combination
Enable an AI agent to recognise a combination, assess whether its constituents can jointly satisfy a stated intent, and decide whether to form, use, compare or revise it.
Research draft, second pass
A second pass drafted this model: the structure a model of this thing needs, and what is known about it in the world. The line under this one says how the second half was obtained - researched against sources, or recalled without web access, in which case nothing here was read anywhere and every claim is a lead to verify. Unreviewed either way.
Researched by: Codex + Grok
Purpose and description
Enable an AI agent to recognise a combination, assess whether its constituents can jointly satisfy a stated intent, and decide whether to form, use, compare or revise it.
A combination is a selection of k distinct elements from a finite set of n elements in which order is irrelevant, so that two selections that differ only by permutation of the same members are the same combination.
It can be Identify constituents and explain why they count as a combination; Compare combinations using explicit identity and equivalence rules; Assess whether a proposed combination satisfies joint constraints; Evaluate joint outcomes against stated baselines; Propose substitutions or membership changes and identify required reassessments; Determine whether formation, use or separation is supported by available evidence.
Distinguishing features
Unlike an arbitrary collection, a combination has an explicit basis for treating its constituents together; merely listing items does not establish that basis.
To distinguish it from a sequence, test whether exchanging constituent positions changes its identity or effect; record the result rather than assuming order is irrelevant.
To distinguish it from a composition or assembly, test whether membership alone suffices or whether constituent roles, proportions and connections are essential.
To distinguish it from a transformation product, test whether the combined constituents remain meaningfully identifiable or whether a new entity requires its own model.
To distinguish an actual combination from a candidate, check whether the required joint presence or relationship has been established.
Scope
+ Constituents and the rule that makes them one combination
+ Whether order, repetition, proportion or arrangement affects identity
+ Conditions under which constituents can be combined
+ Effects and constraints attributable to constituents acting together
+ Permitted substitutions, additions, removals and separations
- The intrinsic properties and lifecycle of each constituent
- The execution history of a process that produces a combination
- The internal architecture of an assembly when independently modelled
- Chemical transformation and the identity of reaction products
- General enumeration of all possible combinations within a search space
Characteristics
- Combination basis
- Selection, joint use, mixture, arrangement, or another explicitly defined basis; provisional categories Determines what it means for constituents to count as combined.
- Constituent membership
- Identified constituents with roles and occurrence counts where relevant Allows an agent to distinguish the combination from its alternatives and detect missing constituents.
- Constituent count
- Number of occurrences and number of distinct constituents, counted separately Distinguishes repeated participation from distinct membership.
- Identity sensitivity
- For order, repetition, proportion and arrangement: identity-defining, non-defining, or unresolved Prevents an agent from treating materially different combinations as interchangeable.
- Realisation state
- Proposed, partially established, established, dissolved, or unknown Separates a description of possible joint participation from an existing combination.
- Compatibility assessment
- Supported, conditional, conflicting, or unassessed, for a specified context Controls whether formation or use is justified under the applicable constraints.
- Joint effect
- Outcome-specific measure with units, context and a constituent-alone or alternative-combination baseline Makes claims about the value or consequence of combining constituents assessable.
- Separation feasibility
- Feasible, conditionally feasible, infeasible, or unknown under stated recovery criteria Determines whether the combination can be revised or undone while preserving required constituent properties.
Also called
Where this came from
wikidata · CC0 1.0
Also registered as vr.tr.combination-3
Drafted structure
Bundle to layer to finding to question, as the second pass will find it: 6 bundles · 11 layers · 18 findings · 28 questions.
Combination basis Establishes what togetherness means for this combination and where its boundary lies.
An agent cannot recognise a combination from constituent co-occurrence alone.
Basis of togetherness
Defines the selection, relationship or joint use that constitutes the combination.
Constitutive rule
Record the rule under which constituents count as combined and any intent used to interpret it.
- What makes these constituents a combination rather than an incidental collection? definition
- Who or what establishes this combination rule, and where is it documented? provenance
Combination boundary
Separates participating constituents from context, enabling resources and resulting entities.
Participation boundary
Record which entities participate and which merely support, surround or result from their combination.
- Which entities are constituents, and which are only containers, tools, conditions or outputs? boundary
- When should the resulting whole be recognised as a separately modelled entity? boundary
Membership and equivalence Captures constituent participation and the rules for recognising the same or an equivalent combination.
The same constituent names may describe different combinations when multiplicity, roles or arrangement matter.
Constituent participation
Identifies members, repeated occurrences and their contributions.
Membership and roles
Record each constituent's identity, participation count and role where the combination distinguishes roles.
- Which distinct constituents participate, and can the same constituent participate more than once? definition
- Which roles must be filled, and which members or roles are optional? boundary
Identity rules
Determines which changes preserve identity or only preserve suitability for a purpose.
Sameness and substitution
Record how ordering, proportion, arrangement and substitution affect sameness and functional equivalence.
- Does changing order, repetition, proportion or arrangement produce a different combination? definition
- Under what purpose-specific criteria can different constituent selections count as equivalent? boundary
Joint feasibility Assesses whether constituents can participate together under the required conditions.
Constituents that are individually acceptable may conflict when combined, including through interactions involving more than two members.
Interaction constraints
Records exclusions, dependencies and limits created by joint participation.
Compatibility conditions
Record supported and unresolved compatibility constraints across the full membership.
- Which constituent pairs or larger subsets conflict, require one another, or impose joint limits? boundary
- What evidence supports compatibility of the full combination beyond checks of individual constituents or pairs? provenance
Establishment conditions
Defines what must hold for a candidate combination to become established and usable.
Formation and readiness
Record necessary joint presence, connections, proportions or timing and whether they currently hold.
- What observable conditions distinguish a proposed, partial and established combination? measurement
- Which unmet conditions must be resolved before this combination can be formed or used? action
Joint outcomes Captures what follows from combining these constituents in a specified context.
An agent needs evidence about joint performance and consequences rather than assuming constituent properties simply add together.
Outcome comparison
Makes the intended benefit or consequence of combination explicit and testable.
Joint effect baseline
Record the outcome of interest and the baseline against which the combination is judged.
- Which joint outcome matters, and how is it measured or otherwise assessed? measurement
- Is the appropriate baseline separate constituent use, omission of a constituent, or another combination? definition
Interaction evidence
Separates observed joint effects from predictions and limits their transfer to other conditions.
Effect support and limits
Record evidence for joint effects, including unintended effects and the conditions within which claims hold.
- What observations or analysis support the claimed effect of combining these constituents? provenance
- Which changes in membership, proportions or context would make that evidence insufficient? boundary
Revision and separation Determines how a combination may change and whether its constituents can be recovered.
Changing one member can alter compatibility, identity and joint effects, while separation may not restore the original constituents.
Membership revision
Assesses additions, removals, substitutions and changes in participation.
Permitted reconfiguration
Record supported revisions and which combination-level judgments each revision invalidates.
- Which constituents may be added, removed or substituted while preserving the required combination rule? action
- Which compatibility and outcome assessments must be repeated after each proposed revision? action
Separation and recovery
Distinguishes ending joint participation from restoring constituent condition.
Dissolution conditions
Record how the combination can cease to hold and what can be recovered afterward.
- What action or condition ends this combination's constitutive relationship? action
- Which constituents can be recovered, in what condition, and what evidence supports that expectation? measurement
Evidence and external alignment What the world already says about this thing, gathered so the model can be checked against it.
A model that cannot be lined up against existing standards, identifiers and practice cannot be adopted by anyone who already uses them.
Reported evidence
Findings from the breadth pass, kept separate from the structural claims.
Kinds and varieties
Reported by the breadth pass; each item needs checking against its source before it becomes normative.
- k-combination (unordered subset of fixed size k)
- combination with repetition (multiset of size k from n types)
- binomial coefficient C(n,k) as the count of k-combinations
- combination lock / mechanical combination (ordered sequence of settings; a misnomer relative to combinatorics)
- pharmaceutical combination product (two or more active ingredients or device-plus-drug in one product)
- statistical combination of evidence (pooling independent results, e.g. Fisher's method)
- logical combination (compound proposition formed by connectives)
- combinatorial combination in games and lotteries (unordered ticket selections)
- Which of these kinds and varieties hold for the sense of combination this model covers, and on what evidence? provenance
Identifiers and schemes
Reported by the breadth pass; each item needs checking against its source before it becomes normative.
- Wikidata - Q184191 (combination in combinatorics) - Distinct from Q5155623 (combination lock) and Q515140 (combination product).
- OEIS / binomial - A007318 (binomial coefficient triangle) / C(n,k) - The count of k-combinations of an n-set.
- FDA product type - combination product (21 CFR 3.2(e)) - US regulatory class, not a combinatorial object.
- Which of these identifiers and schemes hold for the sense of combination this model covers, and on what evidence? provenance
Standards and regulation
Reported by the breadth pass; each item needs checking against its source before it becomes normative.
- ISO 3534-1 (ISO): statistical terms distinguishing combination and permutation in sampling
- NIST DLMF Chapter 26 (NIST): combinatorial analysis, including combinations
- 21 CFR 3.2(e) and 21 CFR Part 4 (US FDA/HHS): definition and current good manufacturing practice for combination products
- Regulation (EC) No 1394/2007 (EU): advanced therapy medicinal products, including combined ATMPs
- Which of these standards and regulation hold for the sense of combination this model covers, and on what evidence? provenance
Real-world use
Reported by the breadth pass; each item needs checking against its source before it becomes normative.
- Lottery and raffle tickets are sold as unordered k-subsets of a number pool; payout depends on matching a combination, not a permutation.
- Clinical and regulatory dossiers treat a drug-eluting stent or a prefilled injector as one combination product, not two independent articles.
- Hash functions, poker hand rankings, and committee-selection algorithms enumerate or sample combinations rather than ordered tuples.
- A mechanical combination lock is opened by an ordered sequence of disc positions; the everyday name contradicts the combinatorial sense.
- Which of these real-world use hold for the sense of combination this model covers, and on what evidence? provenance
Typical measurements
Reported by the breadth pass; each item needs checking against its source before it becomes normative.
- number of k-combinations of an n-set - C(n,k) from 1 (k=0 or k=n) up to C(n, floor(n/2)); e.g. C(52,5)=2,598,960 - 1 (dimensionless count)
- combination size k relative to n - 0 ≤ k ≤ n (without repetition); k ≥ 0 unbounded with repetition - 1 (count of members)
- lottery combination space - C(49,6)=13,983,816 (classic 6/49); C(69,5)×C(26,1) for some multi-pool games - 1 (tickets / equally likely outcomes)
- Which of these typical measurements hold for the sense of combination this model covers, and on what evidence? provenance
Failure modes and hazards
Reported by the breadth pass; each item needs checking against its source before it becomes normative.
- Confusing combination with permutation overcounts by k! and breaks lottery, sampling, and cryptographic enumerations.
- Combination-with-repetition vs without-repetition: using C(n,k) where C(n+k−1,k) is required (or the reverse) mis-sizes the sample space.
- In regulation, failing to classify a co-packaged or integral drug-device article as a combination product leaves GMP, reporting, and lead-center rules unapplied.
- Mechanical 'combination' locks are vulnerable to decoding, impressioning, and residual-manipulation attacks because the secret is an ordered sequence, not an unordered set.
- Which of these failure modes and hazards hold for the sense of combination this model covers, and on what evidence? provenance
Regional variation
Reported by the breadth pass; each item needs checking against its source before it becomes normative.
- English 'combination lock' vs combinatorial 'combination': the lock is an ordered code; some languages (e.g. German Zahlenkombination, French combinaison) keep the same ambiguity.
- Lottery naming: UK National Lottery historically 6/49, US Powerball and Mega Millions use two-pool combinations; 'combination bet' in UK/Irish bookmaking is a parlay of multiple selections.
- Pharmaceutical combination products: US FDA 21 CFR 3.2(e) vs EU 'combined advanced therapy medicinal product' and 'fixed-dose combination' (FDC) medicinal-product language.
- Which of these regional variation hold for the sense of combination this model covers, and on what evidence? provenance
Neighbouring kinds and how to tell them apart
Reported by the breadth pass; each item needs checking against its source before it becomes normative.
- permutation - A permutation of k elements from n is ordered; two sequences with the same members in different order are distinct. Test: if swapping two selected items yields a different object, it is a permutation, not a combination.
- arrangement / tuple - An arrangement is an ordered selection, often allowing a distinguished position; a combination is the underlying unordered set. Test: does position or sequence matter for identity?
- multiset / combination with repetition - A k-combination of an n-set forbids repeated members; a combination with repetition is a multiset of cardinality k. Test: may the same type appear more than once?
- Cartesian product / concatenation - A Cartesian product composes ordered pairs from several sets; a combination selects an unordered subset of one set. Test: are the components typed by source set and ordered?
- combination product (regulatory) - A regulatory combination product is a single article that combines drug, device, and/or biologic constituents; it is not an unordered k-subset. Test: is the object a marketed medical article under 21 CFR 3.2(e) (or EU analogue), or a combinatorial selection?
- combination lock code - The secret is an ordered sequence of settings (a word in a mixed alphabet), not a combination in the combinatorial sense. Test: does reordering the digits change whether the lock opens?
- Which of these neighbouring kinds and how to tell them apart hold for the sense of combination this model covers, and on what evidence? provenance
Sources
- Enumerative Combinatorics, Volume 1 - Standard combinatorial definition of k-combinations as unordered subsets and the binomial count C(n,k).
- NIST Digital Library of Mathematical Functions, §26.3 Combinatorial Analysis - Authoritative statement of combinations without and with repetition and their generating functions.
- Combination Products - Regulatory sense of combination as a drug-device, biologic-device, or drug-biologic product.
- ISO 3534-1: Statistics - Vocabulary and symbols - Part 1: General statistical terms and terms used in probability - Statistical vocabulary distinguishing combinations from permutations in sampling.
What the second pass must settle
- Does this registry entry mean mathematical selection without order, practical joint use, or a broader relation encompassing both?
- Which existing Vercy models own collection, composition, assembly, mixture and conjunction, and where do their boundaries place combination?
- Can a valid combination contain zero or one constituent, and does the registry permit repeated constituents?
- Does combination identity depend on constituent types, particular instances, or a context-specific choice between them?
- Should combinations whose constituents lose their independent identity remain within this model or be linked to a transformation or product model?