feat: add Morse inequalities eval problem#300
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§40 of Knill's "Some Fundamental Theorems in Mathematics" (Marston Morse, 1934). For a Morse function f on a closed smooth finite-dimensional manifold M, the alternating partial sums of critical-point counts c_k(f) dominate those of Betti numbers b_k(M). Mathlib has the smooth-manifold framework, mfderiv, higher Fréchet derivatives, and singularHomologyFunctor but no Morse functions, critical-point counts, Betti numbers, or the Morse inequalities. The Challenge ships seven helper definitions. Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>
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This PR adds the Morse inequalities as a new lean-eval challenge problem — §40 of Oliver Knill's Some Fundamental Theorems in Mathematics (Marston Morse, 1934).
For a Morse function
fon a closed smooth finite-dimensional manifoldMand everyk ∈ ℕ:∑_{j≤k} (−1)^{k−j} c_j(f) ≥ ∑_{j≤k} (−1)^{k−j} b_j(M)where
c_j(f)is the number of Morse-index-jcritical points andb_j(M)is thej-th Betti number.mathlib has the smooth-manifold framework,
mfderiv, higher Fréchet derivatives, andsingularHomologyFunctor— but no Morse functions, Morse index, critical-point counts, or Betti numbers as a named definition. The Challenge ships seven auxiliary definitions:IsCriticalPoint,localHessian,IsNondegenerateCritical,IsMorseFunction,morseIndex,morseCount,bettiNumber,alternatingPartialSum.🤖 Prepared with Claude Code