Open mathematics glossary

A source-boundary glossary for reading Conjecture Fieldbook and interpreting open mathematics without mistaking a source placement, finite computation, or partial result for proof.

Fieldbook records, labels, and learning tools

Atlas: Fieldbook usage. The revision-pinned discovery index of open and recently solved mathematics in Conjecture Fieldbook. It contains source placements and a smaller set of reviewed dossiers.
Source placement: Fieldbook usage. One row extracted from a particular location in the pinned source snapshot. A placement is not guaranteed to represent a unique mathematical problem; multiple rows can refer to the same underlying problem.
Canonical problem: Fieldbook usage. A single, normalized identity for a mathematical problem across names, sources, and repeated mentions. Fieldbook explicitly does not treat every source placement as a canonical problem.
Revision-pinned record: Fieldbook usage. A record tied to an exact revision of its source, so readers can inspect the version from which the record was derived.
Source snapshot: Fieldbook usage. A frozen version of source material captured at a particular revision and retrieval date. Later changes to the source do not silently rewrite the snapshot.
Stable product ID: Fieldbook usage. An internal identifier assigned by Fieldbook to keep a record addressable inside the product. It is not a universal identifier for the mathematical problem.
Reviewed dossier: Fieldbook usage. A richer, editorially reviewed problem record with a clear statement, status, field, depth, source trail, and learning guidance. It is distinct from an automatically extracted source placement.
Editorial dossier: Fieldbook usage. The full reading view for a reviewed problem. Its Overview, Context, and Formal levels present the same problem at increasing levels of mathematical precision.
Overview: Fieldbook usage. The dossier level that explains why a problem matters in accessible language.
Context: Fieldbook usage. The dossier level that situates the problem among variants, solved cases, failed approaches, or neighboring questions.
Formal: Fieldbook usage. The dossier level that states the problem with mathematical notation and explicit hypotheses.
Primary source: Fieldbook usage. The principal outbound source attached to a record. It is a starting point for verification, not a claim that no other or earlier source matters.
Source trail: Fieldbook usage. The chain of links and citations a reader can follow from a Fieldbook summary to the material supporting it.
Source context: Fieldbook usage. A linked source that gives the background needed to interpret a local discussion or learning prompt.
Provenance: Fieldbook usage. Information about where content came from, which revision was used, what Fieldbook changed, and which parts are editorial additions.
Status: Fieldbook usage. The product’s current label for the state of a record, such as Open or Solved since 1995. Status is subject to source quality and editorial review.
Open placement: Fieldbook usage. A source placement classified as unresolved in the pinned index. It is not a guarantee that no later solution, correction, or reclassification exists.
Solved since 1995: Fieldbook usage. A source placement associated with a problem reported solved from 1995 onward. The label describes the index category; readers should verify the cited result.
Notable list: Fieldbook usage. A curated or source-derived collection of related problems. A list can contain overlaps, variants, or records at different levels of specificity.
Field: Fieldbook usage. The broad area of mathematics assigned to a record, such as Number theory, Analysis, or Topology.
Depth: Fieldbook usage. Editorial guidance about the background and abstraction a reader is likely to need. Depth is a learning aid, not a measure of importance or proof difficulty.
Gateway: Fieldbook usage. A depth label for a problem whose statement can be approached with relatively little specialized background, even if solving it is extremely difficult.
Deep: Fieldbook usage. A depth label for a problem that normally requires substantial university-level background in its field.
Very deep: Fieldbook usage. A depth label for a problem whose statement or surrounding literature depends on advanced, specialized theory.
Tag: Fieldbook usage. A descriptive keyword used to group records by concept, method, or topic. Tags support discovery; they are not a formal mathematical classification.
Audience mode: Fieldbook usage. A Student, Teacher, or Researcher view that changes the emphasis of learning guidance. It does not change the underlying mathematical claim.
Reading path: Fieldbook usage. An optional route through prerequisites, reading steps, sources, and notes for one problem.
Reading sequence: Fieldbook usage. A checklist of suggested steps for approaching a problem. Completing the sequence records progress; it does not certify mastery.
Suggested prerequisite: Fieldbook usage. A topic recommended before a reading step. Prerequisites are editorial guidance, not requirements supplied by the pinned source.
Field note: Fieldbook usage. A private note linked to a problem, source, experiment, or reading step.
Revision: Fieldbook usage. A saved state of a local note. Revisions make it possible to review how an idea changed over time.
Local revision history: Fieldbook usage. The sequence of note revisions stored in the current browser profile. It is local history, not a cloud backup or public version-control record.
Installed lab: Fieldbook usage. A bundled, reviewed interactive experiment that runs in the browser and is linked to a mathematical problem.
Local experiment: Fieldbook usage. A computation performed on the user’s device. Unless a lab explicitly says otherwise, no remote mathematical computation is involved.
Input boundary: Fieldbook usage. The exact range, dataset, or assumptions a lab accepts. Results outside that boundary have not been tested by the displayed run.
Method: Fieldbook usage. The procedure a lab uses to transform inputs into displayed results.
Readout: Fieldbook usage. A value, plot, count, or other result produced by a lab for the selected input.
Interpretation boundary: Fieldbook usage. A plain-language statement of what a lab result does and does not establish.
Static dataset: Fieldbook usage. A fixed table bundled with the product rather than fetched or recomputed from a live external service.
Demonstration data: Fieldbook usage. Clearly labeled sample content used to show how a feature works. It is not a real public discussion, scholarly review, or user submission.
Local discussion: Fieldbook usage. A private discussion draft stored in the current browser profile. In the beta release it is not published, shared, identity-verified, or peer-reviewed.

Claims, proof, and evidence

Claim: Mathematics. A statement presented for consideration. A claim becomes a theorem only after a valid proof establishes it within the stated assumptions.
Conjecture: Mathematics. A precisely stated mathematical claim believed or suspected to be true but not yet proved in its full stated form.
Hypothesis: Mathematics. Either an assumption used in a theorem or argument, or—when part of a proper name such as the Riemann hypothesis—an unproved mathematical claim.
Open problem: Mathematics. A mathematical question for which no generally accepted solution is known under the stated formulation.
Theorem: Mathematics. A mathematical statement whose truth has been established by a proof.
Proof: Mathematics. A logically valid argument that derives a conclusion from stated assumptions, definitions, axioms, and previously established results.
Counterexample: Mathematics. A single example that satisfies a claim’s assumptions but not its conclusion. One valid counterexample disproves a universal claim.
Disproof: Mathematics. An argument—often a counterexample—that establishes that a claim is false as stated.
Partial result: Mathematics. A proved result that resolves only part of a problem, such as a restricted range, a special class, or one direction of an equivalence.
Special case: Mathematics. A narrower version of a general problem obtained by adding assumptions or restricting the objects under study.
Equivalent formulation: Mathematics. A statement proved to be true exactly when another statement is true. Proving either one would prove the other.
Classification: Mathematics. A description of all objects of a given kind up to a stated notion of equivalence.
Peer review: Mathematics. Research practice. Evaluation of scholarly work by other qualified researchers. Peer review is a quality-control process, not an infallible guarantee of correctness.
Proof registry: Mathematics. Research practice. A system intended to record and track claimed proofs and their review status. Fieldbook’s broad catalog is explicitly not a proof registry.
Educational evidence: Fieldbook usage. A computation or visualization that supports learning about a displayed finite case. It does not establish a general conjecture unless accompanied by a valid general proof.

Mathematical fields

Algebra: Mathematics. The study of operations, equations, and abstract structures such as groups, rings, fields, vector spaces, and algebras.
Analysis: Mathematics. The study of limits, continuity, change, approximation, and infinite processes. Calculus, real analysis, complex analysis, and functional analysis are major parts of the field.
Functional analysis: Mathematics. The study of vector spaces of functions and the operators acting on them, especially Banach and Hilbert spaces.
Combinatorics: Mathematics. The study of finite or discrete structures, including counting, arrangements, extremal questions, and discrete configurations.
Geometry: Mathematics. The study of shape, size, position, distance, curvature, and the structures that preserve those properties.
Graph theory: Mathematics. The study of graphs: collections of vertices joined by edges. Graphs model relationships, networks, routes, and many discrete structures.
Number theory: Mathematics. The study of integers and related arithmetic structures, including primes, divisibility, Diophantine equations, and number fields.
Additive number theory: Mathematics. The part of number theory that studies how integers can be represented as sums and how addition shapes sets of numbers.
Topology: Mathematics. The study of properties preserved by continuous deformation, such as connectedness, compactness, and the presence of holes.
Model theory: Mathematics. The study of mathematical structures through the formal languages and logical theories used to describe them.
Formal language: Mathematics. A set of finite strings formed from a specified alphabet according to stated rules. Formal languages are used in logic, automata theory, and theoretical computer science.
Dynamical system: Mathematics. A rule describing how a state changes over time, either in discrete steps or continuously.
Diophantine equation: Mathematics. An algebraic equation for which solutions are sought in integers or, in broader usage, rational or algebraic numbers.

Featured problem and lab concepts

Prime number: Mathematics. A positive integer greater than 1 with exactly two positive divisors: 1 and itself.
Prime gap: Mathematics. The difference between consecutive prime numbers.
Twin primes: Mathematics. A pair of prime numbers that differ by 2, such as 11 and 13.
Matrix: Mathematics. A rectangular array of entries used to represent data, linear transformations, systems of equations, and other mathematical structures.
Hadamard matrix: Mathematics. A square matrix with entries \(+1\) and \(-1\) whose rows are mutually orthogonal.
Combinatorial design: Mathematics. An arrangement of a finite set into subsets that satisfies specified balance or incidence conditions.
Perfect cuboid: Mathematics. A rectangular box whose three edge lengths, three face diagonals, and space diagonal are all positive integers. No example is known.
Operator: Mathematics. A mapping that transforms one mathematical object into another. In analysis, operators commonly act on functions or vectors.
Linear operator: Mathematics. An operator \(T\) that preserves addition and scalar multiplication: \(T(ax+by)=aT(x)+bT(y)\).
Bounded linear operator: Mathematics. A linear operator whose output norm is at most a fixed constant times the input norm. Between normed spaces, boundedness is equivalent to continuity.
Hilbert space: Mathematics. A complete vector space equipped with an inner product, allowing geometric ideas such as angles, orthogonality, and projection.
Banach space: Mathematics. A complete normed vector space: every Cauchy sequence converges to a point in the space.
Invariant subspace: Mathematics. A subspace \(M\) preserved by an operator \(T\), meaning \(T(M)\subseteq M\).
Non-trivial invariant subspace: Mathematics. An invariant subspace that is neither the zero subspace nor the entire ambient space.
Compact operator: Mathematics. An operator that maps bounded sets into sets whose closures are compact. Compact operators often behave like infinite-dimensional analogues of matrices.
Self-adjoint operator: Mathematics. An operator that equals its adjoint on the appropriate domain. Self-adjoint operators are central in spectral theory and mathematical physics.
Eigenvalue: Mathematics. A scalar \(\lambda\) for which a nonzero vector \(v\) satisfies \(T(v)=\lambda v\).
Spectrum: Mathematics. The set of scalar values for which an operator fails to have a suitably bounded inverse. In finite dimensions, the spectrum is the set of eigenvalues.
Resolvent: Mathematics. For an operator \(A\), the inverse \((A-\lambda I)^{-1}\) when that inverse exists. The values of \(\lambda\) where it does not exist form the spectrum.
Compact resolvent: Mathematics. An operator has compact resolvent when \((A-\lambda I)^{-1}\) is compact for at least one—and therefore every appropriate—\(\lambda\) in the resolvent set.
Trace formula: Mathematics. An identity that relates spectral data, such as eigenvalues, to geometric, arithmetic, or dynamical data.
Spectral counting law: Mathematics. An estimate or asymptotic formula for the number of eigenvalues or spectral values below a given threshold.
Complex number: Mathematics. A number of the form \(a+bi\), where \(a\) and \(b\) are real and \(i^2=-1\).
Real part: Mathematics. For \(a+bi\), the real part is \(a\), written \(\operatorname{Re}(a+bi)=a\).
Imaginary part: Mathematics. For \(a+bi\), the imaginary part is \(b\), written \(\operatorname{Im}(a+bi)=b\).
Zero of a function: Mathematics. An input at which a function’s value is zero.
Riemann zeta function: Mathematics. A complex function initially defined for \(\operatorname{Re}(s)>1\) by \(\zeta(s)=\sum_{n=1}^{\infty}n^{-s}\), then extended to nearly the whole complex plane by analytic continuation.
Trivial zero: Mathematics. A zero of the Riemann zeta function at a negative even integer.
Non-trivial zero: Mathematics. A zero of the Riemann zeta function in the critical strip \(0<\operatorname{Re}(s)<1\).
Critical strip: Mathematics. The region \(0<\operatorname{Re}(s)<1\) of the complex plane, where the non-trivial zeros of the Riemann zeta function lie.
Critical line: Mathematics. The vertical line \(\operatorname{Re}(s)=\tfrac12\) inside the critical strip.
Imaginary height: Mathematics. The imaginary coordinate \(t\) of a complex point \(s=\sigma+it\). A zeta-zero plot may use this value as its vertical position.
Riemann hypothesis: Mathematics. The conjecture that every non-trivial zero of the Riemann zeta function has real part \(\tfrac12\), so all such zeros lie on the critical line.
Collatz trajectory: Mathematics. The sequence produced by repeatedly applying the Collatz rule: divide an even positive integer by 2; replace an odd one by \(3n+1\).
Goldbach decomposition: Mathematics. A representation of an even integer as the sum of two prime numbers.
Computational complexity: Mathematics. The study of the resources—such as time or memory—required to solve computational problems as input size grows.
P versus NP: Mathematics. The open question of whether every problem whose proposed solution can be checked efficiently can also be solved efficiently.
Graph minor: Mathematics. A graph obtained from another graph by deleting vertices, deleting edges, and contracting edges.
No-three-in-line problem: Mathematics. The problem of placing as many points as possible on an \(n\times n\) grid with no three selected points lying on one straight line.
Manifold: Mathematics. A space that, near each point, resembles ordinary Euclidean space of a fixed dimension.
Four-manifold: Mathematics. A manifold that locally resembles four-dimensional Euclidean space.
Homeomorphism: Mathematics. A continuous bijection with a continuous inverse. Homeomorphic spaces are topologically equivalent.
Diffeomorphism: Mathematics. A smooth bijection with a smooth inverse. Diffeomorphic manifolds have the same smooth structure.
Smooth structure: Mathematics. A consistent system of coordinate charts that makes differentiation possible on a manifold.
Polynomial map: Mathematics. A map whose coordinate functions are polynomials.
Jacobian matrix: Mathematics. The matrix of first partial derivatives of a differentiable map.
Jacobian determinant: Mathematics. For a map between spaces of the same dimension, the determinant of its Jacobian matrix. It describes local change in oriented volume.
Polynomial inverse: Mathematics. An inverse map whose coordinate functions are also polynomials.
Characteristic zero: Mathematics. A property of a field in which repeatedly adding \(1\) never produces \(0\). The rational, real, and complex numbers all have characteristic zero.

Notebook, storage, and privacy

Local-first: Fieldbook usage. The product performs core work and stores user-created content on the user’s device before relying on any remote service.
Saved on this device: Fieldbook usage. Notes, progress, preferences, revisions, and discussion drafts are stored in this browser profile. The label does not mean the data has been backed up.
Offline ready: Fieldbook usage. Installed product assets and local data can support core use without a network connection after the required files have been cached.
Browser profile: Fieldbook usage. The browser’s local environment for history, settings, extensions, and site storage. Data saved in one profile may not appear in another.
IndexedDB: Fieldbook usage. A browser database API for storing and retrieving structured data on the user’s device. Fieldbook uses it for local notes, progress, revisions, preferences, reading lists, and discussion drafts.
Data eviction: Fieldbook usage. Removal of browser-stored site data because the user cleared it, private browsing ended, storage pressure occurred, or the browser applied its storage policy.
Cloud synchronization: Fieldbook usage. Automatic copying and reconciliation of data through a remote service across devices or profiles. The public Fieldbook release does not provide cloud document synchronization.
Export: Fieldbook usage. A file created from data stored in the product so it can be preserved or used elsewhere.
Plaintext: Fieldbook usage. Data stored without encryption. Fieldbook’s JSON, TeX, and lesson-plan exports are plaintext unless the encrypted vault export is explicitly selected.
JSON: Fieldbook usage. JavaScript Object Notation, a text format for structured data. Fieldbook can export local data in JSON for portability and backup.
TeX: Fieldbook usage. A typesetting system widely used for technical and mathematical documents.
LaTeX: Fieldbook usage. A document-preparation system built on TeX that uses commands to structure text and typeset mathematics.
Encrypted vault export: Fieldbook usage. An export option intended to protect file contents with encryption. The user must preserve any required secret or recovery material.
Voice recognition: Fieldbook usage. Conversion of spoken audio into text by the browser’s speech service. Depending on the browser, audio may be sent to the browser vendor after consent.
Analytics consent: Fieldbook usage. The user’s explicit choice to allow privacy-limited product analytics. Without consent—or without a configured measurement ID—the Fieldbook analytics script does not load.
Measurement ID: Fieldbook usage. An identifier that tells an analytics library which configured property should receive permitted events. It is not a note ID or a user-created local identifier.
Hosting record: Fieldbook usage. Routine request metadata processed by hosting or network providers for delivery and security, such as IP address, requested path, user agent, and timestamp.

Definitions describe Fieldbook usage or provide concise mathematical orientation. Consult the linked source trail and primary literature before relying on a term in formal work.