
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          "citation": "The measure of the critical values of differentiable maps (original1942)",
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      "statement": "# Sard: differentiability thresholds for null critical-value sets\n\nFor a C^q map from a region of R^m to R^n, Sard proves measure-zero results for the images of points where the derivative has less than maximal rank. If m>n, q≥m−n+1 gives zero n-dimensional Lebesgue measure; if m≤n, C^1 suffices for the rank-deficient image to have zero m-dimensional Hausdorff measure.\n\n## Scientific scope\n\nThe original defines critical rank as less than min(m,n); for m<n this is not the modern convention rank<n for every point. Retain the distinct measure dimensions and integer differentiability threshold. The proof uses A. P. Morse's decomposition theorem and Whitney's earlier critical-set example; these are inherited, not newly proved here as Sard's sole inventions. No claim that the critical-point set is null or empty, that arbitrary C^1 maps with m>n suffice, or that the result covers arbitrary Sobolev or infinite-dimensional maps.\n\n## Reading status\n\nThis is a source-linked WRS catalogue note. A full explanatory guide and independent scientific reconstruction have not been completed for this entry. Read the identified original work for its complete narrative and arguments.\n\n## People and systems\n\nArthur Sard\n\nSource-listed authors for the exact contribution; experimental subjects are not autonomous scientific authors. Source support and predecessor work retain separate roles. Individual effort shares and unresolved identities are not inferred.\n\n\n\n## Original sources\n\n- [The measure of the critical values of differentiable maps (original1942)](https://www.ams.org/journals/bull/1942-48-12/S0002-9904-1942-07811-6/S0002-9904-1942-07811-6.pdf)\n",
      "summary": "For a C^q map from a region of R^m to R^n, Sard proves measure-zero results for the images of points where the derivative has less than maximal rank. If m>n, q≥m−n+1 gives zero n-dimensional Lebesgue measure; if m≤n, C^1 suffices for the rank-deficient image to have zero m-dimensional Hausdorff measure.",
      "title": "Sard: differentiability thresholds for null critical-value sets"
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    "id": "sard-critical-value-nullity-1942",
    "entry_id": "sard-critical-value-nullity-1942",
    "revision": 1,
    "title": "Sard: differentiability thresholds for null critical-value sets",
    "context": "Differential analysis and geometry",
    "track": "Exact theorem and reusable mathematical method",
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    "what_changed": "For a C^q map from a region of R^m to R^n, Sard proves measure-zero results for the images of points where the derivative has less than maximal rank. If m>n, q≥m−n+1 gives zero n-dimensional Lebesgue measure; if m≤n, C^1 suffices for the rank-deficient image to have zero m-dimensional Hausdorff measure.",
    "scope": "For a C^q map from a region of R^m to R^n, Sard proves measure-zero results for the images of points where the derivative has less than maximal rank. If m>n, q≥m−n+1 gives zero n-dimensional Lebesgue measure; if m≤n, C^1 suffices for the rank-deficient image to have zero m-dimensional Hausdorff measure. The original defines critical rank as less than min(m,n); for m<n this is not the modern convention rank<n for every point. Retain the distinct measure dimensions and integer differentiability threshold. The proof uses A. P. Morse's decomposition theorem and Whitney's earlier critical-set example; these are inherited, not newly proved here as Sard's sole inventions. No claim that the critical-point set is null or empty, that arbitrary C^1 maps with m>n suffice, or that the result covers arbitrary Sobolev or infinite-dimensional maps.",
    "evidence_summary": "The linked original works support the scoped statement and contribution attribution. WRS has not independently replayed experiments or reconstructed a full mathematical proof. The original defines critical rank as less than min(m,n); for m<n this is not the modern convention rank<n for every point. Retain the distinct measure dimensions and integer differentiability threshold. The proof uses A. P. Morse's decomposition theorem and Whitney's earlier critical-set example; these are inherited, not newly proved here as Sard's sole inventions. No claim that the critical-point set is null or empty, that arbitrary C^1 maps with m>n suffice, or that the result covers arbitrary Sobolev or infinite-dimensional maps.",
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