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Exception to a proposed general rule
A counterexample is a specific example that contradicts a claim, hypothesis, or generalization. In logic a counterexample disproves a universally stated
Counterexample
About polynomials in several variables
are invertible. On July 19, 2026, Levent Alpöge presented an explicit counterexample in three variables which he credited to the large language model Claude
Jacobian_conjecture
Witsenhausen's counterexample, shown in the figure below, is a deceptively simple toy problem in decentralized stochastic control. It was formulated by
Witsenhausen's_counterexample
Smallest example which falsifies a claim
In mathematics, a minimal counterexample is the smallest example which falsifies a claim. It is also sometimes called a minimal criminal, smallest criminal
Minimal_counterexample
Book by Lynn Steen
Counterexamples in Topology (1970, 2nd ed. 1978) is a book on mathematics by topologists Lynn Steen and J. Arthur Seebach, Jr. In the process of working
Counterexamples_in_Topology
Philosophical argument
known as Frankfurt counterexamples or Frankfurt-style cases) were presented by philosopher Harry Frankfurt in 1969 as counterexamples to the principle of
Frankfurt_cases
Are certain algebras finitely generated
2 {\displaystyle n\leq 2} . Masayoshi Nagata constructed the first counterexamples in 1958, one of which he announced at the 1958 International Congress
Hilbert's_fourteenth_problem
Technique for symbolic model checking and logic calculi
Counterexample-guided abstraction refinement (CEGAR) is a technique for symbolic model checking. It is also applied in modal logic tableau calculi algorithms
Counterexample-guided abstraction refinement
Counterexample-guided_abstraction_refinement
American-Turkish mathematician (born 1992)
and Pagano in 2024. On July 19, 2026, Alpöge presented an explicit counterexample to the Jacobian conjecture in three-dimensional space, stating that
Levent_Alpöge
Logical fallacy
in which one modifies a prior claim in response to a counterexample by asserting the counterexample is excluded by definition. Rather than admitting error
No_true_Scotsman
Potential counterexample to the generalized Riemann hypothesis
after Edmund Landau and Carl Ludwig Siegel, is a type of potential counterexample to the generalized Riemann hypothesis, on the zeros of Dirichlet L-functions
Siegel_zero
5 Pro to discover a counterexample for Borsuk's conjecture in dimension n = 63, the smallest dimension for which a counterexample is known. In September
List of mathematical discoveries by artificial intelligence
List_of_mathematical_discoveries_by_artificial_intelligence
Large number coined by Ronald Graham
contain no such subgraph if, for example, the bottom edge in the present subgraph were replaced by a blue edge – thus proving by counterexample that N* > 3.
Graham's_number
Method of proof in mathematics
disproved by giving a counterexample, as in classical mathematics. However, it is also possible to give a Brouwerian counterexample to show that the statement
Constructive_proof
Conditions for switching order of integration in calculus
Fubini's theorem gives the conditions under which a double integral can be computed as an iterated integral, i.e. by integrating in one variable at a time
Fubini's_theorem
Counterexample in algebraic geometry
scheme if every orbit is contained in an affine open subscheme; the counterexample above shows that this technical condition cannot be dropped. For quasi-projective
Hironaka's_example
American mathematician
constructed a counterexample that disproved it. Her work involved using fractals and other tools and originally resulted in a more complex counterexample before
Hannah_Cairo
Subset of a topological space whose closure is compact
non-compact homogeneous spaces (specifically spaces of lattices). As a counterexample take any finite neighbourhood of the particular point of an infinite
Relatively_compact_subspace
Transmission of knowledge and skills
occasionally fall outside their parameters. The difficulty of dealing with counterexamples not covered by precise definitions can be avoided by offering less
Education
Mathematical problem solved in 1967
functions that will converge to the counterexample to the common fixed point problem. Although the discovery of counterexamples by Boyce and Huneke meant that
Common_fixed_point_problem
Conjecture in group theory
conjecture is false. While there are no counterexamples known, there are numerous potential counterexamples. It is known that the Zeeman conjecture on
Andrews–Curtis_conjecture
2013 book by J. M. Stoyanov
Counterexamples in Probability is a mathematics book by Jordan M. Stoyanov. Intended to serve as a supplemental text for classes on probability theory
Counterexamples in Probability
Counterexamples_in_Probability
1976 book by Imre Lakatos
counterexample to a lemma (a so-called 'local counterexample') and a counterexample to the specific conjecture under attack (a 'global counterexample'
Proofs_and_Refutations
Unproven conjecture in graph theory
conjecture, or $50 for a counterexample; it is one of many conjectures of Erdős. If the conjecture is false, a counterexample would take the form of a
Erdős–Gyárfás_conjecture
Theorem in probability theory
(Xn)n∈ N {\displaystyle \mathbb {N} } and the number N of terms; see the counterexample below for the necessity. Note that assumption (2) is satisfied when
Wald's_equation
Disproved conjecture in multilinear algebra on the rank of symmetric tensors
All known counterexamples are of very large size and exhibit a gap of exactly one between the two ranks; the problem of finding a counterexample of minimal
Comon's_conjecture
Disproven conjecture in number theory
more accurately called "Pólya's problem". The size of the smallest counterexample is often used to demonstrate the fact that a conjecture can be true
Pólya_conjecture
Field of knowledge
if a result or a theory is wrong, this can be proved by providing a counterexample. Similarly to science, theories and results (theorems) are often obtained
Mathematics
Philosophical problem about what constitutes knowledge
knowledge. Attributed to American philosopher Edmund Gettier, Gettier-type counterexamples (called "Gettier-cases") challenge the long-held notion of knowledge
Gettier_problem
1986 book by Romano and Siegel
Counterexamples in Probability and Statistics is a mathematics book by Joseph P. Romano and Andrew F. Siegel. It began as Romano's senior thesis at Princeton
Counterexamples in Probability and Statistics
Counterexamples_in_Probability_and_Statistics
Mathematical concept
statement that is not universally true for all ordinals must have a minimal counterexample. In fact, this principle is also true for arbitrary well-ordered sets
Transfinite_induction
Theorem in homological algebra
The snake lemma is a tool used in mathematics, particularly homological algebra, to construct long exact sequences. The snake lemma is valid in every abelian
Snake_lemma
Gives conditions that guarantee the max–min inequality holds with equality
In the mathematical area of game theory and of convex optimization, a minimax theorem is a theorem that claims that max x ∈ X min y ∈ Y f ( x , y ) = min
Minimax_theorem
y)=(2x^{2}-y)(y-x^{2}).} It was proposed by Giuseppe Peano in 1899 as a counterexample to a conjectured criterion for the existence of maxima and minima of
Peano_surface
Pathological embedding of the sphere in 3D space
realizing his error, he constructed the horned sphere as a definitive counterexample. Alexander's genius was in realizing that the "wildness" Antoine had
Alexander_horned_sphere
Seven mathematical problems with a US$1 million prize for each solution
American artificial intelligence company OpenAI presented a proposed counterexample to the Navier–Stokes existence and smoothness problem, saying that they
Millennium_Prize_Problems
Chinese mathematician
are now known as the Wall–Sun–Sun primes that guided the search for counterexamples to Fermat's Last Theorem. Zhi-Hong Sun's homepage Archived 2006-07-25
Zhi-Hong_Sun
Paradox arising from an incorrect proof
Two differently colored horses, providing a counterexample to the general theorem
All_horses_are_the_same_color
Equations of motion for viscous fluids
claimed counterexample to the existence and smoothness problem. The announcement was followed by a priority dispute, and the claimed counterexample has yet
Navier–Stokes_equations
Type of category in mathematics
In mathematics, especially in category theory, a closed monoidal category (or a monoidal closed category) is a category that is both a monoidal category
Closed_monoidal_category
Counterexample to the converse of the intermediate value theorem
mathematical function created by British mathematician John H. Conway as a counterexample to the converse of the intermediate value theorem. In other words, it
Conway's_base_13_function
On lengths of shortest paths in convex polytopes
43-dimensional polytope of 86 facets with a diameter of more than 43. The counterexample has no direct consequences for the analysis of the simplex method, as
Hirsch_conjecture
In algebraic geometry, a divisorial scheme is a scheme admitting an ample family of line bundles, as opposed to an ample line bundle. In particular, a
Divisorial_scheme
Disproved conjecture in number theory
1344 involving sums of four fourth powers; this, however, is not a counterexample because no term is isolated on one side of the equation. He also provided
Euler's sum of powers conjecture
Euler's_sum_of_powers_conjecture
Economic observation that capital does not flow from rich to poor countries
In economics, the Lucas paradox or the Lucas puzzle is the observation that capital does not flow from developed countries to developing countries despite
Lucas_paradox
Problem in number theory on equal totients
exists a counterexample to the conjecture, then a positive proportion (in the sense of asymptotic density) of the integers are likewise counterexamples. Although
Carmichael's totient function conjecture
Carmichael's_totient_function_conjecture
Japanese mathematician
counterexamples for some seemingly plausible statements in commutative algebra and algebraic geometry, earning him the nickname "Mr. Counterexample"
Masayoshi_Nagata
Cryptography number with no hidden properties
In cryptography, nothing-up-my-sleeve numbers are any numbers which, by their construction, are above suspicion of hidden properties. They are used in
Nothing-up-my-sleeve_number
Feature of Yiddish expressing parody or skepticism
Shm-reduplication or schm-reduplication is a form of reduplication originating in Yiddish in which the original word or its first syllable (the base) is
Shm-reduplication
statistically independent. However, this is untrue, as can be demonstrated by counterexample. Likewise, it is sometimes mistakenly thought that a linear combination
Misconceptions about the normal distribution
Misconceptions_about_the_normal_distribution
Ring in abstract algebra
In mathematics, specifically abstract algebra, an Artinian ring (sometimes Artin ring) is a ring that satisfies the descending chain condition on (one-sided)
Artinian_ring
Family of tetrahedra on an integer lattice
In geometry, the Reeve tetrahedra are a family of polyhedra with vertices at ( 0 , 0 , 0 ) , ( 1 , 0 , 0 ) , ( 0 , 1 , 0 ) , ( 1 , 1 , r ) , {\displaystyle
Reeve_tetrahedra
Three groups
Thompson in some unpublished handwritten notes in 1965 as a possible counterexample to the von Neumann conjecture. Of the three, F is the most widely studied
Thompson_groups
Ordered field where every nonnegative element is a square
In mathematics, a Euclidean field is an ordered field K for which every non-negative element is a square: that is, x ≥ 0 in K implies that x = y2 for some
Euclidean_ordered_field
Graph theory concept
the conjecture is false, K1,2,2,2 would necessarily be its smallest counterexample. A related conjecture by Michael Fellows, now solved, concerns planar
Planar_cover
Trail in a graph that visits each edge once
directed graph with all even degrees that is not Eulerian, serving as a counterexample to the statement that a sufficient condition for a directed graph to
Eulerian_path
Impossibility for separate objects to have all their properties in common
properties. Max Black has argued against the identity of indiscernibles by counterexample. Notice that to show that the identity of indiscernibles is false, it
Identity_of_indiscernibles
Indicator function of rational numbers
Dirichlet. It is an example of a pathological function which provides counterexamples to many situations. The Dirichlet function is nowhere continuous. We
Dirichlet_function
1994 suicide of Nirvana singer and guitarist
high dose could be survived, although he does not rule out whether a counterexample might exist. Another component is Grant's belief that Cobain's note
Suicide_of_Kurt_Cobain
Continuous function that is not absolutely continuous
that is continuous, but not absolutely continuous. It is a notorious counterexample in analysis, because it challenges naive intuitions about continuity
Cantor_function
also be used to disprove conjectures or proposed theorems by finding counterexamples. Mathematics is verifiable, because once a formal proof is written
Mathematics and artificial intelligence
Mathematics_and_artificial_intelligence
Conjugate transpose of an operator in infinite dimensions
In mathematics, specifically in operator theory, each linear operator A {\displaystyle A} on an inner product space defines a Hermitian adjoint (or adjoint)
Hermitian_adjoint
On unit fractions adding to 4/n
n} that are prime numbers, because any composite counterexample would have a smaller counterexample among its prime factors. Computer searches have verified
Erdős–Straus_conjecture
Proposition in mathematics that is unproven
insufficient for establishing the conjecture's veracity, since a single counterexample could immediately bring down the conjecture. Mathematical journals sometimes
Conjecture
Theorem in functional analysis
In linear algebra and functional analysis, the min-max theorem, or variational theorem, or Courant–Fischer–Weyl min-max principle, is a result that gives
Min-max_theorem
Proposition in complex analysis
true that f {\displaystyle f} will necessarily be differentiable. A counterexample in two dimensions is given by f ( x , y ) = 2 x 2 y + y 3 x 2 + y 2
Osgood's_lemma
Type of commutative ring in mathematics
In mathematics, a Cohen–Macaulay ring is a commutative ring with some of the algebro-geometric properties of a smooth variety, such as local equidimensionality
Cohen–Macaulay_ring
Conjecture in number theory
a monetary prize for a peer-reviewed proof of this conjecture or a counterexample. The value of the prize has increased several times and is currently
Beal_conjecture
Theorem concerning uniform convergence
(A)<\infty } is necessary. To see this, it is simple to construct a counterexample when μ is the Lebesgue measure: consider the sequence of real-valued
Egorov's_theorem
Conjecture in probabilistic combinatorics
original disproof is based on a small counterexample in the 3-uniform hypergraph case with 10 vertices; the counterexample for the standard bunkbed conjecture
Bunkbed_conjecture
counterexample. Schweitzer's construction was then modified by Jenny Harrison in 1988 to make a C 2 + δ {\displaystyle C^{2+\delta }} counterexample for
Seifert_conjecture
Disproven mathematical theory concerning Banach-Tarski and amenable groups
within the class of linear groups. The historically first potential counterexample is Thompson group F. While its amenability is a wide-open problem, the
Von_Neumann_conjecture
Proposed constraint in theoretical linguistics
languages form a natural class of counterexamples to the FOFC. Thus, it must be then investigated whether such counterexamples do indeed violate FOFC, and if
Final-over-final_constraint
Local-global result for when an element in a number field is an nth power
appeared. He said he had a counterexample to a lemma which had been used in the proof. An hour or two later, he produced a counterexample to the theorem itself
Grunwald–Wang_theorem
Theorem on the orders of subgroups
In the mathematical field of group theory, Lagrange's theorem states that if H is a subgroup of any finite group G, then | H | {\displaystyle |H|} is a
Lagrange's theorem (group theory)
Lagrange's_theorem_(group_theory)
Condition for a linear operator to be open
if either space is assumed to be only a normed vector space; see § Counterexample. The proof is based on the following lemmas, which are also somewhat
Open mapping theorem (functional analysis)
Open_mapping_theorem_(functional_analysis)
Distinction between what is and what ought to be
is how Hume and the later positivists conceived of facts. Several counterexamples have been offered by philosophers claiming to show that there are cases
Fact–value_distinction
Delivering multicast packets based on source and destination addressing
Source-specific multicast (SSM) is a method of delivering multicast packets in which the only packets that are delivered to a receiver are those originating
Source-specific_multicast
Mathematical conjecture
Colbrook and George Stepaniants published a preprint establishing a counterexample in two dimensions to the Pompeiu and Schiffer problems, thereby disproving
Pompeiu_problem
In commutative algebra, a complete intersection ring is a commutative ring similar to the coordinate rings of varieties that are complete intersections
Complete_intersection_ring
Principle in John Locke's metaphysics and theory of identity
entrance of another body into the place it possesses. There are some counterexamples to Locke's thesis in the philosophical literature. Locke introduces
Locke's place-time-kind principle
Locke's_place-time-kind_principle
Singularities of holomorphic functions extend infinitely outward
In the theory of functions of several complex variables, Hartogs's extension theorem is a statement about the singularities of holomorphic functions of
Hartogs's_extension_theorem
Type of topological group in mathematics
In mathematics, a locally compact group is a topological group G for which the underlying topology is locally compact and Hausdorff. Locally compact groups
Locally_compact_group
Graphics mode on the Super NES video game console
Mode 7 is a graphics mode on the Super Nintendo Entertainment System (Super NES) video game console that allows a background layer to be rotated and scaled
Mode_7
Disproved conjecture in ring theory
Noetherian rings. In 2026, the AI model GPT-6 Astra found an explicit counterexample which disproves the conjecture, after being prompted by Tom Adamczewski
Köthe_conjecture
Category of mathematical proof
possible counterexamples to be invalid: at least one of the items on a list of possible counterexamples must actually be a valid counterexample to the impossibility
Proof_of_impossibility
Discussions and claims of differences in intelligence along racial lines
Du Bois, and the poet Paul Laurence Dunbar stood as high-profile counterexamples to widespread stereotypes of black intellectual inferiority. In the
Race_and_intelligence
Rule of logical inference
paradoxes of material implication). The general form of McGee-type counterexamples to modus ponens is simply P , P → ( Q → R ) {\displaystyle P,P\rightarrow
Modus_ponens
Curved triangle with constant width
Reuleaux's original motivation for studying the Reuleaux triangle was as a counterexample, showing that three single-point contacts may not be enough to fix a
Reuleaux_triangle
Theorem in number theory
general if the extension is abelian but not cyclic. Hasse gave the counterexample that 3 is a local norm everywhere for the extension Q ( − 3 , 13 ) /
Hasse_norm_theorem
Classic counterexample in topology
Lynn Arthur; Seebach, J. Arthur Jr (1995) [First published 1978], Counterexamples in Topology (Dover reprint of 1978 ed.), Mineola, NY: Dover Publications
Infinite_broom
Open problem on 3x+1 and x/2 functions
holds for all a, then the first counterexample, if it exists, cannot be b modulo 2k. For instance, the first counterexample must be odd because f(2n) = n
Collatz_conjecture
generators that serves as a counterexample to Naimark's problem. More precisely, they showed that the existence of a counterexample generated by ℵ 1 {\displaystyle
Naimark's_problem
Fundamental theorem in mathematical logic
in a model, then one of the model's natural numbers is a counterexample. If this counterexample existed within the standard natural numbers, its existence
Gödel's_completeness_theorem
Written work often reflecting the author's personal point of view
Understanding and Thomas Malthus's An Essay on the Principle of Population are counterexamples. In some countries, such as the United States and Canada, essays have
Essay
Process of making software accessible worldwide
In computing, internationalization and localization (American) or internationalisation and localisation (Commonwealth), often abbreviated i18n and l10n
Internationalization and localization
Internationalization_and_localization
American philosopher (1929–2023)
action. In the field of ethics, Frankfurt gave various influential counterexamples, so-called Frankfurt cases, against the principle that moral responsibility
Harry_Frankfurt
Planar maps require at most four colors
computer-aided proof. This came after many false proofs and mistaken counterexamples in the preceding decades. The Appel–Haken proof proceeds by analyzing
Four_color_theorem
Cubic graph with 10 vertices and 15 edges
and 15 edges. It is a small graph that serves as a useful example and counterexample for many problems in graph theory. The Petersen graph is named after
Petersen_graph
Disproven graph theory
Tutte (1946), who constructed a counterexample with 25 faces, 69 edges and 46 vertices. Several smaller counterexamples, with 21 faces, 57 edges and 38
Tait's_conjecture
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