{"schema":"https://ver.cy/schemas/card/1.0.0","id":"vr.tr.natural-density","code":"thing-q752723","url":"https://ver.cy/models/thing/q752723/","name":"natural density","alternateNames":[],"kind":"thing","status":"published","version":"0.2.0-wave.3","language":"en","classifiers":{"family":"Thing Registry","category":"Cross-cutting context","entryKind":"thing","plane":"XCT","domain":["XCT.QTY"],"industry":[],"navPath":"","tags":[],"facets":{}},"whatItIs":"A measure in number theory of how large a subset of the natural numbers is, defined as the limit, if it exists, of the proportion of numbers up to n that belong to the set as n grows, with upper and lower densities when the limit does not exist; even numbers have density one half, primes have density zero, and natural density underlies results such as Szemeredi theorem and the study of arithmetic progressions.","purpose":"Let an agent explain natural density, relay its definition, properties and examples from mathematical references, describe related densities, and distinguish it from probability and other density notions.","scope":{"in":[],"out":[],"boundaries":[]},"distinguishingFeatures":["Limit definition","Finite additivity only","Upper and lower variants","Combinatorial applications"],"structure":{"bundles":[{"id":"understand","name":"Understand","description":"What natural density is.","layers":[{"id":"definition","name":"Definition","description":"Definition.","findings":[{"id":"definition-finding","name":"Definition","description":"Definition.","questions":[{"text":"How is natural density defined, and when does it exist?","kind":"definition"},{"text":"Is the question about natural density or a probability or other density notion?","kind":"boundary"}]}]},{"id":"examples","name":"Examples","description":"Examples.","findings":[{"id":"examples-finding","name":"Examples","description":"Examples.","questions":[{"text":"What are the densities of even numbers, primes, squarefree numbers and other classic sets?","kind":"measurement"},{"text":"Which entry fits the specific set?","kind":"action"}]}]}]},{"id":"properties","name":"Properties","description":"Properties.","layers":[{"id":"properties","name":"Properties","description":"Properties.","findings":[{"id":"properties-finding","name":"Properties","description":"Properties.","questions":[{"text":"What properties does natural density have, and why is it only finitely additive?","kind":"provenance"},{"text":"Which references are standard?","kind":"provenance"}]}]},{"id":"variants","name":"Variants","description":"Variants.","findings":[{"id":"variants-finding","name":"Variants","description":"Variants.","questions":[{"text":"How do upper, lower, logarithmic, Schnirelmann and Banach densities differ?","kind":"definition"},{"text":"Which entry fits the specific variant?","kind":"action"}]}]}]},{"id":"apply","name":"Apply","description":"Applications.","layers":[{"id":"combinatorics","name":"Combinatorics","description":"Additive combinatorics.","findings":[{"id":"combinatorics-finding","name":"Combinatorics","description":"Combinatorics.","questions":[{"text":"How does density appear in Szemeredi theorem and Erdos conjectures on arithmetic progressions?","kind":"provenance"},{"text":"Which sources are cited?","kind":"provenance"}]}]},{"id":"analytic","name":"Analytic","description":"Analytic number theory.","findings":[{"id":"analytic-finding","name":"Analytic","description":"Analytic.","questions":[{"text":"How does the prime number theorem give the density of primes and related results?","kind":"provenance"},{"text":"Which entry fits prime number theorem?","kind":"action"}]}]}]},{"id":"context","name":"Context","description":"History and teaching.","layers":[{"id":"history","name":"History","description":"History.","findings":[{"id":"history-finding","name":"History","description":"History.","questions":[{"text":"How did the concept develop in the work of Schnirelmann, Erdos and others?","kind":"provenance"},{"text":"Which entry fits the history of number theory?","kind":"action"}]}]},{"id":"teaching","name":"Teaching","description":"Teaching.","findings":[{"id":"teaching-finding","name":"Teaching","description":"Teaching.","questions":[{"text":"How is natural density taught, and what intuitions help?","kind":"provenance"},{"text":"Which entry fits mathematics education?","kind":"action"}]}]}]}]},"agentConduct":{"may":["Explain natural density and compute it for examples such as even numbers or primes.","Distinguish natural density from upper and lower density and from other notions such as logarithmic density."],"mustNot":["State that a set has a natural density when the limit has not been shown to exist.","Treat density zero as meaning the set is finite or empty.","Present a conjectured density as a proven theorem."],"requiresHuman":["Publishing a claimed new result about densities under a person's name."]},"ethics":{"considerations":["Misstated results mislead students and researchers who build on them.","Credit for results should go to those who proved them."],"affectedParties":["Students and learners","Researchers in number theory"]},"owners":{"steward":"Nobody owns the concept; the mathematical community maintains it.","roles":[],"masterSystems":[]},"relations":[{"target":"Q614112","type":"parent","note":"registry parent class"},{"target":"number of entities","type":"related","note":"in registry terms, as a limiting proportion"},{"target":"additive combinatorics","type":"related","note":"such as Szemeredi theorem"},{"target":"prime number theorem","type":"related","note":"for the density of primes"},{"target":"probability measure","type":"related","note":"which is countably additive"}],"interaction":{"identity":{"applicability":"required","items":["Vercy registry: vr.tr.natural-density","Wikidata: Q752723 (https://www.wikidata.org/wiki/Q752723)"]},"properties":{"applicability":"not-applicable","items":["density of even numbers: 1/2 value","density of primes: 0 value","density of squarefree numbers: 6/pi^2 value - about 0.608"]},"recognition":{"applicability":"optional","items":["Asymptotic proportion of a subset of the naturals","Natural, upper, lower, logarithmic and Schnirelmann densities","Probability measures are countably additive; natural density is not","Not a visible object; a limit of ratios."]},"capabilities":{"applicability":"required","items":["explain the definition","relay examples and properties","describe related densities","distinguish from other notions"]},"hazards":{"applicability":"required","items":["Assuming the limit always exists","Treating density as a probability","Confusing density variants"]},"interfaces":{"applicability":"required","items":["No regulation; standard mathematical definitions"]},"context":{"applicability":"required","items":["Not applicable; a mathematical concept.","natural or asymptotic density","upper and lower asymptotic density","logarithmic density","Schnirelmann density","Banach density"]}},"sources":[{"title":"Wikidata item Q752723: natural density","url":"https://www.wikidata.org/wiki/Q752723","note":"identity and sense of the item"},{"title":"Wikipedia: Natural density","url":"https://en.wikipedia.org/wiki/Natural_density","note":"general description of the item"}],"openQuestions":["Should Schnirelmann density be a separate entry?","How should mathematical references be linked?","How should applications be recorded?"],"resources":{"spec":"/models/things/publications/thing-q752723/spec.json"},"provenance":{"origin":"thing registry research (pass 2)","builtFrom":["models/things/publications/thing-q752723/spec.json"],"providers":["Claude"],"researchStatus":"unreviewed","generatedAt":"2026-09-13T02:31:45Z","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":"filled","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 2026-10-06T19:52:02+00:00: completeness 1.00; sense: Wikidata description; 2 live sources; review by codex: pass","interaction.properties":"Plane XCT: no invented physical properties."},"score":1.0}}