{ "vercy": "1.0-draft", "publication": { "status": "research-draft", "adjudicationStatus": "unreviewed", "publishableCanonical": false, "generatedAt": "2026-09-12T09:34:54Z", "providers": [ "Codex" ], "breadth": "recalled by Codex without web access - no source was read", "missingProviders": [], "pass": 2, "cost": { "grok": { "seconds": 15.2, "error": "Reading additional input from stdin...\nOpenAI Codex v0.153.4\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: 01a094f7-1ea9-7520-a3ca-70f5513fe55e\n--------\nuser\nDescribe what is already known about one registered thing. Answer as JSON only, no prose around it.\n\nThing: ex-tangential quadrilateral\nSense to describe: (none recorded)\nDomain code: XCT.QLT\nAlso known as: (none)\n\n\nContext for this batch of 1001 things:\n# Bat", "usd": 0.0, "recall": true }, "codex": { "seconds": 76.5, "error": "Reading additional input from stdin...\nOpenAI Codex v0.153.4\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: 01a094f7-1eb2-7b10-a050-cf2751e07b1f\n--------\nuser\nYou are drafting a Vercy meta-model for one registered thing. Answer as JSON only, no prose around it.\n\nThing: ex-tangential quadrilateral\nRegistry id: vr.tr.ex-tangential-quadrilateral\nPlane / domain: XCT / XCT.QLT\nRegistry definition: (none recorded)\nNames f" } }, "enrichment": { "pass": 3, "engine": "claude", "producedAt": "2026-10-06T12:00:00Z", "ingestedAt": "2026-10-06T08:41:53+00:00", "fields": [ "agent_conduct", "ethics", "owners", "visual_identification" ] }, "gate": { "passed": true, "version": "1.0", "at": "2026-10-07T10:10:31+00:00", "evidence": [ "completeness 1.00", "sense: Wikidata description", "2 live sources", "review by codex: pass" ] } }, "metaModel": { "id": "THING-Q5419140", "registryId": "vr.tr.ex-tangential-quadrilateral", "name": "ex-tangential quadrilateral", "version": "0.1.0-research.1", "entryKind": "thing", "family": "Thing Registry", "domain": [ "XCT.QLT" ], "status": "research-draft" }, "canonicalUrl": "https://ver.cy/models/thing/q5419140/", "model": { "registry_id": "vr.tr.ex-tangential-quadrilateral", "name": "ex-tangential quadrilateral", "purpose": "Enable an agent to recognise an ex-tangential quadrilateral, verify its external tangent circle, and determine which constructions and deductions its geometry supports.", "definition": "An ex-tangential quadrilateral is a convex quadrilateral whose four sides, extended beyond their endpoints, are tangent to a common circle outside the quadrilateral.", "scope_statement": "This model owns the external four-line tangency condition for vr.tr.ex-tangential-quadrilateral in XCT / XCT.QLT, adopting a nondegenerate convex Euclidean quadrilateral with a finite positive-radius circle tangent to the extensions of all four sides as its working scope.", "in_scope": [ "The defining external-circle condition and explicit treatment of alternative conventions.", "Ordered vertices, side-supporting lines and the locations of all four tangency points.", "Excenter and exradius witnesses, with exact or approximate verification.", "Recognition failures caused by unsuitable contact locations, degeneracy or incomplete evidence.", "Constructions and transformations that preserve the defining tangency condition." ], "out_of_scope": [ "General quadrilateral properties that do not affect external tangency.", "Tangential quadrilaterals classified solely by an incircle touching their side segments.", "Cyclic quadrilaterals classified by a circle through their vertices.", "Triangle excircles or circles touching only three quadrilateral side-supporting lines.", "General conics tangent to four lines and non-Euclidean analogues." ], "distinguishing_features": [ "The working definition requires a simple, strictly convex quadrilateral with four distinct vertices and nonzero area.", "One finite circle of positive radius must be tangent to all four side-supporting lines; four unrelated tangent circles do not qualify.", "Each perpendicular contact foot must lie beyond an endpoint of its corresponding side segment, and the circle must lie outside the quadrilateral.", "An incircle certificate or a circumcircle certificate alone does not establish ex-tangentiality.", "Equal distances from a proposed center to the four lines require additional checks of radius, contact locations and external placement." ], "characteristics": [ { "name": "Definition convention", "kind": "category", "unit_or_values": "strict convex finite-excircle convention; explicitly attributed broader convention", "why_it_matters": "Determines whether concave configurations or limiting cases are eligible." }, { "name": "Ordered vertex geometry", "kind": "relation", "unit_or_values": "A, B, C, D in boundary order, with coordinates or an equivalent exact construction", "why_it_matters": "Fixes adjacency, side segments and the lines whose tangency must be checked." }, { "name": "Quadrilateral admissibility", "kind": "state", "unit_or_values": "strictly convex; concave; self-intersecting; degenerate; unresolved", "why_it_matters": "Prevents application of convex-quadrilateral criteria outside their assumptions." }, { "name": "Excenter", "kind": "relation", "unit_or_values": "candidate point O associated with a particular external tangent circle", "why_it_matters": "Provides the common center used in the four tangency checks." }, { "name": "Exradius", "kind": "measurement", "unit_or_values": "finite r > 0 in the coordinate length unit", "why_it_matters": "Defines the circle and excludes zero-radius or infinite-radius substitutes." }, { "name": "Contact parameters", "kind": "measurement", "unit_or_values": "dimensionless t_i where T_i = V_i + t_i(V_(i+1) - V_i); require t_i < 0 or t_i > 1", "why_it_matters": "Distinguishes a contact on a side extension from one on the side segment." }, { "name": "Tangency residual", "kind": "measurement", "unit_or_values": "max_i |distance(O, line_i) - r| in length units, with a declared tolerance", "why_it_matters": "Measures numerical agreement without presenting approximation as an exact proof." }, { "name": "Recognition status", "kind": "state", "unit_or_values": "exactly certified; numerically supported; disproved; unresolved", "why_it_matters": "Controls whether an agent may assert membership or must request additional evidence." } ], "affordances": [ "Classify a supplied quadrilateral under an explicitly selected definition.", "Construct candidate excenters from equal-distance or angle-bisector constraints and verify the resulting circles.", "Produce a certificate containing the center, radius, four perpendicular contact feet and exterior checks.", "Diagnose a failed candidate by identifying the offending line, contact location or convexity condition.", "Generate further instances through Euclidean similarities and verify any additional imposed constraints." ], "agent_conduct": { "may": [ "State the definition and known properties of ex-tangential quadrilaterals with sources.", "Draw and check examples for learners." ], "must_not": [ "Present a conjecture about these shapes as a proven theorem.", "Confuse it with tangential quadrilaterals, whose incircle lies inside.", "Credit a result to a mathematician without a source." ], "requires_human": [ "Publishing new claimed results under a person's name." ] }, "ethics": { "considerations": [ "Correct attribution matters to mathematicians and students.", "Errors in teaching material mislead learners." ], "affected_parties": [ "Students", "Teachers", "Mathematicians" ] }, "owners": { "steward": "Nobody: a mathematical concept held in common.", "master_systems": [] }, "visual_identification": [ "A drawing of a convex four-sided figure with a circle outside it touching the extensions of all four sides." ], "what_it_is_for": [ "Ex-tangential quadrilaterals occur in Euclidean geometry problems involving circles tangent to extended polygon sides.", "They are used to investigate tangent lengths, angle relationships and geometric construction constraints." ], "failure_modes_and_hazards": [ "Confusing an external tangent circle with an inscribed circle can invalidate geometric deductions.", "Accepting apparent tangency in a sketch without mathematical verification can propagate errors into constructions or software." ], "facets_not_applicable": { "interfaces": "No formal standard, law or official classification specifically applies to this mathematical object." } }, "sources": [ { "id": "wikidata", "title": "Wikidata item Q5419140: ex-tangential quadrilateral", "url": "https://www.wikidata.org/wiki/Q5419140", "url_status": "live", "what_it_supports": "identity and sense of the item", "checked_by": "wikidata-api" }, { "id": "wikipedia-en", "title": "Wikipedia: Ex-tangential quadrilateral", "url": "https://en.wikipedia.org/wiki/Ex-tangential_quadrilateral", "url_status": "live", "what_it_supports": "general description of the item", "checked_by": "wikidata-api sitelink" } ], "structure": { "bundles": [ { "id": "external-tangency-definition", "name": "External tangency definition", "description": "Establish the precise circle-contact convention that identifies this registered thing.", "rationale": "External tangency terminology must specify both the quadrilateral class and the permitted contact locations.", "layers": [ { "id": "working-definition", "name": "Working definition", "description": "Record the adopted convex, finite-circle interpretation.", "findings": [ { "id": "four-extensions-one-circle", "name": "Four extensions, one circle", "description": "Adopt the usual convex definition requiring an outside circle tangent to all four side extensions, as described in [Different cases of a tangential quadrilateral theorem](https://www.tandfonline.com/doi/full/10.1080/0020739X.2025.2572374).", "questions": [ { "text": "Does the instance provide one external circle tangent to all four side extensions?", "kind": "definition", "id": "four-extensions-one-circle-q01" }, { "text": "Which source and definition passage justify the convention used for classification?", "kind": "provenance", "id": "four-extensions-one-circle-q02" } ] } ] }, { "id": "alternative-conventions", "name": "Alternative conventions", "description": "Keep broader uses distinguishable from the working definition.", "findings": [ { "id": "convexity-is-a-declared-boundary", "name": "Convexity is a declared boundary", "description": "The terminology is also used for concave objects in [Aebi and Cairns, Lattice Equable Quadrilaterals III](https://arxiv.org/abs/2111.06453). Such usage requires an attributed alternative convention rather than automatic inclusion.", "questions": [ { "text": "Does the supplying source permit concave quadrilaterals, and what contact condition does it use?", "kind": "boundary", "id": "convexity-is-a-declared-boundary-q01" }, { "text": "Should this candidate be assessed under the working definition or retained as an attributed alternative usage?", "kind": "action", "id": "convexity-is-a-declared-boundary-q02" } ] } ] } ] }, { "id": "quadrilateral-and-contact-geometry", "name": "Quadrilateral and contact geometry", "description": "Capture enough geometric information to distinguish sides, their extensions and the external circle.", "rationale": "A drawing of four lines and a circle does not by itself identify the intended quadrilateral or certify valid contacts.", "layers": [ { "id": "ordered-boundary", "name": "Ordered boundary", "description": "Establish the polygon before testing its supporting lines.", "findings": [ { "id": "admissible-vertex-cycle", "name": "Admissible vertex cycle", "description": "Record A, B, C, D in boundary order and check distinctness, nonzero side lengths, simplicity and strict convexity.", "questions": [ { "text": "What coordinates or exact construction establish the boundary order A, B, C, D?", "kind": "measurement", "id": "admissible-vertex-cycle-q01" }, { "text": "Do any repeated vertices, collinear consecutive triples or crossing sides invalidate the working scope?", "kind": "boundary", "id": "admissible-vertex-cycle-q02" } ] } ] }, { "id": "extension-contact-map", "name": "Extension contact map", "description": "Locate the perpendicular foot on each supporting line relative to the finite side.", "findings": [ { "id": "four-exterior-contact-feet", "name": "Four exterior contact feet", "description": "For each side V_i V_(i+1), record its contact foot T_i and parameter t_i; an extension contact requires t_i outside [0,1].", "questions": [ { "text": "What is each contact parameter, and beyond which endpoint does its contact lie?", "kind": "measurement", "id": "four-exterior-contact-feet-q01" }, { "text": "Does any contact lie on a side segment or at a vertex, requiring rejection under the working convention?", "kind": "boundary", "id": "four-exterior-contact-feet-q02" } ] } ] } ] }, { "id": "excircle-recognition-certificate", "name": "Excircle recognition certificate", "description": "Turn a proposed external tangent circle into a checkable membership decision.", "rationale": "Recognition needs a geometric witness and a stated level of certainty, especially when coordinates are approximate.", "layers": [ { "id": "center-radius-witness", "name": "Center and radius witness", "description": "Verify a common positive distance to all four supporting lines.", "findings": [ { "id": "common-distance-and-exterior-placement", "name": "Common distance and exterior placement", "description": "Check distance(O, line_i) = r for every side line with finite r > 0, then check that the circle is external to the polygon and that all contact parameters are admissible.", "questions": [ { "text": "What center and positive radius satisfy all four line-distance equations?", "kind": "measurement", "id": "common-distance-and-exterior-placement-q01" }, { "text": "What geometric test certifies that the circle lies outside the quadrilateral?", "kind": "boundary", "id": "common-distance-and-exterior-placement-q02" } ] } ] }, { "id": "proof-and-numerical-status", "name": "Proof and numerical status", "description": "Distinguish exact certificates from tolerance-dependent evidence.", "findings": [ { "id": "qualified-membership-decision", "name": "Qualified membership decision", "description": "Record whether the equalities and inequalities are proved exactly or checked approximately; a failed candidate circle alone does not disprove the existence of another.", "questions": [ { "text": "Are all required equalities and strict inequalities exact, interval-certified or merely within numerical tolerance?", "kind": "measurement", "id": "qualified-membership-decision-q01" }, { "text": "If membership is rejected, does the evidence exclude every eligible circle or only the current candidate?", "kind": "action", "id": "qualified-membership-decision-q02" } ] } ] } ] }, { "id": "construction-and-preservation", "name": "Construction and preservation", "description": "Support building and modifying instances while retaining a valid external tangency certificate.", "rationale": "An agent must know which geometric operations produce candidates and which preserve established membership.", "layers": [ { "id": "candidate-construction", "name": "Candidate construction", "description": "Use line-distance constraints to locate potential excenters.", "findings": [ { "id": "bisector-branch-validation", "name": "Bisector branch validation", "description": "For intersecting line pairs, equal-distance loci are their angle bisectors. Intersections of selected branches supply candidates that still require the remaining tangency and extension checks.", "questions": [ { "text": "Which angle-bisector branches or equivalent signed-distance equations generated each candidate center?", "kind": "action", "id": "bisector-branch-validation-q01" }, { "text": "Which candidates survive all four line-distance, contact-location and exterior-placement checks?", "kind": "measurement", "id": "bisector-branch-validation-q02" } ] } ] }, { "id": "geometry-preserving-operations", "name": "Geometry-preserving operations", "description": "Track transformations of both the quadrilateral and its circle witness.", "findings": [ { "id": "similarity-preserves-certificate", "name": "Similarity preserves the certificate", "description": "Translations, rotations, reflections and uniform scaling by a nonzero factor preserve circular tangency and extension membership; lengths scale by the absolute scale factor.", "questions": [ { "text": "Is the proposed transformation a Euclidean similarity, and how does it transform the center, radius and contact points?", "kind": "action", "id": "similarity-preserves-certificate-q01" }, { "text": "If vertices are moved independently or scaling is nonuniform, what fresh verification establishes ex-tangentiality?", "kind": "action", "id": "similarity-preserves-certificate-q02" } ] } ] } ] }, { "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-confidence-notes", "name": "Check these first", "description": "Recalled without web access and unsourced; every item is a lead to verify.", "evidence": [ "This describes the conventional convex Euclidean case; extensions of the terminology to nonconvex or crossed quadrilaterals should be checked before inclusion.", "No sources were consulted; specialized classifications and algebraic characterizations are omitted." ], "questions": [ { "id": "evidence-confidence-notes-q01", "text": "Which of these check these first hold for the sense of ex-tangential quadrilateral this model covers, and on what evidence?", "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.", "evidence": [ "Studied in Euclidean geometry through circle tangency, angle bisectors, and relations among quadrilateral side lengths." ], "questions": [ { "id": "evidence-real-world-use-q01", "text": "Which of these real-world use hold for the sense of ex-tangential quadrilateral this model covers, and on what evidence?", "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.", "evidence": [ "Confusing tangency to a side's supporting line with tangency to the side segment itself.", "Confusing an exterior circle tangent to all four extended sides with a circle tangent to only three supporting lines." ], "questions": [ { "id": "evidence-failure-modes-and-hazards-q01", "text": "Which of these failure modes and hazards hold for the sense of ex-tangential quadrilateral this model covers, and on what evidence?", "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.", "evidence": [ { "name": "tangential quadrilateral", "difference": "A tangential quadrilateral has an incircle tangent to all four side segments; an ex-tangential quadrilateral has an exterior circle tangent to their extensions." }, { "name": "cyclic quadrilateral", "difference": "Its four vertices lie on a common circle; ex-tangentiality concerns tangency of extended sides to a circle." } ], "questions": [ { "id": "evidence-neighbours-q01", "text": "Which of these neighbouring kinds and how to tell them apart hold for the sense of ex-tangential quadrilateral this model covers, and on what evidence?", "kind": "provenance" } ] } ] } ] } ] }, "openQuestions": [ "Which authoritative definition should the registry adopt, and should broader concave usages be represented as variants or linked neighbouring concepts?", "Should infinite-radius limiting configurations be recorded as excluded boundary cases, and which sources explicitly formalise that convention?", "Which side-length characterisations are necessary and sufficient under the selected convention, with what exclusions and vertex-order assumptions?", "What existence and uniqueness theorem governs eligible excircles for every admitted configuration?", "Which numerical tolerances or interval methods reliably distinguish valid instances from near-degenerate candidates when input coordinates are approximate?" ], "statistics": { "bundles": 5, "layers": 9, "findings": 12, "questions": 20 } }