{ "model": { "rank": 5905, "code": "thing-q208225", "model_id": "vr.tr.pythagorean-triple", "name": "Pythagorean triple", "purpose": "Enable an AI agent to recognise an exact Pythagorean triple, assess its arithmetic properties, and determine valid transformations and right-triangle uses.", "family": "Thing Registry", "category": "Cross-cutting context", "status": "research-draft", "kind": "thing", "plane": "XCT", "domain": "XCT.QLT", "industry": "", "version": "", "url": "/models/thing/q208225/", "tier": 2, "score": 64, "payload": { "layer": "wikidata", "aliases": [ "primitive Pythagorean triple" ], "aliasCount": 1, "merged": 1, "knownIn": 64, "facets": null, "markers": [], "lexicalClass": "", "senseRank": null, "alsoRegisteredAs": null, "source": { "dataset": "wikidata", "item": "Q208225", "url": "https://www.wikidata.org/wiki/Q208225", "license": "CC0 1.0" } }, "research": { "vercy": "1.0-draft", "publication": { "status": "research-draft", "adjudicationStatus": "unreviewed", "publishableCanonical": false, "generatedAt": "2026-09-07T10:00:46Z", "providers": [ "Codex", "Grok" ], "missingProviders": [], "pass": 2, "cost": { "grok": { "usd": 0.00739874, "turns": 1, "seconds": 110.4, "error": null }, "codex": { "seconds": 72.4, "error": "Reading additional input from stdin...\nOpenAI Codex v0.153.3\n--------\nworkdir: R:\\02_PROJECTS\\02_Meta_Models_Platforms\\Ver.cy\\current\\thing-registry-backlog\nmodel: gpt-6-astra\nprovider: openai\napproval: never\nsandbox: read-only\nreasoning effort: none\nreasoning summaries: none\nsession id: 01a07b4c-24e3-7003-b415-6cb15d57805b\n--------\nuser\nYou are drafting a Vercy meta-model for one registered thing. Answer as JSON only, no prose around it.\n\nThing: Pythagorean triple\nRegistry id: vr.tr.pythagorean-triple\nPlane / domain: XCT / XCT.QLT\nRegistry definition: (none recorded)\nNames folded into this en" } } }, "metaModel": { "id": "THING-Q208225", "registryId": "vr.tr.pythagorean-triple", "name": "Pythagorean triple", "version": "0.1.0-research.1", "entryKind": "thing", "family": "Thing Registry", "domain": [ "XCT.QLT" ], "status": "research-draft" }, "canonicalUrl": "https://ver.cy/models/thing/q208225/", "model": { "registry_id": "vr.tr.pythagorean-triple", "name": "Pythagorean triple", "purpose": "Enable an AI agent to recognise an exact Pythagorean triple, assess its arithmetic properties, and determine valid transformations and right-triangle uses.", "definition": "A Pythagorean triple is a triple of positive integers (a, b, c) satisfying a² + b² = c², equivalently the side lengths of a right-angled triangle with integer sides.", "scope_statement": "This model owns three positive integers satisfying a² + b² = c², their leg and hypotenuse roles, equivalence under exchanging legs, primitive or scaled status, and evidence supporting recognition and use; nonpositive conventions remain an explicit scope question.", "in_scope": [ "Exact integer membership and verification of the Pythagorean equation", "Leg and hypotenuse roles, leg ordering, and canonical representation", "Greatest common divisor, primitive reduction, and integer scaling", "Euclidean generation parameters and their validity conditions", "Right-triangle interpretation and exact derived quantities" ], "out_of_scope": [ "General real-valued right triangles and their measurement uncertainty", "Rational points on the unit circle as independently modelled objects", "General Diophantine equations and their solution spaces", "Physical triangles, surveying observations, and construction tolerances", "Higher-dimensional integer configurations such as Euler bricks" ], "distinguishing_features": [ "All three entries must be positive integers under the adopted convention; a right triangle with an irrational side is not a Pythagorean triple.", "After identifying c as the largest entry, a² + b² must equal c² exactly; approximate equality is insufficient.", "The hypotenuse c is strictly greater than either leg; zero-length and degenerate cases are excluded under the adopted convention.", "A common divisor greater than one makes the triple nonprimitive, not invalid: both (3, 4, 5) and (6, 8, 10) qualify.", "Exchanging the two legs preserves the underlying triple; multiplying all entries by an integer greater than one produces a distinct triple in the same primitive family." ], "characteristics": [ { "name": "Canonical side values", "kind": "measurement", "unit_or_values": "Dimensionless positive integers a < b < c, with a and b the legs", "why_it_matters": "Provides a consistent representation for exact checking and comparison." }, { "name": "Pythagorean residual", "kind": "measurement", "unit_or_values": "Integer a² + b² − c²; zero is required", "why_it_matters": "Separates exact membership from near misses." }, { "name": "Verification state", "kind": "state", "unit_or_values": "Unverified candidate | verified triple | rejected candidate", "why_it_matters": "Prevents an unchecked proposal or failed computation from being treated as a recognised triple." }, { "name": "Common scale factor", "kind": "measurement", "unit_or_values": "Positive integer d = gcd(a, b, c)", "why_it_matters": "Determines whether the triple is primitive and supplies its reduction factor." }, { "name": "Primitiveness", "kind": "category", "unit_or_values": "Primitive when d = 1 | nonprimitive when d > 1", "why_it_matters": "Controls which parity and generation conditions can be applied directly." }, { "name": "Primitive representative", "kind": "relation", "unit_or_values": "Canonical triple (a/d, b/d, c/d)", "why_it_matters": "Links scaled triples without collapsing their distinct integer values." }, { "name": "Euclidean generation witness", "kind": "relation", "unit_or_values": "Integers k ≥ 1 and m > n ≥ 1, gcd(m, n) = 1, m and n of opposite parity; legs k(m² − n²), 2kmn and hypotenuse k(m² + n²)", "why_it_matters": "Supports constructive generation and independent verification, allowing leg exchange." }, { "name": "Right-triangle quantities", "kind": "measurement", "unit_or_values": "Perimeter a + b + c in abstract length units; area ab/2 in squared abstract length units", "why_it_matters": "Supports geometric constraints while keeping physical units separate from integer membership." } ], "affordances": [ "Verify or reject a candidate using exact integer arithmetic.", "Canonicalise leg order and detect equivalent representations.", "Reduce a valid triple to its primitive representative or generate positive integer multiples.", "Construct triples from admissible Euclidean parameters and check a proposed generation witness.", "Complete two proposed side values by testing whether the required third value is a positive integer.", "Compute exact perimeter and area and assess specified right-triangle constraints." ] }, "sources": [ { "id": "wikipedia-pythagorean-triple", "title": "Pythagorean triple", "url": "https://en.wikipedia.org/wiki/Pythagorean_triple", "what_it_supports": "Standard definition, primitive versus non-primitive distinction, Euclid's formula, and the 3-4-5 generating examples.", "url_status": "live" }, { "id": "mathworld-pythagorean-triple", "title": "Pythagorean Triple", "url": "https://mathworld.wolfram.com/PythagoreanTriple.html", "what_it_supports": "Generation formulae, primitive-triple conditions on (m, n), and enumerative properties used for kinds and measurements.", "url_status": "live" }, { "id": "euclid-elements", "title": "Euclid's Elements, Book X (lemma preceding Proposition 29) and the integer right-triangle construction", "url": "", "what_it_supports": "Classical generating construction that all primitive triples arise from two integers of opposite parity." }, { "id": "sierpinski-pythagorean-triangles", "title": "Pythagorean Triangles", "url": "", "what_it_supports": "Monograph treatment of primitive and non-primitive triples, almost-isosceles families, and shared-leg configurations." } ], "structure": { "bundles": [ { "id": "integer-membership", "name": "Integer membership", "description": "Establishes whether the proposed values constitute a Pythagorean triple under the adopted convention.", "rationale": "Integer membership and exact equality are the decisive recognition conditions.", "layers": [ { "id": "side-domain-and-roles", "name": "Side domain and roles", "description": "Identifies admissible entries and assigns the hypotenuse and leg roles.", "findings": [ { "id": "positive-integer-side-assignment", "name": "Positive integer side assignment", "description": "A candidate must contain three positive integers, with the greatest value assigned to the hypotenuse.", "questions": [ { "text": "Are all three supplied values exactly known positive integers?", "kind": "definition", "id": "positive-integer-side-assignment-q01" }, { "text": "Which value is the hypotenuse, and does the supplied role assignment agree with it being the greatest value?", "kind": "boundary", "id": "positive-integer-side-assignment-q02" } ] } ] }, { "id": "exact-equation-check", "name": "Exact equation check", "description": "Records the arithmetic evidence that admits or rejects the candidate.", "findings": [ { "id": "zero-pythagorean-residual", "name": "Zero Pythagorean residual", "description": "Recognition requires an exactly zero residual; rounding and arithmetic overflow cannot establish membership.", "questions": [ { "text": "What is the exact integer value of a² + b² − c²?", "kind": "measurement", "id": "zero-pythagorean-residual-q01" }, { "text": "What exact calculation or symbolic argument establishes the result without rounding or overflow?", "kind": "provenance", "id": "zero-pythagorean-residual-q02" }, { "text": "Does the available evidence justify verifying the candidate, rejecting it, or leaving it unverified?", "kind": "action", "id": "zero-pythagorean-residual-q03" } ] } ] } ] }, { "id": "primitive-family", "name": "Primitive family", "description": "Distinguishes representation equivalence from the relationship between primitive triples and their multiples.", "rationale": "An agent must avoid both duplicate leg permutations and the mistaken collapse of different scaled triples.", "layers": [ { "id": "leg-exchange-equivalence", "name": "Leg exchange equivalence", "description": "Normalises interchangeable leg positions while retaining any application-specific labels.", "findings": [ { "id": "canonical-leg-order", "name": "Canonical leg order", "description": "Sorting the legs gives a stable identity for the underlying triple; an application may separately distinguish horizontal and vertical legs.", "questions": [ { "text": "What is the canonical representation with the smaller leg first and the hypotenuse last?", "kind": "definition", "id": "canonical-leg-order-q01" }, { "text": "Do two records differ only by leg exchange, or do their integer values differ?", "kind": "boundary", "id": "canonical-leg-order-q02" } ] } ] }, { "id": "common-factor-reduction", "name": "Common factor reduction", "description": "Determines primitiveness and identifies the primitive family.", "findings": [ { "id": "primitive-core-and-scale", "name": "Primitive core and scale", "description": "Dividing by the greatest common divisor yields a primitive triple; the divisor records the original triple's integer scale.", "questions": [ { "text": "What are d = gcd(a, b, c) and the canonical reduced triple (a/d, b/d, c/d)?", "kind": "measurement", "id": "primitive-core-and-scale-q01" }, { "text": "Does a comparison target have the same primitive representative, and what scale factor relates their side values?", "kind": "boundary", "id": "primitive-core-and-scale-q02" }, { "text": "Would the proposed rescaling preserve positive integer entries as well as the Pythagorean equation?", "kind": "action", "id": "primitive-core-and-scale-q03" } ] } ] } ] }, { "id": "arithmetic-construction", "name": "Arithmetic construction", "description": "Supports parameter-based generation and arithmetic consistency checks.", "rationale": "A triple can be constructively certified and extended into a family without relying on a list of examples.", "layers": [ { "id": "euclidean-parameters", "name": "Euclidean parameters", "description": "Relates the triple to the standard positive integer parameterisation.", "findings": [ { "id": "admissible-generation-witness", "name": "Admissible generation witness", "description": "With coprime opposite-parity m > n ≥ 1, Euclid's formulas produce a primitive triple; multiplication by k ≥ 1 gives its positive integer multiples.", "questions": [ { "text": "Which k, m and n reproduce the two legs and hypotenuse exactly, allowing the legs to exchange positions?", "kind": "measurement", "id": "admissible-generation-witness-q01" }, { "text": "Do m and n satisfy positivity, ordering, coprimality and opposite-parity conditions?", "kind": "boundary", "id": "admissible-generation-witness-q02" }, { "text": "Was this witness supplied by a source, recovered from the triple, or used to generate it?", "kind": "provenance", "id": "admissible-generation-witness-q03" } ] } ] }, { "id": "primitive-parity-checks", "name": "Primitive parity checks", "description": "Applies necessary arithmetic properties to the primitive representative.", "findings": [ { "id": "primitive-side-consistency", "name": "Primitive side consistency", "description": "In a primitive triple, exactly one leg is even, the hypotenuse is odd, and the sides are pairwise coprime; these checks supplement exact equation verification.", "questions": [ { "text": "Does the primitive representative have exactly one even leg and an odd hypotenuse?", "kind": "measurement", "id": "primitive-side-consistency-q01" }, { "text": "Are the primitive representative's three pairwise greatest common divisors all one?", "kind": "measurement", "id": "primitive-side-consistency-q02" }, { "text": "If a necessary check fails, is the primitive classification wrong or is the candidate itself invalid?", "kind": "boundary", "id": "primitive-side-consistency-q03" } ] } ] } ] }, { "id": "right-triangle-use", "name": "Right-triangle use", "description": "Connects verified integer triples to exact geometric quantities and constrained completion tasks.", "rationale": "Recognition should support concrete decisions about side completion, size constraints and geometric interpretation.", "layers": [ { "id": "exact-geometric-quantities", "name": "Exact geometric quantities", "description": "Derives right-triangle quantities from the verified side roles.", "findings": [ { "id": "area-perimeter-and-scale", "name": "Area, perimeter and scale", "description": "The associated right triangle has perimeter a + b + c and area ab/2; applying a physical length scale requires a separate application interpretation.", "questions": [ { "text": "What are the exact perimeter and area of the associated right triangle?", "kind": "measurement", "id": "area-perimeter-and-scale-q01" }, { "text": "Are these abstract side values or integer counts of an explicitly stated common length unit?", "kind": "boundary", "id": "area-perimeter-and-scale-q02" }, { "text": "Does the triple meet the application's specified perimeter, area or side-length bounds?", "kind": "action", "id": "area-perimeter-and-scale-q03" } ] } ] }, { "id": "integer-side-completion", "name": "Integer side completion", "description": "Determines whether two proposed side values can be completed to an admissible triple.", "findings": [ { "id": "perfect-square-completion", "name": "Perfect-square completion", "description": "Two legs require a positive integer square root of their squared sum; a hypotenuse and leg require a positive integer square root of their squared difference.", "questions": [ { "text": "Are the known values both legs, or is one explicitly the hypotenuse?", "kind": "definition", "id": "perfect-square-completion-q01" }, { "text": "Is the required squared sum or positive squared difference a perfect square?", "kind": "measurement", "id": "perfect-square-completion-q02" }, { "text": "Which completions remain after enforcing side roles, positivity and any primitiveness requirement?", "kind": "action", "id": "perfect-square-completion-q03" } ] } ] } ] }, { "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.", "rationale": "A model that cannot be lined up against existing standards, identifiers and practice cannot be adopted by anyone who already uses them.", "layers": [ { "id": "reported-evidence", "name": "Reported evidence", "description": "Findings from the breadth pass, kept separate from the structural claims.", "findings": [ { "id": "evidence-kinds", "name": "Kinds and varieties", "description": "Reported by the breadth pass; each item needs checking against its source before it becomes normative.", "evidence": [ "primitive Pythagorean triple (gcd(a, b, c) = 1; equivalently pairwise coprime, with legs of opposite parity)", "non-primitive (imprimitive) triple: a positive-integer multiple k(a, b, c) of a primitive triple, k ≥ 2", "Euclid-generated triple a = k(m² − n²), b = k(2mn), c = k(m² + n²) for integers m > n ≥ 1 and k ≥ 1 (this parametrizes all triples, up to swapping the legs)", "primitive Euclid-generated triple: the same with k = 1, gcd(m, n) = 1, and m, n of opposite parity", "ordered triple versus unordered pair of legs: (3, 4, 5) distinguished from (4, 3, 5) when orientation or which leg is even is recorded", "almost-isosceles triple: legs consecutive integers, |a − b| = 1 (e.g. 20, 21, 29)", "common-leg or twin families: distinct triples that share a leg, used in catalogs and tiling arguments" ], "questions": [ { "id": "evidence-kinds-q01", "text": "Which of these kinds and varieties hold for the sense of Pythagorean triple this model covers, and on what evidence?", "kind": "provenance" } ] }, { "id": "evidence-identifiers", "name": "Identifiers and schemes", "description": "Reported by the breadth pass; each item needs checking against its source before it becomes normative.", "evidence": [ { "scheme": "Wikidata item", "value_or_pattern": "Q214371", "note": "Item for the mathematical notion Pythagorean triple (not the Pythagorean theorem)." }, { "scheme": "Wikipedia title", "value_or_pattern": "Pythagorean triple", "note": "Canonical English encyclopaedia lemma." } ], "questions": [ { "id": "evidence-identifiers-q01", "text": "Which of these identifiers and schemes hold for the sense of Pythagorean triple this model covers, and on what evidence?", "kind": "provenance" } ] }, { "id": "evidence-real-world-use", "name": "Real-world use", "description": "Reported by the breadth pass; each item needs checking against its source before it becomes normative.", "evidence": [ "Laying out a right angle on site with a 3-4-5 (or scaled) rope, tape, or story pole in carpentry, masonry, and surveying.", "School and contest number theory: listing primitives, applying Euclid's formula, and proving that one leg is divisible by 3, one by 4, and the hypotenuse or a leg by 5.", "Generating all rational points on the unit circle via (a/c, b/c), hence all rational-slope right triangles.", "Integer lattice geometry: axis-aligned or grid vectors of integer length whose sum of squares is a square.", "Historical tables of integer right triangles (e.g. Plimpton 322; lists in the Indian Śulbasūtras) used as computational examples rather than as a modern product class." ], "questions": [ { "id": "evidence-real-world-use-q01", "text": "Which of these real-world use hold for the sense of Pythagorean triple this model covers, and on what evidence?", "kind": "provenance" } ] }, { "id": "evidence-measurements", "name": "Typical measurements", "description": "Reported by the breadth pass; each item needs checking against its source before it becomes normative.", "evidence": [ { "quantity": "legs a, b", "typical_range": "positive integers; smallest 3 and 4; one even, one odd in every primitive triple", "unit": "1 (dimensionless integer length)" }, { "quantity": "hypotenuse c", "typical_range": "positive integer ≥ 5; primitives have c ≡ 1 (mod 4)", "unit": "1 (dimensionless integer length)" }, { "quantity": "common factor k", "typical_range": "1 for a primitive triple; unbounded for general triples", "unit": "1" }, { "quantity": "Euclid parameters m > n ≥ 1", "typical_range": "small positive integers in tables; coprime and opposite parity when generating primitives", "unit": "1" }, { "quantity": "area ab/2", "typical_range": "positive integer ≥ 6 (attained at 3-4-5), always integral because one leg is even", "unit": "1 (square of the length unit)" }, { "quantity": "inradius (a + b − c)/2", "typical_range": "positive integer ≥ 1", "unit": "1 (same length unit as the sides)" } ], "questions": [ { "id": "evidence-measurements-q01", "text": "Which of these typical measurements hold for the sense of Pythagorean triple this model covers, and on what evidence?", "kind": "provenance" } ] }, { "id": "evidence-failure-modes-and-hazards", "name": "Failure modes and hazards", "description": "Reported by the breadth pass; each item needs checking against its source before it becomes normative.", "evidence": [ "A 3-4-5 layout stretched or marked inaccurately yields an angle that is not 90°, so the triple is used as if it guaranteed squareness when the physical lengths are wrong.", "Floating-point tests of a² + b² = c² on measured or computed lengths both miss true triples and accept near-misses.", "Integer overflow when forming a², b², or c² in fixed-width arithmetic, especially 32-bit, silently accepts or rejects triples.", "Admitting 0 or negative entries (valid as integer solutions of a² + b² = c²) breaks the geometric reading as a triangle.", "Counting (3, 4, 5) and (4, 3, 5) twice, or omitting non-primitive multiples, depending on whether the catalog is ordered and primitive-only." ], "questions": [ { "id": "evidence-failure-modes-and-hazards-q01", "text": "Which of these failure modes and hazards hold for the sense of Pythagorean triple this model covers, and on what evidence?", "kind": "provenance" } ] }, { "id": "evidence-regional-variation", "name": "Regional variation", "description": "Reported by the breadth pass; each item needs checking against its source before it becomes normative.", "evidence": [ "English school usage sometimes says \"Pythagorean triplet\" rather than \"triple.\"", "Chinese sources treat the same objects as 勾股数 (gougu numbers) under the 勾股 theorem, not under Pythagoras' name.", "Some number-theory conventions fix a odd, b even, and a < c, so the ordered triple is unique for each unordered triangle; others list both leg orders.", "Babylonian sexagesimal tables and Vedic Śulbasūtra lists record specific triples without the modern primitive/non-primitive vocabulary." ], "questions": [ { "id": "evidence-regional-variation-q01", "text": "Which of these regional variation hold for the sense of Pythagorean triple this model covers, and on what evidence?", "kind": "provenance" } ] }, { "id": "evidence-neighbours", "name": "Neighbouring kinds and how to tell them apart", "description": "Reported by the breadth pass; each item needs checking against its source before it becomes normative.", "evidence": [ { "name": "Pythagorean theorem", "difference": "The theorem is the real identity relating the sides of any right triangle; a Pythagorean triple is an integer solution of that equation." }, { "name": "rational-sided right triangle", "difference": "Sides may be rational rather than integral; clearing a common denominator yields a Pythagorean triple, and conversely every triple scales to a rational point on the unit circle." }, { "name": "Heronian triangle", "difference": "Integer sides and integer area, but the triangle need not be right-angled; a Pythagorean triple always gives a Heronian triangle, the converse fails (e.g. 5-5-6 or 5-5-4, 13-14-15)." }, { "name": "Pythagorean quadruple (and n-tuples)", "difference": "Sum of three or more integer squares equals a square (a² + b² + c² = d², etc.), not two." }, { "name": "Eisenstein triple", "difference": "Positive integers satisfying a² − ab + b² = c², the side lengths of a 60° integer triangle, not a 90° one." }, { "name": "integer-sided (scalene) triangle that is not right", "difference": "The sides fail a² + b² = c² (and the two cyclic permutations); the numerical test is exact on integers." } ], "questions": [ { "id": "evidence-neighbours-q01", "text": "Which of these neighbouring kinds and how to tell them apart hold for the sense of Pythagorean triple this model covers, and on what evidence?", "kind": "provenance" } ] } ] } ] } ] }, "openQuestions": [ "Does the registry intend only positive integer triples, or also signed and zero-containing solutions requiring an explicit extension of scope?", "Does registry identity treat leg permutations as one thing, or must ordered triples be represented separately for particular tasks?", "Should rationally proportional triples be linked through primitive representatives here, or through a neighbouring model of triangle similarity?", "What evidence must consuming agents retain for verification: exact residual calculations, generation witnesses, formal proofs, or a specified combination?" ], "statistics": { "bundles": 5, "layers": 9, "findings": 15, "questions": 29 } }, "draft": { "generator": "vr.draft.v3", "status": "draft-generated", "researched": false, "archetype": "abstract concept", "method": "Written from the archetype playbook - what this kind of thing needs beyond identity and provenance - and from the structure that recurred across 6,333 models already researched by two engines. Applied to this entry by rule. No source was read for this thing and no claim here is researched. This entry carries no facets of its own, so they were inferred from its domain - a guess about a whole domain applied to one thing.", "facetsInferred": true, "nextPass": "A researcher replaces this draft with a sourced specification. Treat every sentence below as a proposal to argue with.", "purpose": "Give an agent a durable, checkable way to recognise a Pythagorean triple, record what state it is in, and decide what may be done with it.", "whatItIs": "Enable an AI agent to recognise an exact Pythagorean triple, assess its arithmetic properties, and determine valid transformations and right-triangle uses.", "characteristics": { "substance": "abstract", "origin": "conceptual", "agency": "inert" }, "whatYouCanDoWithIt": [ "observed and measured" ], "distinguishingFeatures": [ "Names folded into this entry, which a task may need to split apart again: primitive Pythagorean triple.", "1 finer distinctions are held as aliases rather than separate entries, because telling them apart needs a task that asks for it.", "Described in 64 Wikipedia languages, which is a measure of how widely the thing is known, not of how important it is." ], "openQuestionsForResearch": [ "Which of the bundles below does a real task actually need, and which are ceremony?", "What does this thing have that the facets do not capture at all?", "Which neighbouring kind is most often confused with a Pythagorean triple, and on what evidence are they told apart?" ], "whatItIsMadeOf": "an abstraction with no physical instance", "physicalCharacter": [ "Does nothing on its own; everything it does, something else did to it.", "These come from the domain this entry sits in rather than from the entry itself, so treat them as a first guess about the whole domain applied to one thing." ], "whatCanBeDoneWithIt": [ "observe it, measure it, record its state" ], "howItIsRecognised": [ "Nothing to see. What is recognised is an instance of it, and which instances count is exactly what is argued about." ], "relatedModels": [ { "relation": "covers", "note": "Finer kinds folded into this entry because telling them apart needs a task that asks for it. Each is a model waiting to be split out when one does.", "targets": [ "primitive Pythagorean triple" ] } ], "standing": "Described in 64 Wikipedia languages, which measures how widely it is written about rather than how important or how common it is. 1 finer distinctions are held inside this entry as names rather than as separate models.", "structure": { "bundles": [ { "id": "identity-and-classification", "name": "Identity, naming and classification", "description": "How an agent tells one Pythagorean triple from another, and a Pythagorean triple from things that resemble it.", "rationale": "Recognition comes before every other claim. Without stable identity nothing else in the model can be trusted to be about the same thing twice.", "layers": [ { "id": "naming-and-identifiers", "name": "Names and identifiers", "description": "The names this thing goes by and the identifiers that survive translation and time.", "findings": [ { "id": "preferred-name-and-aliases", "name": "Preferred name, aliases and local names", "description": "Which name to use, which names mean the same thing, and which merely sound similar.", "questions": [ { "id": "preferred-name-and-aliases-q01", "text": "What identifies and describes the name of a Pythagorean triple, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "preferred-name-and-aliases-q02", "text": "Who or what asserted this about the name of a Pythagorean triple, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "preferred-name-and-aliases-q03", "text": "What may an agent decide or do once the name of a Pythagorean triple is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] }, { "id": "stable-identifiers", "name": "Stable identifiers and external keys", "description": "Identifiers that keep pointing at this kind of thing across systems and languages.", "questions": [ { "id": "stable-identifiers-q01", "text": "What identifies and describes an identifier for a Pythagorean triple, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "stable-identifiers-q02", "text": "Who or what asserted this about an identifier for a Pythagorean triple, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "stable-identifiers-q03", "text": "What may an agent decide or do once an identifier for a Pythagorean triple is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] }, { "id": "classification-and-granularity", "name": "Classification and granularity", "description": "Where a Pythagorean triple sits among kinds, and how finely a task needs to cut it.", "findings": [ { "id": "kind-and-parents", "name": "Kind, parents and neighbouring kinds", "description": "The classes this thing belongs to and the ones it is next to.", "questions": [ { "id": "kind-and-parents-q01", "text": "What identifies and describes the kind of a Pythagorean triple, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "kind-and-parents-q02", "text": "Who or what asserted this about the kind of a Pythagorean triple, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "kind-and-parents-q03", "text": "What may an agent decide or do once the kind of a Pythagorean triple is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] }, { "id": "distinguishing-features", "name": "Distinguishing features", "description": "What separates a Pythagorean triple from the things most often confused with it.", "questions": [ { "id": "distinguishing-features-q01", "text": "What identifies and describes what distinguishes a Pythagorean triple, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "distinguishing-features-q02", "text": "Who or what asserted this about what distinguishes a Pythagorean triple, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "distinguishing-features-q03", "text": "What may an agent decide or do once what distinguishes a Pythagorean triple is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] } ] }, { "id": "state-and-lifecycle", "name": "State and lifecycle", "description": "The states a Pythagorean triple passes through and the events that move it between them.", "rationale": "Most decisions about a thing depend on what state it is in now, which is a claim with a time on it, not a property.", "layers": [ { "id": "lifecycle-stages", "name": "Lifecycle stages", "description": "From coming into existence to ceasing to be one of these.", "findings": [ { "id": "stages-and-transitions", "name": "Stages and transitions", "description": "The stages worth naming and what moves a Pythagorean triple between them.", "questions": [ { "id": "stages-and-transitions-q01", "text": "What identifies and describes the lifecycle of a Pythagorean triple, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "stages-and-transitions-q02", "text": "Who or what asserted this about the lifecycle of a Pythagorean triple, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "stages-and-transitions-q03", "text": "What may an agent decide or do once the lifecycle of a Pythagorean triple is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] }, { "id": "observations-and-status", "name": "Observations and current status", "description": "What is observed about a Pythagorean triple, how often and by whom.", "findings": [ { "id": "observation-record", "name": "Observation record", "description": "How an observation of a Pythagorean triple is recorded so that it can be superseded rather than overwritten.", "questions": [ { "id": "observation-record-q01", "text": "What identifies and describes an observation of a Pythagorean triple, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "observation-record-q02", "text": "Who or what asserted this about an observation of a Pythagorean triple, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "observation-record-q03", "text": "What may an agent decide or do once an observation of a Pythagorean triple is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] } ] }, { "id": "definitions-in-contest", "name": "Definitions and who holds them", "description": "What Pythagorean triple is taken to mean, and by whom.", "rationale": "When a field disagrees about a concept, the disagreement is the content. A model that picks one definition silently destroys the information.", "layers": [ { "id": "competing-definitions", "name": "Competing definitions", "description": "The main readings and the traditions behind them.", "findings": [ { "id": "definition-map", "name": "Definitions and their holders", "description": "Each definition with the school or body that holds it.", "questions": [ { "id": "definition-map-q01", "text": "Which definitions of Pythagorean triple are in use, and which tradition or body holds each?", "kind": "definition" }, { "id": "definition-map-q02", "text": "What turns on the difference between them in practice?", "kind": "boundary" } ] } ] }, { "id": "operationalisation", "name": "Operationalisation", "description": "How it is measured or applied when it has to be.", "findings": [ { "id": "operational-record", "name": "Measures and proxies", "description": "Instruments and indicators used to stand in for it.", "questions": [ { "id": "operational-record-q01", "text": "How is Pythagorean triple operationalised or measured in practice, and by what instrument?", "kind": "measurement" }, { "id": "operational-record-q02", "text": "What does that operationalisation leave out, and when does that matter?", "kind": "boundary" } ] } ] } ] }, { "id": "instances-and-use", "name": "Instances, use and consequence", "description": "What counts as an instance of Pythagorean triple and what follows from calling something that.", "rationale": "Applying a concept is an act with consequences, so a model must say what the label licenses and what it does not.", "layers": [ { "id": "instances", "name": "What counts as an instance", "description": "Clear cases, borderline cases and non-cases.", "findings": [ { "id": "instance-tests", "name": "Tests for an instance", "description": "What would settle whether something falls under it.", "questions": [ { "id": "instance-tests-q01", "text": "What would settle whether something is an instance of Pythagorean triple?", "kind": "boundary" }, { "id": "instance-tests-q02", "text": "Which borderline cases are argued about, and on what grounds?", "kind": "definition" } ] } ] }, { "id": "consequence", "name": "Consequence of application", "description": "Rights, duties or decisions that follow from the label.", "findings": [ { "id": "consequence-record", "name": "What the label licenses", "description": "What an agent may do once something is classified this way.", "questions": [ { "id": "consequence-record-q01", "text": "What follows practically once something is treated as Pythagorean triple?", "kind": "action" }, { "id": "consequence-record-q02", "text": "What must an agent not infer from the label alone?", "kind": "action" } ] } ] } ] }, { "id": "provenance-and-evidence", "name": "Provenance, evidence and time", "description": "Where every claim about a Pythagorean triple came from and when it held.", "rationale": "A claim without a source and a time cannot be superseded, only overwritten, and an agent that overwrites loses the ability to explain itself.", "layers": [ { "id": "source-and-authority", "name": "Source and authority", "description": "Who said it, on what evidence, and how strongly.", "findings": [ { "id": "claim-provenance", "name": "Claim provenance and confidence", "description": "The authority behind each claim about a Pythagorean triple and how confident it is.", "questions": [ { "id": "claim-provenance-q01", "text": "What identifies and describes a claim about a Pythagorean triple, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "claim-provenance-q02", "text": "Who or what asserted this about a claim about a Pythagorean triple, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "claim-provenance-q03", "text": "What may an agent decide or do once a claim about a Pythagorean triple is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] }, { "id": "time-and-versions", "name": "Time, versions and supersession", "description": "When a claim was true, when it was learnt, and what replaced it.", "findings": [ { "id": "validity-and-supersession", "name": "Validity period and supersession", "description": "How an old claim about a Pythagorean triple is retired without being erased.", "questions": [ { "id": "validity-and-supersession-q01", "text": "What identifies and describes the validity of a claim about a Pythagorean triple, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "validity-and-supersession-q02", "text": "Who or what asserted this about the validity of a claim about a Pythagorean triple, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "validity-and-supersession-q03", "text": "What may an agent decide or do once the validity of a claim about a Pythagorean triple is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] } ] } ] }, "statistics": { "bundles": 5, "layers": 10, "findings": 12, "questions": 32 } } } }