Kemnitz's Conjecture¶

In additive number theory, Kemnitz's conjecture states that every set of integer lattice points in the plane has a large subset whose centroid is also a lattice point.

Version: v1 · 2026-09-28 · History
Domain-specific #: 10236
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomains: Additive Number Theory, Combinatorics → Mathematics

Core Idea¶

Kemnitz's Conjecture is treated here as the recurring mathematics, logic, and statistics identity summarized by this source-grounded definition: In additive number theory, Kemnitz's conjecture states that every set of integer lattice points in the plane has a large subset whose centroid is also a lattice point.

In additive number theory, Kemnitz's conjecture states that every set of integer lattice points in the plane has a large subset whose centroid is also a lattice point. It was proved independently in the autumn of 2003 by Christian Reiher, then an undergraduate student, and Carlos di Fiore, then a high school student. The exact formulation of this conjecture is as follows.

Let n be a natural number and S a set of 4n-3 lattice points in plane. Then there exists a subset S_1 \subseteq S with n points such that the centroid of all points from S_1 is also a lattice point. Kemnitz's conjecture was formulated in 1983 by Arnfried Kemnitz as a generalization of the Erdős–Ginzburg–Ziv theorem, an analogous one-dimensional result stating that every 2n-1 integers have a subset of size n whose average is an integer.

For Kemnitz's Conjecture, the abstraction is narrower than the article's general subject matter: a positive case must preserve In additive number theory, Kemnitz's conjecture states that every set of integer lattice points in the plane has a large subset whose centroid is also a lattice point. Retaining only the name, a familiar example, or a downstream effect is insufficient. The specialist roles and tests remain anchored in mathematics, logic, and statistics, which is why this identity is domain-specific rather than prime.

Structural Signature¶

Sig role-phrases:

Defining carrier — It was proved independently in the autumn of 2003 by Christian Reiher, then an undergraduate student, and Carlos di Fiore, then a high school student.
Constitutive relation — Kemnitz's conjecture was formulated in 1983 by Arnfried Kemnitz as a generalization of the Erdős–Ginzburg–Ziv theorem, an analogous one-dimensional result stating that every 2n-1 integers have a subset of size n whose average is an integer.
Operating condition — In additive number theory, Kemnitz's conjecture states that every set of integer lattice points in the plane has a large subset whose centroid is also a lattice point.
Recognition evidence — Let n be a natural number and S a set of 4n-3 lattice points in plane.
Admissible variation — Then there exists a subset S_1 \subseteq S with n points such that the centroid of all points from S_1 is also a lattice point.
Characteristic consequence — In 2000, Lajos Rónyai proved a weakened form of Kemnitz's conjecture for sets with 4n-2 lattice points.
Failure boundary — Then, in 2003, Christian Reiher proved the full conjecture using the Chevalley–Warning theorem.

What It Is Not¶

Not the whole field of mathematics, logic, and statistics. The node requires the specific identity stated by In additive number theory, Kemnitz's conjecture states that every set of integer lattice points in the plane has a large subset whose centroid is also a lattice point.
Not an over-broad reading. In additive number theory, Kemnitz's conjecture states that every set of integer lattice points in the plane has a large subset whose centroid is also a lattice point.
Not an over-broad reading. It was proved independently in the autumn of 2003 by Christian Reiher, then an undergraduate student, and Carlos di Fiore, then a high school student.
Not an over-broad reading. Let n be a natural number and S a set of 4n-3 lattice points in plane.
Not automatically Integer points in convex polyhedra. Retrieval proximity does not establish equivalence; the two identities must be compared by carrier, operation, and failure boundary.

Scope of Application¶

Kemnitz's Conjecture applies literally inside mathematics, logic, and statistics wherever the source-defined carrier and relation can be established. Its documented habitats include:

Documented setting. In additive number theory, Kemnitz's conjecture states that every set of integer lattice points in the plane has a large subset whose centroid is also a lattice point.
Documented setting. It was proved independently in the autumn of 2003 by Christian Reiher, then an undergraduate student, and Carlos di Fiore, then a high school student.
Documented setting. Let n be a natural number and S a set of 4n-3 lattice points in plane.
Documented setting. Then there exists a subset S_1 \subseteq S with n points such that the centroid of all points from S_1 is also a lattice point.
Documented setting. Kemnitz's conjecture was formulated in 1983 by Arnfried Kemnitz as a generalization of the Erdős–Ginzburg–Ziv theorem, an analogous one-dimensional result stating that every 2n-1 integers have a subset of size n whose average is an integer.
Documented setting. In 2000, Lajos Rónyai proved a weakened form of Kemnitz's conjecture for sets with 4n-2 lattice points.

Outside mathematics, logic, and statistics, the name should be retained only when these same operational conditions survive; otherwise the comparison belongs to the broader parent Theory or should be marked as analogy.

Clarity¶

A clear use of Kemnitz's Conjecture names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is In additive number theory, Kemnitz's conjecture states that every set of integer lattice points in the plane has a large subset whose centroid is also a lattice point. The strongest recognition evidence in the frozen account is: Let n be a natural number and S a set of 4n-3 lattice points in plane. A report should distinguish that evidence from a proxy, consequence, or common implementation. It should also state the qualification In additive number theory, Kemnitz's conjecture states that every set of integer lattice points in the plane has a large subset whose centroid is also a lattice point. so that a reader can reproduce the classification rather than infer it from topical resemblance.

Manages Complexity¶

Kemnitz's Conjecture compresses multiple mathematics, logic, and statistics details into a stable diagnostic relation. The source shows both the central mechanism—kemnitz's conjecture was formulated in 1983 by Arnfried Kemnitz as a generalization of the Erdős–Ginzburg–Ziv theorem, an analogous one-dimensional result stating that every 2n-1 integers have a subset of size n whose average is an integer.—and the practical consequence—in 2000, Lajos Rónyai proved a weakened form of Kemnitz's conjecture for sets with 4n-2 lattice points. This compression makes cases comparable while leaving parameters, conventions, exceptions, and evidential quality explicit. It is lossy by design: local history and implementation details may be omitted only when they do not alter the defining relation.

Abstract Reasoning¶

Type the carrier. Identify the mathematics, logic, and statistics entities to which the claim applies.
State the relation. Use the source-grounded identity: In additive number theory, Kemnitz's conjecture states that every set of integer lattice points in the plane has a large subset whose centroid is also a lattice point.
Check operation and conditions. In additive number theory, Kemnitz's conjecture states that every set of integer lattice points in the plane has a large subset whose centroid is also a lattice point.
Demand recognition evidence. Let n be a natural number and S a set of 4n-3 lattice points in plane.
Test variation. Change an implementation or setting while preserving then there exists a subset S_1 \subseteq S with n points such that the centroid of all points from S_1 is also a lattice point.
Run the collapse test. Remove the defining operation; if the label still seems equally apt, only a topic or correlate was retained.
Reduce cautiously. When the specialist conditions cannot be carried, route the residual comparison to Theory.

Knowledge Transfer¶

Within the home domain. Knowledge about Kemnitz's Conjecture transfers literally when a new case preserves the same carrier type, relation, and recognition test. In additive number theory, Kemnitz's conjecture states that every set of integer lattice points in the plane has a large subset whose centroid is also a lattice point. It was proved independently in the autumn of 2003 by Christian Reiher, then an undergraduate student, and Carlos di Fiore, then a high school student.

Beyond the home domain. No canonical parent is asserted for Kemnitz's Conjecture. An outside case receives the specialist name only when the same typed roles and rejection conditions can be filled literally; otherwise the comparison remains an analogy pending later graph densification.

Examples¶

Canonical¶

In additive number theory, Kemnitz's conjecture states that every set of integer lattice points in the plane has a large subset whose centroid is also a lattice point. This case is canonical because it supplies a concrete carrier and lets the defining relation be checked rather than merely named.

Mapped back: carrier → the entities in the documented case; operation → In additive number theory, Kemnitz's conjecture states that every set of integer lattice points in the plane has a large subset whose centroid is also a lattice point; recognition evidence → Let n be a natural number and S a set of 4n-3 lattice points in plane

Applied / In Practice¶

It was proved independently in the autumn of 2003 by Christian Reiher, then an undergraduate student, and Carlos di Fiore, then a high school student. The applied case shows how the identity is used under a second setting or qualification while keeping the same operative relation.

Mapped back: changed setting → the applied context; invariant → In additive number theory, Kemnitz's conjecture states that every set of integer lattice points in the plane has a large subset whose centroid is also a lattice point; boundary → the case exits the class when in additive number theory, Kemnitz's conjecture states that every set of integer lattice points in the plane has a large subset whose centroid is also a lattice point

Structural Tensions¶

T1 — Stable identity versus admissible variation. In additive number theory, Kemnitz's conjecture states that every set of integer lattice points in the plane has a large subset whose centroid is also a lattice point. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Which changes preserve the defining relation, and which replace it?

T2 — Recognition versus proxy. It was proved independently in the autumn of 2003 by Christian Reiher, then an undergraduate student, and Carlos di Fiore, then a high school student. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Does the cited evidence establish the identity or only a correlated sign?

T3 — Definition versus implementation. Let n be a natural number and S a set of 4n-3 lattice points in plane. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Is the observed implementation constitutive, optional, or merely common?

T4 — Scope versus overextension. Then there exists a subset S_1 \subseteq S with n points such that the centroid of all points from S_1 is also a lattice point. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Can every claimed application fill the same typed roles without metaphor?

T5 — Transfer versus domain accent. It was proved independently in the autumn of 2003 by Christian Reiher, then an undergraduate student, and Carlos di Fiore, then a high school student. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Does the receiving case instantiate Kemnitz's Conjecture literally, co-instantiate Theory, or only resemble it?

T6 — Autonomy versus reduction. Kemnitz's conjecture was formulated in 1983 by Arnfried Kemnitz as a generalization of the Erdős–Ginzburg–Ziv theorem, an analogous one-dimensional result stating that every 2n-1 integers have a subset of size n whose average is an integer. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: What does Kemnitz's Conjecture distinguish that the broader parent Theory leaves together?

Structural–Framed Character¶

Kemnitz's Conjecture is structural-leaning. Its structural side is the repeatable organization summarized by In additive number theory, Kemnitz's conjecture states that every set of integer lattice points in the plane has a large subset whose centroid is also a lattice point. Its framed side is the mathematics, logic, and statistics vocabulary that fixes the carrier, evidence, exceptions, and admissible transformations.

Evaluative weight: the identity can be stated descriptively even when applications carry practical stakes. Human-practice dependence: the source-grounded carrier determines whether the relation exists independently or is constituted by a practice. Institutional origin: disciplinary conventions stabilize the name and test. Vocabulary portability: In additive number theory, Kemnitz's conjecture states that every set of integer lattice points in the plane has a large subset whose centroid is also a lattice point. Import versus recognition: literal transfer requires the same mechanism; shape alone is analogy.

Its portable skeleton is Theory. Its character: a recurring specialist identity whose thin organization can be abstracted, while its operational meaning remains domain-bound.

Structural Core vs. Domain Accent¶

What is skeletal. In additive number theory, Kemnitz's conjecture states that every set of integer lattice points in the plane has a large subset whose centroid is also a lattice point. The stable skeleton is the typed relation expressed in that definition and the entry's recognition and collapse tests. The source identifies these operative conditions: It was proved independently in the autumn of 2003 by Christian Reiher, then an undergraduate student, and Carlos di Fiore, then a high school student. Kemnitz's conjecture was formulated in 1983 by Arnfried Kemnitz as a generalization of the Erdős–Ginzburg–Ziv theorem, an analogous one-dimensional result stating that every 2n-1 integers have a subset of size n whose average is an integer. It further constrains recognition and variation through: In additive number theory, Kemnitz's conjecture states that every set of integer lattice points in the plane has a large subset whose centroid is also a lattice point. Let n be a natural number and S a set of 4n-3 lattice points in plane.

What is domain-bound. mathematics, logic, and statistics supplies the operative entities, technical vocabulary, warrants, and exceptions that make Kemnitz's Conjecture literal. Its documented scope includes the condition that In additive number theory, Kemnitz's conjecture states that every set of integer lattice points in the plane has a large subset whose centroid is also a lattice point. Another bounded application condition is that It was proved independently in the autumn of 2003 by Christian Reiher, then an undergraduate student, and Carlos di Fiore, then a high school student. These are not decorative examples; they determine which carrier and evidence can fill the abstraction's roles.

Why no parent is asserted. Removing those specialist details does not currently yield one live catalog node that is a necessary genus for every instance. The entry is therefore approved as unparented rather than attached by topical resemblance. Its collapse evidence remains specific—Then there exists a subset S1 \subseteq S with n points such that the centroid of all points from S1 is also a lattice point.—and future graph densification may discover a defensible relation only if it preserves that boundary.

Approved unparented node. No current live node supplies a defensible necessary genus or structural prerequisite for Kemnitz's Conjecture. The reviewed identity is: In additive number theory, Kemnitz's conjecture states that every set of integer lattice points in the plane has a large subset whose centroid is also a lattice point. The accelerated suggestion was declined because topical or lexical similarity does not establish hierarchy; the node is admitted without a parent pending later graph densification.
Related reasoning operations. Evidence, representation, comparison, classification, transformation, or evaluation may participate in particular cases, but participation does not make any one of them a necessary parent of every instance.

Neighborhood in Abstraction Space¶

Kemnitz's Conjecture sits in a sparse region of the domain-specific corpus (65^th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Number Theory & Packing Conjectures (5 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08

Not to Be Confused With¶

Theory. The parent omits the specialist differentia. Tell: Can the case establish In additive number theory, Kemnitz's conjecture states that every set of integer lattice points in the plane has a large subset whose centroid is also a lattice point?
Integer points in convex polyhedra. The study of integer points in convex polyhedra is motivated by questions such as "how many nonnegative integer-valued solutions does a system of linear equations with nonnegative coefficients have" or "how many solutions does an integer linear program have". Tell: Which entry's carrier, operation, and failure condition are satisfied?
Ruzsa–Szemerédi Problem. An extremal-combinatorics problem asking how dense an n-vertex graph can be when every edge lies in exactly one triangle, equivalently how many triples can be chosen with no three supported on six vertices. Tell: Which entry's carrier, operation, and failure condition are satisfied?
Cannonball problem. The Diophantine problem of finding integers for which a square pyramidal number is also a perfect square, equivalently when the sum of consecutive squares from one to n is itself square. Tell: Which entry's carrier, operation, and failure condition are satisfied?
A measurement, proxy, or consequence. Those may provide evidence without being the identity. Tell: Would Kemnitz's Conjecture remain present if the detector or downstream effect changed?
A metaphorical analogue. A similar shape outside mathematics, logic, and statistics lacks the specialist mechanism. Tell: Do the native roles transfer literally, or only the parent Theory?

References¶

Frozen Wikipedia discovery revision: https://en.wikipedia.org/wiki/Kemnitz%27s_conjecture (revision 1329482731).

The frozen Wikipedia revision is discovery provenance. The retained source set was reviewed for identity, formal or operational relation, and scope. The encyclopedia's structural synthesis is bounded to those claims; a thin authority surface is recorded as a nonblocking source-strengthening repair rather than concealed.