{"schema":"https://ver.cy/schemas/card/1.0.0","id":"vr.tr.discrete-mathematics","code":"thing-q121416","url":"https://ver.cy/models/thing/q121416/","name":"discrete mathematics","alternateNames":[],"kind":"thing","status":"published","version":"0.1.0-research.1","language":"en","classifiers":{"family":"Thing Registry","category":"Information and virtual systems","entryKind":"thing","plane":"INF","domain":["INF.KNW"],"industry":[],"navPath":"","tags":[],"facets":{}},"whatItIs":"Discrete mathematics is the field of mathematics concerned with discrete structures, including finite sets, integers, graphs and finite sequences, studied through methods such as logical proof, combinatorial counting and recursive construction.","purpose":"Enable an agent to recognise discrete mathematics as a field, assess the scope and justification of its knowledge, and select appropriate discrete representations and reasoning methods.","scope":{"in":["Discrete structures such as graphs, finite sequences, sets, relations, partially ordered sets and combinatorial configurations","Questions of existence, enumeration, structure, extremal behaviour and discrete optimisation","Proof techniques including induction, bijection, invariants, contradiction and combinatorial probabilistic arguments","Relationships between mathematical results, constructive procedures and computational complexity","Declared curricular, research and classification conventions that determine the field's boundaries"],"out":["Continuous analysis and differential equations as independent fields, while retaining their use in discrete problems","The full scope of logic, number theory, algebra and probability beyond their identified discrete-mathematical contributions","Software engineering and concrete implementations of algorithms","Empirical application domains whose situations are represented using discrete structures","Practitioners, departments, societies and publications as independently modelled people, organisations or information objects"],"boundaries":[]},"distinguishingFeatures":["A problem centrally concerns distinguishable elements and combinatorial relationships, rather than merely using digitally stored numbers.","The subject includes infinite discrete structures; finite size alone neither defines membership nor establishes the field's boundary.","A contribution may establish existence, counting or structural properties without supplying an algorithm, distinguishing the field from a purely computational treatment.","An application enters this model through its discrete abstraction and mathematical questions, rather than through the application domain's full empirical behaviour.","Membership of neighbouring subjects such as logic or number theory is recorded against an explicit curricular or research convention, rather than assumed from the label alone."],"structure":{"bundles":[{"id":"field-scope-and-placement","name":"Field scope and placement","description":"Locate discrete mathematics within mathematical knowledge using explicit conventions and evidence.","layers":[{"id":"subject-boundaries","name":"Subject boundaries","description":"Determine which discrete subjects the selected convention includes.","findings":[{"id":"core-and-overlapping-subjects","name":"Core and overlapping subjects","description":"Record the treatment of combinatorics, graph theory and neighbouring areas such as logic, number theory, algebra and theoretical computer science.","questions":[{"text":"Under the chosen convention, which subjects constitute discrete mathematics and which are supporting or overlapping fields?","id":"core-and-overlapping-subjects-q01","kind":"boundary"},{"text":"Does the intended scope describe an introductory curriculum, a research field or another explicitly named use?","id":"core-and-overlapping-subjects-q02","kind":"definition"}]}]},{"id":"classification-and-community-evidence","name":"Classification and community evidence","description":"Connect field placement to documented classifications and institutional usage.","findings":[{"id":"documented-field-placement","name":"Documented field placement","description":"Record verified classification mappings and the scope statements of relevant curricula, handbooks, journals or learned societies.","questions":[{"text":"Which inspected classification scheme and edition places the relevant discrete subjects, and does it treat them as one field or several categories?","id":"documented-field-placement-q01","kind":"provenance"},{"text":"Which defining handbook, curriculum or research-community scope statement supports the proposed boundary?","id":"documented-field-placement-q02","kind":"provenance"}]}]}]},{"id":"discrete-structures-and-representations","name":"Discrete structures and representations","description":"Specify the mathematical objects and representation choices on which discrete reasoning operates.","layers":[{"id":"objects-and-relations","name":"Objects and relations","description":"Identify the elements and structural rules of the discrete setting.","findings":[{"id":"structure-signature","name":"Structure signature","description":"Record the chosen structure family, its defining constraints and its finite or infinite setting.","questions":[{"text":"Are the objects graphs, hypergraphs, orders, relations, words or another discrete structure, and what conditions define admissible instances?","id":"structure-signature-q01","kind":"definition"},{"text":"Is the underlying set finite or infinite, and which arguments depend on that distinction?","id":"structure-signature-q02","kind":"boundary"}]}]},{"id":"equivalence-and-encoding","name":"Equivalence and encoding","description":"Separate mathematical identity from labels and implementation representations.","findings":[{"id":"representation-sensitive-claims","name":"Representation-sensitive claims","description":"Record whether objects are distinguished by labels, isomorphism or another equivalence, and how encodings preserve relevant structure.","questions":[{"text":"When do two labelled graphs, sequences or configurations count as the same mathematical object?","id":"representation-sensitive-claims-q01","kind":"definition"},{"text":"Which properties and size parameters are preserved or changed by the selected encoding?","id":"representation-sensitive-claims-q02","kind":"measurement"}]}]}]},{"id":"combinatorial-questions-and-results","name":"Combinatorial questions and results","description":"Distinguish the main questions asked about discrete structures and the forms their answers take.","layers":[{"id":"existence-and-enumeration","name":"Existence and enumeration","description":"Specify whether a problem asks for admissible objects, their number or their classification.","findings":[{"id":"existence-counting-and-classification-target","name":"Existence, counting and classification target","description":"Record the admissibility conditions and whether the requested answer is a witness, count, recurrence, generating function, estimate or classification.","questions":[{"text":"Is the task to prove existence, construct an example, count objects or classify them up to a stated equivalence?","id":"existence-counting-and-classification-target-q01","kind":"definition"},{"text":"For enumeration, is the result exact or asymptotic, and over which parameters and admissible objects does it range?","id":"existence-counting-and-classification-target-q02","kind":"measurement"}]}]},{"id":"extremal-and-structural-behaviour","name":"Extremal and structural behaviour","description":"Capture bounds, optimal configurations and structural consequences.","findings":[{"id":"bounds-and-attaining-configurations","name":"Bounds and attaining configurations","description":"Specify the constrained family, quantity being bounded and conditions under which equality or an optimum is attained.","questions":[{"text":"What discrete quantity is minimised or maximised, and which constraints define the feasible structures?","id":"bounds-and-attaining-configurations-q01","kind":"definition"},{"text":"Is the bound tight, and are attaining configurations known for all stated parameter values or only some?","id":"bounds-and-attaining-configurations-q02","kind":"measurement"}]}]}]},{"id":"proof-and-epistemic-status","name":"Proof and epistemic status","description":"Record how discrete-mathematical claims are justified and where their validity ends.","layers":[{"id":"proof-strategy-and-assumptions","name":"Proof strategy and assumptions","description":"Connect a claim to its proof mechanism and required hypotheses.","findings":[{"id":"discrete-proof-obligations","name":"Discrete proof obligations","description":"Record the assumptions and critical steps of induction, bijective counting, invariants, extremal arguments or probabilistic existence proofs.","questions":[{"text":"Which proof mechanism establishes the claim, and what are its essential obligations, such as an induction base and step or a bijection's inverse?","id":"discrete-proof-obligations-q01","kind":"definition"},{"text":"Which inspected proof source supports the claim, and what hypotheses or exceptional cases does it require?","id":"discrete-proof-obligations-q02","kind":"provenance"}]}]},{"id":"computational-evidence-and-counterexamples","name":"Computational evidence and counterexamples","description":"Assess what enumeration, search and machine-assisted reasoning actually establish.","findings":[{"id":"verification-coverage","name":"Verification coverage","description":"Separate sampled examples, exhaustive bounded checks, independently checkable certificates and justified reductions to finite verification.","questions":[{"text":"Which instances were checked, and what establishes coverage of the claimed finite range or reduction?","id":"verification-coverage-q01","kind":"measurement"},{"text":"What additional proof, certificate verification or counterexample search is needed before the claim can support the proposed use?","id":"verification-coverage-q02","kind":"action"}]}]}]},{"id":"construction-computation-and-application","name":"Construction, computation and application","description":"Connect discrete knowledge to usable procedures and defensible abstractions of other problems.","layers":[{"id":"constructive-and-algorithmic-consequences","name":"Constructive and algorithmic consequences","description":"Distinguish existence from construction and evaluate computational requirements where established.","findings":[{"id":"existence-to-procedure","name":"Existence to procedure","description":"Record whether a result yields an explicit construction or algorithm and what guarantees accompany it.","questions":[{"text":"Does the result establish existence only, or supply a procedure that produces an admissible object?","id":"existence-to-procedure-q01","kind":"action"},{"text":"For an associated algorithm, what input encoding, size parameters, computational model and proven resource bounds apply?","id":"existence-to-procedure-q02","kind":"measurement"}]}]},{"id":"application-abstraction-and-transfer","name":"Application abstraction and transfer","description":"Record how external tasks become discrete problems and which conclusions transfer back.","findings":[{"id":"discrete-model-fidelity","name":"Discrete model fidelity","description":"Make explicit the interpretation of graph vertices, edges, order relations or combinatorial constraints in the application.","questions":[{"text":"What do the discrete elements and relations represent, and which relevant features of the original task are omitted?","id":"discrete-model-fidelity-q01","kind":"boundary"},{"text":"Which assumptions must be checked before a colouring, matching, ordering or other discrete solution can guide action in the application?","id":"discrete-model-fidelity-q02","kind":"action"}]}]}]},{"id":"evidence-and-external-alignment","name":"Evidence and external alignment","description":"What the world already says about this thing, gathered so the model can be checked against it.","layers":[{"id":"reported-evidence","name":"Reported evidence","description":"Findings from the breadth pass, kept separate from the structural claims.","findings":[{"id":"evidence-confidence-notes","name":"Check these first","description":"Recalled without web access and unsourced; every item is a lead to verify.","questions":[{"text":"Which of these check these first hold for the sense of discrete mathematics this model covers, and on what evidence?","id":"evidence-confidence-notes-q01","kind":"provenance"}]},{"id":"evidence-kinds","name":"Kinds and varieties","description":"Recalled without web access and unsourced; every item is a lead to verify.","questions":[{"text":"Which of these kinds and varieties hold for the sense of discrete mathematics this model covers, and on what evidence?","id":"evidence-kinds-q01","kind":"provenance"}]},{"id":"evidence-identifiers","name":"Identifiers and schemes","description":"Recalled without web access and unsourced; every item is a lead to verify.","questions":[{"text":"Which of these identifiers and schemes hold for the sense of discrete mathematics this model covers, and on what evidence?","id":"evidence-identifiers-q01","kind":"provenance"}]},{"id":"evidence-real-world-use","name":"Real-world use","description":"Recalled without web access and unsourced; every item is a lead to verify.","questions":[{"text":"Which of these real-world use hold for the sense of discrete mathematics this model covers, and on what evidence?","id":"evidence-real-world-use-q01","kind":"provenance"}]},{"id":"evidence-failure-modes-and-hazards","name":"Failure modes and hazards","description":"Recalled without web access and unsourced; every item is a lead to verify.","questions":[{"text":"Which of these failure modes and hazards hold for the sense of discrete mathematics this model covers, and on what evidence?","id":"evidence-failure-modes-and-hazards-q01","kind":"provenance"}]},{"id":"evidence-regional-variation","name":"Regional variation","description":"Recalled without web access and unsourced; every item is a lead to verify.","questions":[{"text":"Which of these regional variation hold for the sense of discrete mathematics this model covers, and on what evidence?","id":"evidence-regional-variation-q01","kind":"provenance"}]},{"id":"evidence-neighbours","name":"Neighbouring kinds and how to tell them apart","description":"Recalled without web access and unsourced; every item is a lead to verify.","questions":[{"text":"Which of these neighbouring kinds and how to tell them apart hold for the sense of discrete mathematics this model covers, and on what evidence?","id":"evidence-neighbours-q01","kind":"provenance"}]}]}]}]},"agentConduct":{"may":["Teach and explain discrete mathematics topics, with correct proofs.","Check counting, graph and logic arguments a user submits and point out errors."],"mustNot":["Present a flawed or incomplete proof as correct.","Complete assessed coursework for a student who is meant to do it alone.","Attribute a theorem or result to the wrong person."],"requiresHuman":["Using a discrete model to make decisions about people, such as in scheduling or allocation."]},"ethics":{"considerations":["Discrete methods underpin cryptography and algorithms that decide outcomes for people, so errors spread widely."],"affectedParties":["Students","Users of software built on it"]},"owners":{"steward":"Nobody owns it; it is held in common as mathematics.","roles":[],"masterSystems":[]},"relations":[{"target":"Q837863","type":"parent","note":"registry parent class"}],"interaction":{"identity":{"applicability":"required","items":["Vercy registry: vr.tr.discrete-mathematics","Wikidata: Q121416 (https://www.wikidata.org/wiki/Q121416)"]},"properties":{"applicability":"not-applicable","items":["Boundary convention: Named curricular, research, bibliographic or institutional convention, with version or date where available","Structure family: Graphs and hypergraphs; sets and relations; orders; sequences and words; combinatorial configurations; other specified discrete structures","Underlying size regime: Finite; countably infinite; other explicitly justified infinite setting; unspecified","Identification rule: Labelled objects; equality of representations; isomorphism classes; equivalence under a specified action or relation","Mathematical objective: Existence; enumeration; classification; extremal bound; optimisation; construction; decidability; complexity","Justification status: Definition; conjecture; proved under stated assumptions; computationally checked over a stated range; refuted; unresolved","Constructive and computational content: Links to constructions or algorithms, with input model, correctness conditions and any established resource bounds","Problem size parameters: Dimensionless counts such as vertices n, edges m, sequence length or input bit length, each explicitly defined"]},"recognition":{"applicability":"not-applicable","items":[]},"capabilities":{"applicability":"required","items":["Classify a topic or problem within discrete mathematics using an explicit boundary convention.","Translate a stated problem into a graph, relation, order, sequence or other discrete structure and record what the translation preserves.","Select proof or counting methods suited to the structure, assumptions and desired result.","Assess whether a claim is supported by proof, a restricted computation, examples or a counterexample.","Identify whether a result supplies existence, an explicit construction or an algorithm with established resource requirements.","Connect topics to verified classifications, curricula and research sources while recording disputed or overlapping membership."]},"hazards":{"applicability":"required","items":["Off-by-one and counting errors that propagate into software.","Proofs that miss edge cases."]},"interfaces":{"applicability":"required","items":["The Mathematics Subject Classification, maintained by Mathematical Reviews of the American Mathematical Society and zbMATH, classifies relevant research under subjects including combinatorics, mathematical logic and graph theory."]},"context":{"applicability":"required","items":["Discrete mathematics is used in computing, network analysis, scheduling and other settings involving distinct objects or countable possibilities.","It provides methods for proving statements, counting arrangements and reasoning about graphs, logical systems and algorithms."]}},"sources":[{"title":"Wikidata item Q121416: discrete mathematics","url":"https://www.wikidata.org/wiki/Q121416","note":"identity and sense of the item"},{"title":"Wikipedia: Discrete mathematics","url":"https://en.wikipedia.org/wiki/Discrete_mathematics","note":"general description of the item"}],"openQuestions":["Which researched definition and boundary convention should anchor this registry entry: introductory curricular usage, research usage or an explicitly layered combination?","Which classification schemes and editions provide defensible mappings for the field and its constituent subjects without implying a single universal classification code?","How much logic, elementary number theory, discrete probability and algebra should this entry own, and which neighbouring registry models should own their broader treatment?","Does an existing Vercy world model already cover this field or a materially equivalent concept, requiring a link instead of a separate publication?","Which inspected handbooks, reviews and institutional scope statements best support the treatment of infinite discrete structures and methods borrowed from continuous mathematics?"],"resources":{"spec":"/models/things/publications/thing-q121416/spec.json"},"provenance":{"origin":"thing registry research (pass 2)","builtFrom":["models/things/publications/thing-q121416/spec.json"],"providers":["Codex"],"researchStatus":"unreviewed","generatedAt":"2026-09-07T14:06:53Z","builder":"tools/build_cards.py@1.0.0"},"completeness":{"sections":{"classifiers":"filled","whatItIs":"filled","purpose":"filled","distinguishingFeatures":"filled","structure":"filled","agentConduct":"filled","ethics":"filled","owners":"filled","relations":"filled","interaction.identity":"filled","interaction.properties":"not-applicable","interaction.recognition":"not-applicable","interaction.capabilities":"filled","interaction.hazards":"filled","interaction.interfaces":"filled","interaction.context":"filled","sources":"filled"},"notes":{"_gate":"Published through the thing publication gate 1.0 on 2026-10-07T10:10:25+00:00: completeness 1.00; sense: Wikidata description; 2 live sources; review by codex: pass","interaction.properties":"Plane INF: no invented physical properties.","interaction.recognition":"Discrete mathematics is an abstract field that cannot be identified by visual appearance."},"score":1.0}}