{"schema":"https://ver.cy/schemas/card/1.0.0","id":"vr.tr.exponentiation","code":"thing-q33456","url":"https://ver.cy/models/thing/q33456/","name":"exponentiation","alternateNames":[],"kind":"thing","status":"published","version":"0.1.0-research.1","language":"en","classifiers":{"family":"Thing Registry","category":"Cross-cutting context","entryKind":"thing","plane":"XCT","domain":["XCT.QLT"],"industry":[],"navPath":"","tags":[],"facets":{}},"whatItIs":"Exponentiation is the binary operation that sends a base a and an exponent b to a power a^b, defined as iterated multiplication of a when b is a natural number and extended to integer, rational, real and complex exponents by taking inverses, roots and the identity a^b = exp(b Log a), which is single-valued for positive real a and in general multi-valued once Log is.","purpose":"Enable an AI agent to recognise an exponentiation, determine whether it is defined under the applicable mathematical conventions, and choose justified evaluation, comparison or transformation actions.","scope":{"in":["Identification of the base, exponent and grouping in a power expression.","Operand domains and the mathematical structure that gives exponentiation meaning.","Conditions of definedness, including zero, negative bases and noninteger exponents.","Single-valued, branch-selected or multivalued interpretations where applicable.","Conditions for evaluating, comparing and rewriting powers.","Exact and approximate results, including numerical limits and error requirements."],"out":["General multiplication and the full theory of the algebraic structure containing the base.","Logarithms and root extraction as independently modelled operations.","Exponential functions considered as functions with domains, graphs, derivatives and integrals.","General equation solving, including complete solution sets for exponential equations.","Applications such as compound interest, radioactive decay and cryptographic protocols.","Programming-language operator specifications except where they determine the meaning or evaluation of a recorded power."],"boundaries":[]},"distinguishingFeatures":["A power assigns different roles to its operands: in a^b, a is the base and b is the exponent; swapping them generally changes the result.","For a positive integer exponent n, a^n denotes a product of n copies of a, whereas n times a denotes repeated addition.","Exponentiation starts with a base and an exponent; root extraction and logarithms instead seek an operand from a specified power value.","A power expression records an application of an operation; treating x ↦ a^x as a function additionally requires identifying x as a variable and specifying its domain.","The symbol ^ establishes exponentiation only when the notation or execution context assigns it that meaning; in some contexts it denotes a different operation."],"structure":{"bundles":[{"id":"power-expression","name":"Power expression","description":"Records what expression is being interpreted as exponentiation and how its operands are bound.","layers":[{"id":"operand-binding","name":"Operand binding","description":"Separates the base and exponent from surrounding signs and operations.","findings":[{"id":"base-exponent-grouping","name":"Base, exponent and grouping","description":"Record the two operands and preserve grouping rather than inferring it from a flattened string.","questions":[{"text":"Which complete expression is the base, and which complete expression is the exponent?","id":"base-exponent-grouping-q01","kind":"definition"},{"text":"Do minus signs or nested powers belong inside an operand or outside this exponentiation?","id":"base-exponent-grouping-q02","kind":"boundary"}]}]},{"id":"notation-context","name":"Notation context","description":"Establishes how the source notation denotes a power.","findings":[{"id":"operator-interpretation","name":"Operator interpretation","description":"Record the notation or language convention that makes the expression an exponentiation.","questions":[{"text":"What source or notation convention establishes that the operator denotes exponentiation?","id":"operator-interpretation-q01","kind":"provenance"},{"text":"How does that convention group an unparenthesised expression such as a^b^c?","id":"operator-interpretation-q02","kind":"definition"}]}]}]},{"id":"domain-and-definition","name":"Domain and definition","description":"Records the setting and exponent class that determine the meaning of the power.","layers":[{"id":"ambient-structure","name":"Ambient structure","description":"Identifies where the operands live and where a result is permitted to live.","findings":[{"id":"operand-result-domains","name":"Operand and result domains","description":"Record the mathematical setting, relevant operations and any permitted extension of the result domain.","questions":[{"text":"In which domains or structures are the base, exponent and intended result interpreted?","id":"operand-result-domains-q01","kind":"definition"},{"text":"May evaluation extend a real-valued expression into complex values, or must it remain within the declared domain?","id":"operand-result-domains-q02","kind":"boundary"}]}]},{"id":"exponent-definition","name":"Exponent-dependent definition","description":"Connects the exponent class to the definition being used.","findings":[{"id":"power-construction","name":"Power construction","description":"Record whether the power uses repeated multiplication, an inverse, a root-based convention or another specified extension.","questions":[{"text":"What is known about the exponent's class, and what definition gives the power meaning for that class?","id":"power-construction-q01","kind":"definition"},{"text":"For a negative integer exponent, is the base invertible in the selected setting?","id":"power-construction-q02","kind":"boundary"},{"text":"For a noninteger exponent, which definition or convention authorises the extension beyond integer powers?","id":"power-construction-q03","kind":"provenance"}]}]}]},{"id":"definedness-and-branches","name":"Definedness and branches","description":"Records exceptional inputs and choices that determine whether a power has an admissible value.","layers":[{"id":"exceptional-operands","name":"Exceptional operands","description":"Checks input cases that require more than the ordinary positive-base rule.","findings":[{"id":"zero-and-negative-base-cases","name":"Zero and negative-base cases","description":"Record the actual base and exponent conditions before assigning definedness.","questions":[{"text":"Can the base be zero, and if so is the exponent positive, zero, negative or still unknown?","id":"zero-and-negative-base-cases-q01","kind":"boundary"},{"text":"If both operands are zero, which context-specific convention governs 0^0?","id":"zero-and-negative-base-cases-q02","kind":"definition"},{"text":"For a negative real base and a noninteger exponent, does the selected real or complex interpretation define a value?","id":"zero-and-negative-base-cases-q03","kind":"boundary"}]}]},{"id":"branch-selection","name":"Branch selection","description":"Makes any branch choice or multivalued interpretation explicit.","findings":[{"id":"selected-power-values","name":"Selected power values","description":"Record whether an interpretation selects one value or retains multiple values, together with its governing branch convention.","questions":[{"text":"Is this power interpreted as single-valued, branch-selected or multivalued?","id":"selected-power-values-q01","kind":"definition"},{"text":"If the definition uses a complex logarithm, which logarithm branch or set of logarithm values is used?","id":"selected-power-values-q02","kind":"definition"},{"text":"Would crossing a branch boundary require changing the selected value or refusing the proposed continuation?","id":"selected-power-values-q03","kind":"action"}]}]}]},{"id":"power-laws-and-comparison","name":"Power laws and comparison","description":"Records the conditions under which powers may be rewritten or ordered.","layers":[{"id":"conditional-rewrites","name":"Conditional rewrites","description":"Attaches applicability conditions to each proposed algebraic transformation.","findings":[{"id":"rewrite-obligations","name":"Rewrite obligations","description":"Record the intended identity and establish its conditions before using it.","questions":[{"text":"Which rewrite is proposed: combining equal bases, distributing a power over a product, or replacing a power of a power?","id":"rewrite-obligations-q01","kind":"action"},{"text":"Are both sides defined and equal under the recorded domain, operand assumptions and branch choices?","id":"rewrite-obligations-q02","kind":"boundary"},{"text":"If the bases belong to a structure with noncommutative multiplication, does the rewrite require a commutativity condition?","id":"rewrite-obligations-q03","kind":"boundary"}]}]},{"id":"ordering-and-bounds","name":"Ordering and bounds","description":"Establishes whether available operand information supports comparing or bounding results.","findings":[{"id":"comparison-conditions","name":"Comparison conditions","description":"Record the ordered setting, operand restrictions and direction of monotonicity needed for a comparison.","questions":[{"text":"Does the result domain support the requested order comparison?","id":"comparison-conditions-q01","kind":"boundary"},{"text":"For a positive real base, is it below one, equal to one or above one, and what does that imply when comparing exponents?","id":"comparison-conditions-q02","kind":"measurement"},{"text":"What bounds on the base and exponent justify a bound on this power without exact evaluation?","id":"comparison-conditions-q03","kind":"action"}]}]}]},{"id":"evaluation-and-result","name":"Evaluation and result","description":"Records how the power is to be computed and how its result may safely be used.","layers":[{"id":"evaluation-plan","name":"Evaluation plan","description":"Selects an evaluation approach consistent with the mathematical interpretation and required output.","findings":[{"id":"method-and-target","name":"Method and target","description":"Record whether evaluation should produce an exact value, approximation, enclosure or retained symbolic expression.","questions":[{"text":"Does the intended use require an exact value, a certified bound or a numerical approximation?","id":"method-and-target-q01","kind":"action"},{"text":"Which evaluation method preserves the declared domain and branch interpretation for these operands?","id":"method-and-target-q02","kind":"action"},{"text":"Would retaining the power symbolically preserve information that the available numerical representation would lose?","id":"method-and-target-q03","kind":"boundary"}]}]},{"id":"result-quality","name":"Result quality","description":"Separates mathematical definedness from computational success and approximation quality.","findings":[{"id":"accuracy-and-range","name":"Accuracy and range","description":"Record evaluation status, numerical error and representation limits before accepting the result.","questions":[{"text":"What precision or error bound accompanies the computed power, and does it meet the intended use?","id":"accuracy-and-range-q01","kind":"measurement"},{"text":"Did evaluation encounter overflow, underflow or a domain error, and does that reflect the mathematics or only the implementation?","id":"accuracy-and-range-q02","kind":"boundary"},{"text":"Would uncertain or rounded operands materially change the result or its definedness, requiring higher precision or an explicit enclosure?","id":"accuracy-and-range-q03","kind":"action"}]}]}]},{"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.","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.","questions":[{"text":"Which of these kinds and varieties hold for the sense of exponentiation this model covers, and on what evidence?","id":"evidence-kinds-q01","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.","questions":[{"text":"Which of these identifiers and schemes hold for the sense of exponentiation this model covers, and on what evidence?","id":"evidence-identifiers-q01","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.","questions":[{"text":"Which of these standards and regulation hold for the sense of exponentiation this model covers, and on what evidence?","id":"evidence-standards-and-regulation-q01","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.","questions":[{"text":"Which of these real-world use hold for the sense of exponentiation this model covers, and on what evidence?","id":"evidence-real-world-use-q01","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.","questions":[{"text":"Which of these typical measurements hold for the sense of exponentiation this model covers, and on what evidence?","id":"evidence-measurements-q01","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.","questions":[{"text":"Which of these failure modes and hazards hold for the sense of exponentiation this model covers, and on what evidence?","id":"evidence-failure-modes-and-hazards-q01","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.","questions":[{"text":"Which of these regional variation hold for the sense of exponentiation this model covers, and on what evidence?","id":"evidence-regional-variation-q01","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.","questions":[{"text":"Which of these neighbouring kinds and how to tell them apart hold for the sense of exponentiation this model covers, and on what evidence?","id":"evidence-neighbours-q01","kind":"provenance"}]}]}]}]},"agentConduct":{"may":["Compute powers and explain the rules and their exceptions, such as zero to the power zero.","State which branch or convention is used for complex or negative bases."],"mustNot":["Present an exponential projection as certain when it rests on assumptions that may fail.","Use overflowed or rounded results without flagging the loss of precision.","Mix conventions for complex powers without saying so."],"requiresHuman":["Relying on an exponential growth model for public health or financial decisions."]},"ethics":{"considerations":["Exponential growth is easy to underestimate; honest presentation matters in pandemics and debt.","Misused growth claims can push people into risky investments or panic."],"affectedParties":["People relying on calculations","Learners","The public facing growth projections"]},"owners":{"steward":"Nobody: the operation belongs to mathematics, held in common.","roles":[],"masterSystems":[]},"relations":[{"target":"Q164307","type":"parent","note":"registry parent class"}],"interaction":{"identity":{"applicability":"required","items":["Vercy registry: vr.tr.exponentiation","Wikidata: Q33456 (https://www.wikidata.org/wiki/Q33456)"]},"properties":{"applicability":"not-applicable","items":["Operand roles and grouping: Base expression, exponent expression and explicit parenthesisation","Mathematical setting: Real numbers, complex numbers, modular arithmetic, another explicitly specified structure, or unresolved","Base properties: Known or unknown zero status, sign where ordered, magnitude where defined, and invertibility where required","Exponent class: Positive integer, zero, negative integer, noninteger rational, irrational real, nonreal complex, or unresolved","Definedness: Defined, conditionally defined, undefined under the selected convention, or unresolved","Value interpretation: Single-valued, selected branch, multivalued, or unresolved; record the governing convention","Result representation: Exact symbolic expression, exact numeric value, certified enclosure, approximate numeric value, or unevaluated expression","Numerical error requirement: Absolute tolerance in result units, dimensionless relative tolerance, or requested precision"]},"recognition":{"applicability":"not-applicable","items":[]},"capabilities":{"applicability":"required","items":["Resolve notation into a base, exponent and explicit grouping.","Check definedness and expose assumptions needed to obtain a value.","Evaluate a power exactly or approximately under a declared interpretation.","Apply or reject a power-law rewrite after checking its domain and branch conditions.","Compare or bound powers using justified sign, ordering and magnitude information.","Preserve an unevaluated power or request a missing convention when evaluation would otherwise change its meaning."]},"hazards":{"applicability":"required","items":["Overflow, underflow or rounding error can produce misleading numerical results in downstream calculations.","Ignoring domain restrictions or complex branch choices can invalidate transformations and conclusions.","Confusing exponentiation with a programming language's similarly written operator can cause incorrect software behaviour."]},"interfaces":{"applicability":"required","items":["ISO 80000-2, issued by the International Organization for Standardization, standardises mathematical signs and symbols including notation for powers.","IEEE 754, issued by the Institute of Electrical and Electronics Engineers, specifies floating-point arithmetic and includes recommended power operations relevant to numerical exponentiation."]},"context":{"applicability":"required","items":["Exponentiation is used in algebra, scientific modelling and computation to express powers, scaling and exponential relationships.","Its evaluation requires attention to the number domain, exponent conventions and any choice of complex logarithm branch."]}},"sources":[{"title":"Exponentiation. Wikimedia Foundation","url":"https://en.wikipedia.org/wiki/Exponentiation","note":"Definition via iterated multiplication and its extensions; 0^0 conventions; notation; distinction from the exponential function and from tetration."},{"title":"ISO 80000-2:2019 Quantities and units - Part 2: Mathematics. International Organization for Standardization","url":"","note":"Standard mathematical notation and names for powers, exponents and roots used in scientific writing."},{"title":"IEEE Std 754-2019, IEEE Standard for Floating-Point Arithmetic. IEEE","url":"","note":"Recommended operations pow, pown and powr, including domain conventions such as pow(±0, ±0) = 1 and the treatment of negative bases."},{"title":"The Art of Computer Programming, Volume 2: Seminumerical Algorithms. Donald E. Knuth, Addison-Wesley","url":"","note":"Binary (square-and-multiply) exponentiation, modular exponentiation, and the place of exponentiation in the hyperoperation hierarchy."},{"title":"NIST Digital Library of Mathematical Functions, Chapter 4 (Elementary Functions). National Institute of Standards and Technology","url":"https://dlmf.nist.gov/","note":"Analytic definition of exp and Log and the resulting treatment of real and complex powers, including branch cuts."},{"title":"Wikidata item Q33456: exponentiation","url":"https://www.wikidata.org/wiki/Q33456","note":"identity and sense of the item"}],"openQuestions":["Does the registry intend exponentiation to cover only scalar numerical powers, or also matrix, operator, ordinal and cardinal exponentiation?","Which neighbouring registered entries own exponential functions, logarithms and root extraction, and how should shared assumptions be linked?","Which conventions should be adopted for 0^0 in each supported mathematical and computational context?","Should complex exponentiation default to a specified principal branch, require an explicit branch choice or permit a multivalued result?","Which authoritative mathematical references and execution-environment specifications should substantiate the supported definitions, rewrite conditions and evaluation behaviour?"],"resources":{"spec":"/models/things/publications/thing-q33456/spec.json"},"provenance":{"origin":"thing registry research (pass 2)","builtFrom":["models/things/publications/thing-q33456/spec.json"],"providers":["Codex","Grok"],"researchStatus":"unreviewed","generatedAt":"2026-09-07T09:51:40Z","builder":"tools/build_cards.py@1.0.0"},"completeness":{"sections":{"classifiers":"filled","whatItIs":"filled","purpose":"filled","distinguishingFeatures":"filled","structure":"filled","agentConduct":"filled","ethics":"filled","owners":"filled","relations":"filled","interaction.identity":"filled","interaction.properties":"not-applicable","interaction.recognition":"not-applicable","interaction.capabilities":"filled","interaction.hazards":"filled","interaction.interfaces":"filled","interaction.context":"filled","sources":"filled"},"notes":{"_gate":"Published through the thing publication gate 1.0 on 2026-10-07T10:10:30+00:00: completeness 1.00; sense: Wikidata description; 3 live sources; review by codex: pass","interaction.properties":"Plane XCT: no invented physical properties.","interaction.recognition":"Exponentiation is an abstract operation whose written representations do not constitute a perceptible physical identity."},"score":1.0}}