AWhat 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.
Scientific 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
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.
Contributors
- The named authors in the research record · SCIENCE: Source-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.
Provisional WRS assessment, edition 2026-10-23.3. Independent expert calibration is pending. This grade does not certify correctness or assign ownership.
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