Open mathematics study cards

Use this 88-card learning deck to distinguish mathematical claims, evidence grades, source records, finite experiments, and privacy boundaries.

Study modules

Reading Fieldbook — 26 cards

Record types, provenance labels, depth levels, reading tools, and experimental boundaries.

Atlas: The revision-pinned Fieldbook index of open and recently solved mathematics, containing source placements and a smaller set of reviewed dossiers.
Source placement: One row extracted from a particular location in the pinned source snapshot; multiple placements can refer to the same mathematical problem.
Canonical problem: A single normalized identity for a mathematical problem across repeated mentions, names, and sources.
Revision-pinned record: A record tied to an exact revision of its source so the version used can be inspected.
Source snapshot: A frozen version of source material captured at a particular revision and retrieval date.
Reviewed dossier: A richer editorial record with a clear statement, status, field, depth, source trail, and learning guidance.
Editorial dossier: The full reviewed-problem view whose Overview, Context, and Formal levels present increasing mathematical precision.
Primary source: The principal outbound source attached to a Fieldbook record; it is a starting point for verification.
Source trail: The chain of links and citations from a Fieldbook summary to the material supporting it.
Provenance: Information about where content came from, which revision was used, what changed, and which parts are editorial additions.
Open placement: A source placement classified as unresolved in the pinned index, subject to later correction or reclassification.
Solved since 1995: A source placement associated with a problem reported solved from 1995 onward; the cited result should still be verified.
Depth: Fieldbook guidance about the background and abstraction a reader is likely to need, not a measure of importance.
Gateway: A Fieldbook depth label for a problem whose statement can be approached with relatively little specialized background.
Deep: A Fieldbook depth label for a problem that normally requires substantial university-level background in its field.
Very deep: A Fieldbook depth label for a problem whose statement or literature depends on advanced specialized theory.
Reading path: An optional route through prerequisites, reading steps, sources, and notes for one problem.
Reading sequence: A checklist of suggested steps for approaching a problem; completion records progress but does not certify mastery.
Suggested prerequisite: A topic recommended before a reading step; it is editorial guidance rather than a source-supplied requirement.
Field note: A private note linked to a problem, source, experiment, or reading step.
Local revision history: The sequence of note revisions stored in the current browser profile, not a cloud backup.
Installed lab: A bundled and reviewed interactive experiment that runs in the browser and is linked to a mathematical problem.
Input boundary: The exact range, dataset, or assumptions accepted by a lab.
Interpretation boundary: A statement of what a lab result does and does not establish.
Static dataset: A fixed table bundled with the product rather than fetched or recomputed from a live external service.
Demonstration data: Clearly labeled sample content used to show a feature; it is not a real public post or scholarly review.

Claims and evidence — 11 cards

Conjectures, proofs, counterexamples, special cases, partial results, and peer review.

Conjecture: A precisely stated mathematical claim believed or suspected to be true but not yet proved in its full stated form.
Open problem: A mathematical question for which no generally accepted solution is known under the stated formulation.
Theorem: A mathematical statement whose truth has been established by a proof.
Proof: A logically valid argument deriving a conclusion from stated assumptions, definitions, axioms, and established results.
Counterexample: An example satisfying a universal claim's assumptions but not its conclusion; one valid counterexample disproves that claim.
Disproof: An argument, often a counterexample, that establishes that a claim is false as stated.
Partial result: A proved result resolving only part of a problem, such as a restricted range, special class, or one direction.
Special case: A narrower version of a general problem obtained by adding assumptions or restricting the objects studied.
Equivalent formulation: A statement proved true exactly when another statement is true; proving either one proves the other.
Peer review: Evaluation of scholarly work by qualified researchers; it is quality control, not an infallible guarantee.
Educational evidence: A computation or visualization supporting a displayed finite case without establishing a general conjecture.

Mathematical fields — 13 cards

The major areas used by Atlas, from algebra and analysis to model theory and dynamical systems.

Algebra: The study of operations, equations, and abstract structures such as groups, rings, fields, and vector spaces.
Analysis: The study of limits, continuity, change, approximation, and infinite processes.
Functional analysis: The study of vector spaces of functions and the operators acting on them, especially Banach and Hilbert spaces.
Combinatorics: The study of finite or discrete structures, including counting, arrangements, and extremal questions.
Geometry: The study of shape, size, position, distance, curvature, and structures preserving those properties.
Graph theory: The study of graphs, which are collections of vertices joined by edges.
Number theory: The study of integers and related arithmetic structures, including primes, divisibility, and Diophantine equations.
Additive number theory: The part of number theory studying representations as sums and the effect of addition on sets of numbers.
Topology: The study of properties preserved by continuous deformation, such as connectedness, compactness, and holes.
Model theory: The study of mathematical structures through the formal languages and logical theories used to describe them.
Formal language: A set of finite strings formed from a specified alphabet according to stated rules.
Dynamical system: A rule describing how a state changes over time, either in discrete steps or continuously.
Diophantine equation: An algebraic equation for which solutions are sought in integers or, more broadly, rational or algebraic numbers.

Featured concepts — 28 cards

Concepts behind installed labs and reviewed dossiers, from prime gaps to graph minors.

Prime number: A positive integer greater than 1 with exactly two positive divisors: 1 and itself.
Prime gap: The difference between consecutive prime numbers.
Twin primes: A pair of prime numbers differing by 2, such as 11 and 13.
Hadamard matrix: A square matrix with entries +1 and -1 whose rows are mutually orthogonal.
Perfect cuboid: A rectangular box whose edges, face diagonals, and space diagonal are all positive integers; no example is known.
Bounded linear operator: A linear operator whose output norm is at most a fixed constant times the input norm.
Hilbert space: A complete inner-product space supporting geometric ideas such as angles, orthogonality, and projection.
Banach space: A complete normed vector space in which every Cauchy sequence converges within the space.
Invariant subspace: A subspace M preserved by an operator T, meaning T(M) is contained in M.
Compact operator: An operator mapping bounded sets into sets whose closures are compact.
Self-adjoint operator: An operator equal to its adjoint on the appropriate domain.
Eigenvalue: A scalar lambda for which a nonzero vector v satisfies T(v) = lambda v.
Spectrum: The scalar values for which an operator fails to have a suitably bounded inverse; in finite dimensions these are the eigenvalues.
Riemann zeta function: A complex function initially defined for Re(s) > 1 by the sum of n^(-s), then analytically continued to nearly the whole complex plane.
Critical strip: The region 0 < Re(s) < 1 where the non-trivial zeros of the Riemann zeta function lie.
Critical line: The vertical line Re(s) = 1/2 inside the critical strip.
Riemann hypothesis: The conjecture that every non-trivial zero of the Riemann zeta function has real part 1/2.
Collatz trajectory: The sequence made by repeatedly dividing an even positive integer by 2 or replacing an odd one by 3n + 1.
Goldbach decomposition: A representation of an even integer as the sum of two prime numbers.
Computational complexity: The study of resources such as time or memory required to solve computational problems as input size grows.
P versus NP: The open question of whether every problem whose proposed solution can be checked efficiently can also be solved efficiently.
Graph minor: A graph obtained from another by deleting vertices, deleting edges, and contracting edges.
No-three-in-line problem: The problem of placing as many points as possible on an n by n grid with no three selected points on one straight line.
Four-manifold: A manifold that locally resembles four-dimensional Euclidean space.
Homeomorphism: A continuous bijection with a continuous inverse; homeomorphic spaces are topologically equivalent.
Diffeomorphism: A smooth bijection with a smooth inverse; diffeomorphic manifolds have the same smooth structure.
Jacobian determinant: The determinant of the matrix of first partial derivatives, describing local change in oriented volume.
Polynomial inverse: An inverse map whose coordinate functions are also polynomials.

Notebook and privacy — 10 cards

Browser storage, offline use, exports, voice recognition, analytics consent, and backup boundaries.

IndexedDB: A browser database API for structured data stored on the user's device.
Data eviction: Removal of browser-stored site data because it was cleared, private browsing ended, storage pressure occurred, or browser policy applied.
Saved on this device: The data is stored in the current browser profile; it may be lost and is not automatically backed up.
Offline ready: Core installed assets and local data can be used without a network connection after required files are cached.
Cloud synchronization: Automatic copying and reconciliation of data through a remote service across devices or profiles.
JSON: A text format for structured data used by Fieldbook for portable local-data exports.
LaTeX: A document-preparation system built on TeX and widely used to structure technical text and typeset mathematics.
Encrypted vault export: A Fieldbook export option intended to protect file contents with encryption.
Voice recognition: Conversion of speech to text by the browser's speech service; depending on the browser, audio may be sent to its vendor after consent.
Analytics consent: The user's explicit choice to allow privacy-limited product analytics; without consent the Fieldbook analytics script does not load.

Check your understanding

What is the strongest description of a source placement? Answer: B. A placement records where an item appears in the source snapshot. More than one placement can refer to the same problem.
What does revision pinning make possible? Answer: C. Revision pinning fixes the source version. It does not prove the claim or guarantee that the catalog is complete.
Which record has the strongest Fieldbook editorial treatment? Answer: B. A reviewed dossier adds a structured statement, labels, source trail, and learning guidance.
What turns a mathematical claim into a theorem? Answer: C. Examples can build confidence, but a theorem requires proof.
What can one valid counterexample do? Answer: B. A universal statement says the conclusion holds in every allowed case; one allowed failure is enough to refute it.
A lab checks a conjecture for one million inputs and finds no failure. What follows? Answer: C. A bounded computation establishes only what happened within its stated input and method boundaries.
What does the Riemann hypothesis assert? Answer: B. The conjecture places all non-trivial zeros of the Riemann zeta function on the critical line.
Which statement defines an invariant subspace? Answer: A. The operator maps every vector in the subspace back into that subspace.
What does “saved on this device” mean in Fieldbook? Answer: B. Browser-stored data can still be cleared or evicted. Important work should be exported.
Which term describes a statement true exactly when another statement is true? Answer: B. Proving either member of a genuine equivalence proves the other.
Are Fieldbook depth labels claims supplied by Wikipedia? Answer: B. Fieldbook’s content notice identifies learning stages and prerequisite paths as editorial additions.
Which export is plaintext by default? Answer: D. Fieldbook states that these exports are plaintext unless the encrypted vault export is explicitly chosen.

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