{ "model": { "rank": 5852, "code": "thing-q191884", "model_id": "vr.tr.inverse-function", "name": "inverse function", "purpose": "Enable an AI agent to recognise an inverse function, establish where it reverses a specified function, and decide which inversion, evaluation and substitution operations are justified.", "family": "Thing Registry", "category": "Cross-cutting context", "status": "research-draft", "kind": "thing", "plane": "XCT", "domain": "XCT.QLT", "industry": "", "version": "", "url": "/models/thing/q191884/", "tier": 2, "score": 64, "payload": { "layer": "wikidata", "aliases": [], "aliasCount": 0, "merged": 0, "knownIn": 64, "facets": null, "markers": [], "lexicalClass": "", "senseRank": null, "alsoRegisteredAs": null, "source": { "dataset": "wikidata", "item": "Q191884", "url": "https://www.wikidata.org/wiki/Q191884", "license": "CC0 1.0" } }, "research": { "vercy": "1.0-draft", "publication": { "status": "research-draft", "adjudicationStatus": "unreviewed", "publishableCanonical": false, "generatedAt": "2026-09-07T09:40:35Z", "providers": [ "Codex", "Grok" ], "missingProviders": [], "pass": 2, "cost": { "grok": { "usd": 0.03662362, "turns": 2, "seconds": 124.7, "error": null }, "codex": { "seconds": 78.4, "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: 01a07b3b-caa6-7971-b273-71da92c3202c\n--------\nuser\nYou are drafting a Vercy meta-model for one registered thing. Answer as JSON only, no prose around it.\n\nThing: inverse function\nRegistry id: vr.tr.inverse-function\nPlane / domain: XCT / XCT.REL\nRegistry definition: (none recorded)\nNames folded into this entry:" } } }, "metaModel": { "id": "THING-Q191884", "registryId": "vr.tr.inverse-function", "name": "inverse function", "version": "0.1.0-research.1", "entryKind": "thing", "family": "Thing Registry", "domain": [ "XCT.REL" ], "status": "research-draft" }, "canonicalUrl": "https://ver.cy/models/thing/q191884/", "model": { "registry_id": "vr.tr.inverse-function", "name": "inverse function", "purpose": "Enable an AI agent to recognise an inverse function, establish where it reverses a specified function, and decide which inversion, evaluation and substitution operations are justified.", "definition": "A function g is an inverse of f when the two compose to identity maps; a two-sided inverse exists if and only if f is bijective, in which case it is unique and is the converse relation of f.", "scope_statement": "This model owns the inverse function's relationship to its original function, its domain and codomain, the conditions supporting inversion, any restrictions selecting a branch, and the evidence and operational limits governing its use.", "in_scope": [ "The original function and the mapping that reverses it on explicitly stated sets", "Existence, uniqueness and verification of a two-sided inverse", "Domain restrictions and branch choices that make inversion well-defined", "Symbolic, implicit or computational representations of the inverse", "Validity, accuracy and sensitivity of inverse evaluation" ], "out_of_scope": [ "General properties of functions unrelated to inversion", "Inverse relations that remain multivalued", "Preimage operators acting on sets", "Reciprocal functions and multiplicative inverses", "Generalised inverses, pseudoinverses and inverse problems without a uniquely specified inverse function" ], "distinguishing_features": [ "For a proposed inverse g:B→A of f:A→B, both g(f(x))=x for every x in A and f(g(y))=y for every y in B must hold.", "The notation f⁻¹ denotes reversal of a mapping here; it does not mean the reciprocal function x↦1/f(x).", "An inverse function assigns exactly one original input to each input in its own domain; a multivalued inverse relation fails this test until an appropriate restriction is specified.", "The inverse's domain must match the codomain of the function being inverted; inversion only on the image requires that narrower target set to be explicit.", "An expression such as f⁻¹(S) may denote a preimage of a set even when f has no inverse function, so notation alone does not establish this thing's identity." ], "characteristics": [ { "name": "Original function and declared mapping", "kind": "relation", "unit_or_values": "Reference to f:A→B, including its rule, domain A and codomain B", "why_it_matters": "Invertibility depends on the declared sets as well as the function rule." }, { "name": "Inverse mapping", "kind": "relation", "unit_or_values": "Proposed or established g:B→A, with any restricted original function identified", "why_it_matters": "Identifies exactly which mapping is being reversed." }, { "name": "Invertibility assessment", "kind": "state", "unit_or_values": "Unassessed; established bijective on declared sets; disproved on declared sets; unresolved", "why_it_matters": "Controls whether an agent may treat a candidate as an established inverse." }, { "name": "Composition evidence", "kind": "state", "unit_or_values": "For each composition identity: proved; disproved; tested only; unresolved", "why_it_matters": "Separates a two-sided inverse from a one-sided inverse or a numerically plausible candidate." }, { "name": "Extent of inversion", "kind": "category", "unit_or_values": "Global on the declared mapping; inverse of an explicit restriction; local on specified neighbourhoods", "why_it_matters": "Prevents local or branch-specific validity from being extended without justification." }, { "name": "Branch-defining restriction", "kind": "relation", "unit_or_values": "Selected subset of the original domain, resulting image and endpoint or cut conventions; not applicable", "why_it_matters": "Records the choices that make a candidate inverse single-valued." }, { "name": "Representation", "kind": "category", "unit_or_values": "Explicit expression; implicit defining equation; algorithm; finite lookup; combinations", "why_it_matters": "Determines how an agent can evaluate and verify the inverse." }, { "name": "Evaluation error", "kind": "measurement", "unit_or_values": "Absolute or relative error bound in a declared norm or metric; output units where applicable; unknown", "why_it_matters": "Distinguishes the exact inverse from an approximate evaluation." }, { "name": "Local sensitivity", "kind": "measurement", "unit_or_values": "Declared absolute or relative condition measure at a specified input, with units and assumptions; unknown; not applicable", "why_it_matters": "Shows whether small input or computation errors can cause large changes in the recovered value." } ], "affordances": [ "Test whether a proposed mapping is a two-sided inverse on specified sets", "Construct an inverse candidate by reversing a rule and checking its validity", "Restrict the original domain and identify a corresponding inverse branch", "Evaluate the inverse for an admissible input using a justified representation", "Use the inverse to solve an equation or substitute variables while preserving domain conditions", "Reject, narrow or qualify an inversion when uniqueness, coverage or evaluation accuracy is unsupported" ] }, "sources": [ { "id": "wikipedia-inverse-function", "title": "Inverse function - Wikipedia", "url": "https://en.wikipedia.org/wiki/Inverse_function", "what_it_supports": "Definition via converse relation; uniqueness; left, right and two-sided inverses; invertibility iff bijection.", "url_status": "live" }, { "id": "mathworld-inverse-function", "title": "Inverse Function - Wolfram MathWorld", "url": "https://mathworld.wolfram.com/InverseFunction.html", "what_it_supports": "Invertibility iff bijective; f^{-1} notation clash with powers; multivalued inverses of elementary functions in the complex plane.", "url_status": "live" }, { "id": "eom-inverse-function", "title": "Inverse function - Encyclopedia of Mathematics (EMS Press)", "url": "https://encyclopediaofmath.org/wiki/Inverse_function", "what_it_supports": "Invertible map as unique preimage of cardinality one; distinction from set-valued preimage; left inverse of an injection.", "url_status": "live" }, { "id": "wikidata-q191884", "title": "inverse function (Q191884) - Wikidata", "url": "https://www.wikidata.org/wiki/Q191884", "what_it_supports": "Canonical item, aliases, ISO 80000-2:2019 item 2-12.10, and distinction from the reciprocal function.", "url_status": "live" }, { "id": "iso-80000-2-2019", "title": "ISO 80000-2:2019, Quantities and units - Part 2: Mathematics, item 2-12.10 - International Organization for Standardization", "url": "", "what_it_supports": "Standard notation f^{-1}; inverse defined when f is injective, with domain equal to the range; warning not to confuse with the pointwise reciprocal." }, { "id": "wikipedia-inverse-trig", "title": "Inverse trigonometric functions - Wikipedia", "url": "https://en.wikipedia.org/wiki/Inverse_trigonometric_functions", "what_it_supports": "Named family of restricted inverses; principal-value ranges; ISO 80000-2:2019 mandate of the arc- prefix.", "url_status": "live" } ], "structure": { "bundles": [ { "id": "mapping-contract", "name": "Mapping contract", "description": "Identify the original function, the proposed inverse and the sets between which each acts.", "rationale": "A formula alone cannot identify an inverse because changing the original domain or codomain can change invertibility.", "layers": [ { "id": "original-mapping", "name": "Original mapping", "description": "Establish the exact function whose action is to be reversed.", "findings": [ { "id": "declared-original-function", "name": "Declared original function", "description": "Record the original rule and its domain and codomain, distinguishing the codomain from the actual image.", "questions": [ { "text": "What is the original function f:A→B, including its rule and explicitly declared sets A and B?", "kind": "definition", "id": "declared-original-function-q01" }, { "text": "Does f attain every element of B, or is the proposed inversion actually limited to f(A)?", "kind": "boundary", "id": "declared-original-function-q02" } ] } ] }, { "id": "reverse-mapping", "name": "Reverse mapping", "description": "Identify the candidate inverse and disambiguate its notation.", "findings": [ { "id": "candidate-inverse-identity", "name": "Candidate inverse identity", "description": "Record the reverse mapping and establish that it denotes a function inverse rather than a reciprocal or preimage operator.", "questions": [ { "text": "What mapping g is proposed as the inverse, and do its domain and codomain reverse those of the specified original mapping?", "kind": "definition", "id": "candidate-inverse-identity-q01" }, { "text": "What definition or contextual evidence establishes that f⁻¹ denotes an inverse function here?", "kind": "provenance", "id": "candidate-inverse-identity-q02" } ] } ] } ] }, { "id": "invertibility-and-verification", "name": "Invertibility and verification", "description": "Establish whether the original mapping admits the claimed inverse and whether the candidate satisfies both inverse identities.", "rationale": "An agent must distinguish a verified inverse from a candidate, a one-sided inverse or a mapping contradicted by collisions or missing outputs.", "layers": [ { "id": "existence-and-uniqueness", "name": "Existence and uniqueness", "description": "Assess whether each intended inverse input has exactly one corresponding original input.", "findings": [ { "id": "unique-preimage-coverage", "name": "Unique preimage coverage", "description": "Record evidence for injectivity and surjectivity on the declared sets, including counterexamples and unresolved conditions.", "questions": [ { "text": "What establishes that distinct inputs in A cannot produce the same output under f?", "kind": "definition", "id": "unique-preimage-coverage-q01" }, { "text": "What establishes that every intended inverse input has a preimage, and which inputs remain excluded or unresolved?", "kind": "boundary", "id": "unique-preimage-coverage-q02" }, { "text": "Which proof, theorem with checked hypotheses or exhaustive finite check supports these conclusions?", "kind": "provenance", "id": "unique-preimage-coverage-q03" } ] } ] }, { "id": "two-sided-composition", "name": "Two-sided composition", "description": "Check the proposed inverse in both directions over the full claimed sets.", "findings": [ { "id": "inverse-identities", "name": "Inverse identities", "description": "Record the status and extent of evidence for g∘f=id_A and f∘g=id_B.", "questions": [ { "text": "Does g(f(x))=x hold for every x in A, including any included boundary points?", "kind": "boundary", "id": "inverse-identities-q01" }, { "text": "Does f(g(y))=y hold for every y in B, and is this established exactly or only tested on samples?", "kind": "provenance", "id": "inverse-identities-q02" } ] } ] } ] }, { "id": "restrictions-and-branches", "name": "Restrictions and branches", "description": "Record the restrictions under which inversion becomes single-valued and the limits of each selected branch.", "rationale": "Common inverse expressions reverse restricted functions, and their validity depends on choices that notation may conceal.", "layers": [ { "id": "branch-selection", "name": "Branch selection", "description": "Specify the original-domain restriction and any convention selecting one inverse branch.", "findings": [ { "id": "single-valued-restriction", "name": "Single-valued restriction", "description": "Identify the restricted original function and the reason for selecting its inverse over other possible branches.", "questions": [ { "text": "If f is not injective on its original domain, which subset A' is selected and what is its image f(A')?", "kind": "boundary", "id": "single-valued-restriction-q01" }, { "text": "What task requirement or stated convention selects this restriction rather than another valid inverse branch?", "kind": "provenance", "id": "single-valued-restriction-q02" } ] } ] }, { "id": "locality-and-branch-boundaries", "name": "Locality and branch boundaries", "description": "Define where a local or restricted inverse remains valid and what happens at its limits.", "findings": [ { "id": "branch-validity-region", "name": "Branch validity region", "description": "Record valid neighbourhoods, included endpoints, excluded values and any branch cuts relevant to the mapping.", "questions": [ { "text": "On which exact sets or neighbourhoods is this inverse valid, and which endpoints or cut boundaries are included?", "kind": "boundary", "id": "branch-validity-region-q01" }, { "text": "What must an agent check before extending the inverse's domain or switching to another branch?", "kind": "action", "id": "branch-validity-region-q02" } ] } ] } ] }, { "id": "representation-and-evaluation", "name": "Representation and evaluation", "description": "Describe how the inverse is represented and how admissible inputs produce exact or approximate outputs.", "rationale": "An inverse may exist without an explicit formula, and an evaluation procedure needs its own accuracy and failure criteria.", "layers": [ { "id": "inverse-representation", "name": "Inverse representation", "description": "Connect the mathematical inverse to its expression, defining equation, lookup or algorithm.", "findings": [ { "id": "representation-validity", "name": "Representation validity", "description": "Record how the representation was derived and why it denotes the selected inverse throughout its claimed domain.", "questions": [ { "text": "Is the inverse represented explicitly, implicitly, algorithmically or by a finite lookup?", "kind": "definition", "id": "representation-validity-q01" }, { "text": "What derivation or verification connects this representation to the original function and selected branch?", "kind": "provenance", "id": "representation-validity-q02" } ] } ] }, { "id": "evaluation-assurance", "name": "Evaluation assurance", "description": "Determine whether a particular inverse evaluation is admissible and sufficiently accurate.", "findings": [ { "id": "evaluation-result-and-error", "name": "Evaluation result and error", "description": "Record input admissibility, output accuracy and computational failure conditions without equating a small residual with a small inverse error.", "questions": [ { "text": "What domain, branch, initialisation or convergence checks must pass before the evaluation result can be accepted?", "kind": "action", "id": "evaluation-result-and-error-q01" }, { "text": "What bound supports the error in the returned inverse value, and what assumptions connect any residual |f(x̂)−y| to that error?", "kind": "measurement", "id": "evaluation-result-and-error-q02" } ] } ] } ] }, { "id": "regularity-and-permitted-use", "name": "Regularity and permitted use", "description": "Record the inverse properties and operation-specific conditions needed for reliable downstream use.", "rationale": "Existence alone does not establish continuity, differentiability, numerical stability or the validity of a particular transformation.", "layers": [ { "id": "regularity-and-sensitivity", "name": "Regularity and sensitivity", "description": "Assess continuity, differentiability and sensitivity only where relevant and supported.", "findings": [ { "id": "supported-inverse-behaviour", "name": "Supported inverse behaviour", "description": "Record which regularity properties hold and where recovering an input becomes sensitive to perturbations.", "questions": [ { "text": "Which continuity or differentiability properties of the inverse are established, and on what sets and under which hypotheses?", "kind": "boundary", "id": "supported-inverse-behaviour-q01" }, { "text": "Where is inverse evaluation sensitive to perturbations, and what declared derivative, norm or condition measure supports that assessment?", "kind": "measurement", "id": "supported-inverse-behaviour-q02" } ] } ] }, { "id": "equation-solving-and-transformations", "name": "Equation solving and transformations", "description": "Identify justified uses of the inverse and the conditions that must accompany them.", "findings": [ { "id": "operation-preconditions", "name": "Operation preconditions", "description": "Record when applying the inverse solves the intended equation or supports a transformation without losing branch or domain conditions.", "questions": [ { "text": "Before replacing f(x)=y with x=g(y), how will the agent confirm that y lies in the inverse domain and x is constrained to the selected original domain?", "kind": "action", "id": "operation-preconditions-q01" }, { "text": "Which additional hypotheses must be established for the intended use, such as differentiation or a change of variables?", "kind": "action", "id": "operation-preconditions-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-kinds", "name": "Kinds and varieties", "description": "Reported by the breadth pass; each item needs checking against its source before it becomes normative.", "evidence": [ "two-sided inverse of a bijection", "left inverse of an injection", "right inverse of a surjection", "local inverse (inverse-function theorem)", "principal branch of a multivalued inverse", "inverse trigonometric functions", "inverse hyperbolic functions", "inverse of an analytic function (Lagrange inversion)" ], "questions": [ { "id": "evidence-kinds-q01", "text": "Which of these kinds and varieties hold for the sense of inverse function 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", "value_or_pattern": "Q191884", "note": "Item for inverse function / inverse mapping; distinct from Q3075242 (reciprocal function x ↦ 1/x)." }, { "scheme": "ISO 80000-2:2019", "value_or_pattern": "2-12.10", "note": "Named 'inverse function of f'; symbol f^{-1}." }, { "scheme": "GND", "value_or_pattern": "4186767-1", "note": "Deutsche Nationalbibliothek subject heading linked from Wikidata." }, { "scheme": "Freebase / Google Knowledge Graph", "value_or_pattern": "/m/03vjb", "note": "Freebase mid recorded on Wikidata P646." }, { "scheme": "Wolfram Language", "value_or_pattern": "InverseFunction[f]", "note": "Computer-algebra constructor for a symbolic inverse." }, { "scheme": "Encyclopædia Britannica topic", "value_or_pattern": "topic/inverse-function", "note": "Britannica identifier recorded on Wikidata P1417." } ], "questions": [ { "id": "evidence-identifiers-q01", "text": "Which of these identifiers and schemes hold for the sense of inverse function this model covers, and on what evidence?", "kind": "provenance" } ] }, { "id": "evidence-standards-and-regulation", "name": "Standards and regulation", "description": "Reported by the breadth pass; each item needs checking against its source before it becomes normative.", "evidence": [ "ISO 80000-2:2019 Quantities and units - Part 2: Mathematics, item 2-12.10 (International Organization for Standardization): defines f^{-1} for injective f, sets dom(f^{-1})=ran(f), and forbids reading f^{-1} as the pointwise reciprocal.", "ISO 80000-2:2019 (same part): since the 2019 revision, inverse trigonometric functions are to be written with the arc- prefix (arcsin, arccos, …), not as sin^{-1}." ], "questions": [ { "id": "evidence-standards-and-regulation-q01", "text": "Which of these standards and regulation hold for the sense of inverse function 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": [ "Undoing a named elementary map: logarithm inverts exponentiation, square root inverts squaring on [0,∞), arcsin recovers an angle from a sine ratio in navigation, surveying and computer graphics.", "Change of variables and substitution in calculus, and recovering a coordinate chart locally when the Jacobian is non-singular (inverse-function theorem).", "Computer-algebra and numeric libraries expose InverseFunction, asin/acos/atan (and atan2) as the everyday interface; branch cuts are part of the contract.", "Cryptography and one-way hashing treat invertibility as a security property: a function is used precisely because a feasible inverse is not known.", "Control, signal processing and inverse kinematics invert a forward map (plant, filter, or robot Jacobian) to recover an input or joint angle." ], "questions": [ { "id": "evidence-real-world-use-q01", "text": "Which of these real-world use hold for the sense of inverse function 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": "existence of a two-sided inverse", "typical_range": "yes iff f is bijective; otherwise no", "unit": "dimensionless (boolean)" }, { "quantity": "Jacobian determinant det Df(x) as local-invertibility witness", "typical_range": "nonzero real at points of local invertibility; zero at critical points", "unit": "dimensionless (or product of the coordinate units of f)" }, { "quantity": "derivative of the inverse, (f^{-1})'(y)=1/f'(x) for y=f(x)", "typical_range": "unbounded as f'(x)→0; order-1 when |f'| is order-1", "unit": "reciprocal of the unit of f'" }, { "quantity": "principal range of real arcsine", "typical_range": "[-π/2, π/2]", "unit": "radian" }, { "quantity": "principal range of real arccosine", "typical_range": "[0, π]", "unit": "radian" }, { "quantity": "real domain of arcsine / arccosine", "typical_range": "[-1, 1]", "unit": "dimensionless" } ], "questions": [ { "id": "evidence-measurements-q01", "text": "Which of these typical measurements hold for the sense of inverse function 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": [ "Writing f^{-1} when f is not injective (classic case: x ↦ x² on ℝ), so the converse relation is not a function.", "Reading f^{-1} as the pointwise reciprocal 1/f, which ISO 80000-2 and calculus texts both flag; sin^{-1}x is the usual collision.", "Taking a right inverse for a two-sided inverse: √(x²) equals |x|, not x.", "Crossing a branch cut or leaving the principal range of a multivalued inverse (complex log, arcsin, square root), so f(f^{-1}(z))=z or f^{-1}(f(z))=z fails.", "Inverse-function theorem does not apply where the Jacobian vanishes, so no C¹ local inverse exists through a critical point.", "Numerical inversion becomes ill-conditioned as f'→0; library asin/acos raise domain errors outside [-1,1]." ], "questions": [ { "id": "evidence-failure-modes-and-hazards-q01", "text": "Which of these failure modes and hazards hold for the sense of inverse function 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": [ "French: fonction réciproque is the inverse function; fonction inverse is the reciprocal x ↦ 1/x (Wikidata Q3075242). Spanish usage sometimes follows that pair (función recíproca on the inverse-function item).", "German Umkehrfunktion and Norwegian invers funksjon / omvendt funksjon are the ordinary names; English anti-function is attested but rare.", "Notation split: ISO 80000-2:2019 requires arcsin etc.; English school and programming practice still uses sin^{-1}, asin, acos, atan.", "Set-theory texts demand a bijection onto the stated codomain for a two-sided inverse; ISO 80000-2:2019 and many calculus texts define f^{-1} as soon as f is injective, taking the domain of the inverse to be the image." ], "questions": [ { "id": "evidence-regional-variation-q01", "text": "Which of these regional variation hold for the sense of inverse function 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": "reciprocal (multiplicative inverse) function x ↦ 1/x", "difference": "The reciprocal is pointwise multiplication-inverse; the inverse function is composition-inverse. Test: check whether g(f(x))=x or whether f(x)·g(x)=1." }, { "name": "preimage / inverse image f^{-1}({y})", "difference": "The preimage is a set and may have any cardinality; an inverse function exists only when every relevant preimage is a singleton. Test: count solutions of f(x)=y." }, { "name": "inverse relation (converse of a binary relation)", "difference": "Every function has a converse relation; that converse is an inverse function only if it is itself a function. Test: is the converse single-valued and total on the image?" }, { "name": "inverse of a matrix / linear map", "difference": "Matrix inverse is the inverse function in the special case where f is linear between finite-dimensional spaces. Test: is f linear? If not, no matrix represents it." }, { "name": "inverse problem (recovering causes from observed effects)", "difference": "An inverse problem is a reconstruction task, often ill-posed; it is not the same object as the inverse of a given function. Test: is there a specified map f whose compositional inverse is claimed, or a data-to-cause inference?" }, { "name": "antiderivative", "difference": "Differentiation and indefinite integration are inverse operations up to a constant, not inverse functions of a single map. Test: is uniqueness required, and is the object a function or a family of primitives?" } ], "questions": [ { "id": "evidence-neighbours-q01", "text": "Which of these neighbouring kinds and how to tell them apart hold for the sense of inverse function this model covers, and on what evidence?", "kind": "provenance" } ] } ] } ] } ] }, "openQuestions": [ "Does the registry intend this entry to cover inverse functions across arbitrary sets, or should continuous, differentiable and complex-analytic cases have specialised extensions?", "Should one-sided inverses be represented as neighbouring concepts or as explicitly distinguished cases within this entry?", "How should multiple local inverses and branch conventions be linked to one original function without implying a global inverse exists?", "What evidence standard should distinguish verified symbolic inverses, certified numerical evaluations and candidates supported only by sampling?", "Which sensitivity and error measures should be required for scalar, vector-valued and other computational inverse functions?" ], "statistics": { "bundles": 6, "layers": 11, "findings": 18, "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 inverse function, record what state it is in, and decide what may be done with it.", "whatItIs": "Enable an AI agent to recognise an inverse function, establish where it reverses a specified function, and decide which inversion, evaluation and substitution operations are justified.", "characteristics": { "substance": "abstract", "origin": "conceptual", "agency": "inert" }, "whatYouCanDoWithIt": [ "observed and measured" ], "distinguishingFeatures": [ "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 inverse function, 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": [], "standing": "Described in 64 Wikipedia languages, which measures how widely it is written about rather than how important or how common it is.", "structure": { "bundles": [ { "id": "identity-and-classification", "name": "Identity, naming and classification", "description": "How an agent tells one inverse function from another, and a inverse function 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 inverse function, 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 inverse function, 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 inverse function 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 inverse function, 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 inverse function, 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 inverse function 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 inverse function 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 inverse function, 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 inverse function, 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 inverse function 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 inverse function from the things most often confused with it.", "questions": [ { "id": "distinguishing-features-q01", "text": "What identifies and describes what distinguishes a inverse function, 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 inverse function, 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 inverse function 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 inverse function 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 inverse function between them.", "questions": [ { "id": "stages-and-transitions-q01", "text": "What identifies and describes the lifecycle of a inverse function, 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 inverse function, 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 inverse function 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 inverse function, how often and by whom.", "findings": [ { "id": "observation-record", "name": "Observation record", "description": "How an observation of a inverse function 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 inverse function, 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 inverse function, 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 inverse function 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 inverse function 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 inverse function 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 inverse function 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 inverse function 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 inverse function?", "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 inverse function?", "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 inverse function 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 inverse function and how confident it is.", "questions": [ { "id": "claim-provenance-q01", "text": "What identifies and describes a claim about a inverse function, 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 inverse function, 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 inverse function 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 inverse function is retired without being erased.", "questions": [ { "id": "validity-and-supersession-q01", "text": "What identifies and describes the validity of a claim about a inverse function, 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 inverse function, 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 inverse function 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 } } } }