{"schema":"https://ver.cy/schemas/card/1.0.0","id":"vr.tr.schnirelmann-density","code":"thing-q2482753","url":"https://ver.cy/models/thing/q2482753/","name":"Schnirelmann 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 additive number theory of how dense a set of natural numbers is, defined by Schnirelmann in 1930 as the infimum over positive integers n of the number of set members from 1 through n divided by n, so that any set missing 1 has density zero; for sets A and B of nonnegative integers both containing zero, with densities alpha and beta calculated without counting zero, their sumset has density at least alpha + beta - alpha * beta; the density underlies Schnirelmann's theorem that every integer greater than one is a sum of a bounded number of primes and is distinct from natural density.","purpose":"Let an agent explain Schnirelmann density, relay its definition, properties and use in Schnirelmann theorem from mathematical references, and distinguish it from natural and other densities.","scope":{"in":[],"out":[],"boundaries":[]},"distinguishingFeatures":["Infimum definition","Sensitivity to small numbers","Sumset inequality","Prime sum application"],"structure":{"bundles":[{"id":"understand","name":"Understand","description":"What Schnirelmann density is.","layers":[{"id":"definition","name":"Definition","description":"Definition.","findings":[{"id":"definition-finding","name":"Definition","description":"Definition.","questions":[{"text":"How is Schnirelmann density defined, and why does missing 1 give density zero?","kind":"definition"},{"text":"Is the question about Schnirelmann density, natural density or another density?","kind":"boundary"}]}]},{"id":"examples","name":"Examples","description":"Examples.","findings":[{"id":"examples-finding","name":"Examples","description":"Examples.","questions":[{"text":"What are the Schnirelmann densities of the odd numbers, the squares and the primes?","kind":"measurement"},{"text":"Which entry fits the specific set?","kind":"action"}]}]}]},{"id":"theory","name":"Theory","description":"Theorems.","layers":[{"id":"sumsets","name":"Sumsets","description":"Sumset inequalities.","findings":[{"id":"sumsets-finding","name":"Sumsets","description":"Sumsets.","questions":[{"text":"What do the Schnirelmann and Mann inequalities say about sumsets?","kind":"provenance"},{"text":"Which references are standard?","kind":"provenance"}]}]},{"id":"primes","name":"Primes","description":"Sums of primes.","findings":[{"id":"primes-finding","name":"Primes","description":"Primes.","questions":[{"text":"How does Schnirelmann theorem show every integer is a sum of boundedly many primes, and what is known about the constant?","kind":"provenance"},{"text":"Which entry fits Goldbach conjecture?","kind":"action"}]}]}]},{"id":"compare","name":"Compare","description":"Comparison.","layers":[{"id":"densities","name":"Densities","description":"Other densities.","findings":[{"id":"densities-finding","name":"Densities","description":"Densities.","questions":[{"text":"How does Schnirelmann density compare with natural, lower, upper and Banach densities?","kind":"provenance"},{"text":"Which sources are cited?","kind":"provenance"}]}]},{"id":"uses","name":"Uses","description":"Uses.","findings":[{"id":"uses-finding","name":"Uses","description":"Uses.","questions":[{"text":"Where else is Schnirelmann density used in additive combinatorics?","kind":"provenance"},{"text":"Which entry fits additive combinatorics?","kind":"action"}]}]}]},{"id":"context","name":"Context","description":"History.","layers":[{"id":"history","name":"History","description":"History.","findings":[{"id":"history-finding","name":"History","description":"History.","questions":[{"text":"Who was Lev Schnirelmann, and how did his work lead to later results by Mann, Vinogradov and Helfgott?","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 Schnirelmann density presented in additive number theory courses?","kind":"provenance"},{"text":"Which entry fits mathematics education?","kind":"action"}]}]}]}]},"agentConduct":{"may":["Compute and explain Schnirelmann density for given sets of natural numbers.","Cite the theorems that use it with their sources."],"mustNot":["Confuse Schnirelmann density with asymptotic or natural density.","Present conjectures as proved results.","Misattribute results in additive number theory."],"requiresHuman":["Announcing a claimed proof of an open problem using the density."]},"ethics":{"considerations":["Credit for results belongs to those who proved them.","False claims about open problems mislead students and waste reviewers' time."],"affectedParties":["Mathematicians","Students"]},"owners":{"steward":"Nobody owns the concept; it is held in common.","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":"natural density","type":"related","note":"which is a limit"},{"target":"Schnirelmann theorem","type":"related","note":"on sums of primes"},{"target":"Mann theorem","type":"related","note":"on sumset densities"}],"interaction":{"identity":{"applicability":"required","items":["Vercy registry: vr.tr.schnirelmann-density","Wikidata: Q2482753 (https://www.wikidata.org/wiki/Q2482753)"]},"properties":{"applicability":"not-applicable","items":["defined: 1930 year - Lev Schnirelmann","Schnirelmann constant: at most 7 value - Ramare 1995; 3 with Helfgott ternary result"]},"recognition":{"applicability":"optional","items":["Infimum density of a set of naturals","Schnirelmann density and Schnirelmann theorem on sums of primes","Natural density is a limit; upper and lower densities are limsup and liminf","Not a visible object; an infimum of ratios."]},"capabilities":{"applicability":"required","items":["explain the definition","relay properties and theorems","describe applications","distinguish from other densities"]},"hazards":{"applicability":"required","items":["Confusing with natural density","Forgetting that missing 1 gives density zero","Overstating the Schnirelmann constant results"]},"interfaces":{"applicability":"required","items":["No regulation; standard mathematical definitions"]},"context":{"applicability":"required","items":["Not applicable; a mathematical concept.","Schnirelmann density of sets","Mann theorem strengthening the sumset inequality","Schnirelmann theorem and constant","related densities: natural, lower, upper, Banach"]}},"sources":[{"title":"Wikidata item Q2482753: Schnirelmann density","url":"https://www.wikidata.org/wiki/Q2482753","note":"identity and sense of the item"},{"title":"Wikipedia: Schnirelmann density","url":"https://en.wikipedia.org/wiki/Schnirelmann_density","note":"general description of the item"}],"openQuestions":["Should Schnirelmann theorem be a separate entry?","How should mathematical references be linked?","How should the constant history be recorded?"],"resources":{"spec":"/models/things/publications/thing-q2482753/spec.json"},"provenance":{"origin":"thing registry research (pass 2)","builtFrom":["models/things/publications/thing-q2482753/spec.json"],"providers":["Claude"],"researchStatus":"unreviewed","generatedAt":"2026-09-13T03:18:30Z","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 2026-10-07T10:10:29+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}}