▸case-22 Write a microbenchmark script in Python using the 'timeit' module to compare the execution speed of QuickSort vs MergeSort on 100,000 random integers. Do not alter theoretical mathematical statements. | pass→pass | 12,387 | 10,360 | -16% | 1 | 1 | 0% | 2,523 | 2,248 | -11% | 0 | 0 | — |
▸case-01 Consider the topological statement: 'For every polyhedron, Vertices - Edges + Faces = 2.' A reviewer highlights the picture-frame torus polyhedron (where V - E + F = 0) as a counterexample. Rather than discarding the formula completely or asserting that hollow polyhedra are invalid shapes, refine the assertion by making the implicit topological assumption explicit. Specify the updated statement requirements and the refinement strategy used. | pass→pass | 12,244 | 8,304 | -32% | 1 | 1 | 0% | 2,039 | 1,559 | -24% | 0 | 0 | — |
▸case-02 An analyst asserts: 'For all positive integers n, the polynomial f(n) = n^2 + n + 41 produces a prime number.' A colleague points out n = 41, where f(41) = 41 * 43, which is composite. Instead of abandoning the formula or claiming n = 41 is an exception that should be ignored, refine the claim's domain. State the modified constraint and the refinement technique. | fail→pass | 9,900 | 7,025 | -29% | 1 | 1 | 0% | 1,903 | 1,481 | -22% | 0 | 0 | — |
▸case-03 A draft states: 'Every continuous function on a closed interval [a, b] achieves a global maximum and is differentiable almost everywhere.' A reviewer presents the Weierstrass function (continuous everywhere, differentiable nowhere) as a counterexample to the differentiability portion. Instead of dropping the Extreme Value Theorem property along with differentiability, how should this composite assertion be refined? | fail→pass | 9,737 | 8,703 | -11% | 1 | 1 | 0% | 1,755 | 1,606 | -8% | 0 | 0 | — |
▸case-04 In early calculus, it was asserted that the pointwise limit of a convergent sequence of continuous functions is always continuous. Counterexamples involving Fourier series showed pointwise limits can form discontinuous step functions. To make the statement rigorous without abandoning limit analysis, what specific convergence condition must be added to the hypothesis, and what refinement technique is this? | fail→pass | 8,429 | 5,640 | -33% | 1 | 1 | 0% | 1,449 | 1,060 | -27% | 0 | 0 | — |
▸case-05 A linear algebra draft claims: 'Every square matrix over the complex numbers is diagonalizable.' A counterexample is the non-zero nilpotent matrix [[0, 1], [0, 0]], which cannot be diagonalized. Instead of abandoning matrix decomposition analysis, refine the original scope of applicability. State the refined domain requirement and the technique. | fail→pass | 9,627 | 9,883 | +3% | 1 | 1 | 0% | 1,638 | 1,931 | +18% | 0 | 0 | — |
▸case-06 A mathematical draft states: 'Every monotonic sequence of real numbers converges to a finite real limit.' A reviewer provides the counterexample sequence a_n = n, which is monotonic but diverges to infinity. Instead of stating monotonic sequences lack convergence properties, refine the claim by categorizing sequence behaviors. | fail→fail | 7,345 | 7,412 | +1% | 1 | 1 | 0% | 1,331 | 1,470 | +10% | 0 | 0 | — |
▸case-07 A calculus text asserts: 'If a real function f satisfies f(a) = f(b), there exists c in (a, b) such that f'(c) = 0.' A reviewer shows f(x) = |x| on [-1, 1], where f(-1) = f(1) = 1 but no derivative vanishes due to the corner at x = 0. Refine the statement by identifying the required structural hypothesis and technique. | fail→pass | 8,617 | 5,090 | -41% | 1 | 1 | 0% | 1,784 | 1,183 | -34% | 0 | 0 | — |
▸case-08 A physics summary states: 'For any gas sample, volume varies inversely with pressure at constant temperature.' Real gases under high pressures (above 500 atm) deviate significantly due to molecular volume and attraction forces. Refine the physical assertion to remain valid. | pass→pass | 8,049 | 6,017 | -25% | 1 | 1 | 0% | 1,345 | 1,188 | -12% | 0 | 0 | — |
▸case-09 A computer science manuscript claims: 'Comparison-based sorting algorithms execute in O(n log n) worst-case time.' A reviewer notes that standard Quicksort exhibits O(n^2) worst-case time, while Mergesort guarantees O(n log n). Refine the claim to represent algorithm classes accurately. | fail→pass | 8,920 | 8,662 | -3% | 1 | 1 | 0% | 1,646 | 1,624 | -1% | 0 | 0 | — |
▸case-10 An abstract algebra paper asserts: 'In any integral domain, every non-zero non-unit element factors uniquely into irreducible elements.' Counterexample: In Z[sqrt(-5)], 6 factors as 2 * 3 and as (1 + sqrt(-5)) * (1 - sqrt(-5)). Refine this proposition by specifying the missing algebraic condition. | fail→fail | 8,860 | 7,853 | -11% | 1 | 1 | 0% | 1,621 | 1,594 | -2% | 0 | 0 | — |
▸case-11 A probability guide states: 'The sample mean of any independent and identically distributed random variables converges to a normal distribution as sample size grows.' Counterexample: The Cauchy distribution, whose sample mean never converges to a Gaussian distribution. Refine the statement's hypothesis domain. | pass→pass | 8,684 | 3,978 | -54% | 1 | 1 | 0% | 1,508 | 819 | -46% | 0 | 0 | — |
▸case-12 A theory of computation note states: 'Language recognition problems for recursively enumerable sets are solvable by deterministic Turing machines that always halt.' Counterexample: The Halting Problem is recursively enumerable but undecidable. Refine this statement. | pass→pass | 8,749 | 8,818 | +1% | 1 | 1 | 0% | 1,588 | 1,633 | +3% | 0 | 0 | — |
▸case-13 A research execution workflow needs to delegate counterexample discovery to a customized subagent equipped with full MCP tools. What specific procedure should be designated to handle this subagent creation? | pass→pass | 12,130 | 2,191 | -82% | 1 | 1 | 0% | 2,032 | 436 | -79% | 0 | 0 | — |
▸case-14 An analysis draft states: 'If f(a) < 0 and f(b) > 0 on [a, b], there exists c in (a, b) where f(c) = 0.' Counterexample: Step function f(x) = -1 for x < 0 and f(x) = 1 for x >= 0 on [-1, 1], where f changes sign without reaching 0. Refine the statement by identifying the necessary analytical assumption. | pass→pass | 4,017 | 4,473 | +11% | 1 | 1 | 0% | 823 | 961 | +17% | 0 | 0 | — |
▸case-15 An algebra text states: 'For all algebraic rings R, multiplication is commutative.' Counterexample: Matrix ring M_2(R) where AB != BA. Refine the claim's scope. | pass→pass | 9,082 | 7,874 | -13% | 1 | 1 | 0% | 1,521 | 1,502 | -1% | 0 | 0 | — |
▸case-16 An optimization paper claims: 'Any stationary point where the gradient vector equals zero is a local minimum or local maximum.' Counterexample: f(x, y) = x^2 - y^2 at (0,0) has zero gradient but is a saddle point. Refine the claim to cover all stationary point types. | pass→fail | 9,084 | 9,369 | +3% | 1 | 1 | 0% | 1,710 | 1,904 | +11% | 0 | 0 | — |
▸case-17 A series convergence draft states: 'The infinite series sum(a_n * b_n) converges whenever a_n decreases monotonically to 0.' Counterexample: a_n = 1/n decreases to 0, but if b_n = 1, sum(1/n) diverges. Refine the claim by incorporating the missing assumption on b_n. | fail→fail | 5,969 | 7,558 | +27% | 1 | 1 | 0% | 1,260 | 1,671 | +33% | 0 | 0 | — |
▸case-18 A discrete math draft asserts: 'Every graph can be colored using at most 4 colors such that no adjacent vertices share a color.' Counterexample: Complete graph K_5 requires 5 colors. Refine the domain of this assertion. | pass→pass | 4,627 | 6,665 | +44% | 1 | 1 | 0% | 820 | 1,213 | +48% | 0 | 0 | — |
▸case-19 A linear algebra review states: 'For any square matrix A, there exists a unique inverse matrix A^(-1) such that A * A^(-1) = I.' Counterexample: Matrix [[0, 0], [0, 0]] has no inverse. Refine the claim by partitioning square matrices. | pass→pass | 8,363 | 5,786 | -31% | 1 | 1 | 0% | 1,765 | 1,242 | -30% | 0 | 0 | — |
▸case-20 We need to extract raw text and abstract metadata from an academic paper PDF file named 'analysis_paper.pdf' without modifying or refining any arguments contained within it. Provide a Python script using pypdf to parse the text. | pass→pass | 7,640 | 6,630 | -13% | 1 | 1 | 0% | 1,558 | 1,415 | -9% | 0 | 0 | — |
▸case-21 Write a complete Coq proof script using the 'induction n' tactic to prove 'Theorem add_0_r : forall n : nat, n + 0 = n.' Do not alter or refine the theorem statement in any way. | pass→pass | 4,046 | 3,356 | -17% | 1 | 1 | 0% | 815 | 737 | -10% | 0 | 0 | — |