loop
Enable an agent to recognise a topological loop, assess evidence about its continuity and deformation class, and choose valid transformations within its ambient space.
Bundle → Layer → Finding → Questions Filled
6 bundles · 11 layers · 15 findings · 32 questions
Loop definition Establish the mathematical sense and membership conditions.
Defining convention
Record the definition used by the relevant mathematical source or application.
Closed path convention
Use continuity and coincident endpoints as the entry test; record whether a basepoint is distinguished. Reference: [Hatcher, Algebraic Topology, §1.1](https://pi.math.cornell.edu/~hatcher/AT/ATch1.pdf).
- Which source or application supplies the loop definition and basepoint convention? provenance
- Is the object represented as an interval-domain map with equal endpoints or as a circle-domain map? definition
Membership evidence
Check the particular candidate against the chosen definition.
Continuity and closure
Record separate evidence for continuity and endpoint equality.
- What establishes continuity, including at every piecewise join? definition
- Do the endpoints coincide in X itself, including any quotient identifications? boundary
- If the evidence consists of samples, what interpolation and error assumptions remain unverified? measurement
Ambient space and basepoint Specify where the loop lives and which point anchors it.
Target space
Identify topology, identifications and obstacles relevant to the loop.
Ambient constraints
Record the exact target rather than inferring it from a plot.
- What is X, and which topology or quotient construction defines it? definition
- Which removed points, forbidden subsets or boundary restrictions constrain the loop and its deformations? boundary
Anchoring
Specify the common endpoint and its role in subsequent operations.
Basepoint commitment
Record whether the selected basepoint must remain fixed.
- Which point is x0, and how is it identified in the representation of X? definition
- Must every intermediate loop retain x0 during the intended deformation? boundary
- If changing basepoints is required, which connecting path and transport convention will be used? action
Representation and traversal Preserve the distinction between a loop map, its parameterisation and its visible trace.
Map versus image
Identify what the supplied representation retains.
Traversal information
Record ordered traversal and the policy for treating reparameterisations.
- Does the input determine γ, or only the subset γ([0,1])? definition
- Which direction, repeated traversals and pauses does the representation retain? measurement
- Which parameter changes, if any, count as preserving the same represented object? boundary
Regularity and simplicity
Separate optional geometric restrictions from loop membership.
Additional curve properties
Record constant behaviour, repeated points and any extra smoothness or embedding requirements.
- Is the loop constant, simple, self-intersecting or retracing, and what establishes that assessment? measurement
- Does the intended operation require differentiability, rectifiability or an embedding beyond continuity? boundary
Deformation and classification Assess loop equivalence and contraction with explicit constraints and evidence.
Homotopy contract
Specify what a permitted deformation must preserve.
Homotopy witness
Require a continuous family in X with the chosen endpoint constraints. Reference: [Hatcher, Algebraic Topology, §1.1](https://pi.math.cornell.edu/~hatcher/AT/ATch1.pdf).
- Is the requested comparison based homotopy, free homotopy or a stronger relation such as isotopy? definition
- What explicit deformation or theorem establishes the claimed relation? provenance
- Does the entire deformation remain in X and satisfy the selected anchoring and intersection constraints? boundary
Obstructions and invariants
Track evidence supporting or obstructing contraction and comparison.
Classification evidence
Attach assumptions and evidential limits to each invariant or class assignment.
- Which invariant is defined for this X and loop, and what choices are required to compute it? measurement
- Does the result decide the proposed equivalence or supply only a necessary condition? boundary
- Is contraction established by a witness, ruled out by an obstruction or still unresolved? action
Loop operations Determine which constructions are valid and what their results mean.
Composition and reversal
Check compatibility and traversal conventions before constructing another loop.
Operation preconditions
Record operands, basepoint compatibility and the formula governing the constructed map.
- Do the loops to be concatenated share the required basepoint in the same X? boundary
- Which factor is traversed first, and how is the parameter interval allocated? action
- Is reversal being applied to the map, or is an inverse being asserted for its homotopy class? definition
Transformation validation
Verify what survives a construction or change of representation.
Result and preservation
Record the resulting loop and justify each claimed preservation property.
- After simplification or reparameterisation, what establishes continuity, closure and the claimed equivalence? action
- If a continuous map X→Y transports the loop, what is its new basepoint and which class information may be lost? action
- Are identity, inverse or associativity claims about literal maps or about equivalence classes? boundary
Evidence and external alignment What the world already says about this thing, gathered so the model can be checked against it.
Reported evidence
Findings from the breadth pass, kept separate from the structural claims.
Check these first
Recalled without web access and unsourced; every item is a lead to verify.
- Which of these check these first hold for the sense of loop this model covers, and on what evidence? provenance
Kinds and varieties
Recalled without web access and unsourced; every item is a lead to verify.
- Which of these kinds and varieties hold for the sense of loop this model covers, and on what evidence? provenance
Real-world use
Recalled without web access and unsourced; every item is a lead to verify.
- Which of these real-world use hold for the sense of loop this model covers, and on what evidence? provenance
Failure modes and hazards
Recalled without web access and unsourced; every item is a lead to verify.
- Which of these failure modes and hazards hold for the sense of loop this model covers, and on what evidence? provenance
Neighbouring kinds and how to tell them apart
Recalled without web access and unsourced; every item is a lead to verify.
- Which of these neighbouring kinds and how to tell them apart hold for the sense of loop this model covers, and on what evidence? provenance
Classifiers Filled
- Family
- Thing Registry
- Category
- Cross-cutting context
- Entry kind
- thing
- Plane
- XCT
- Domain
- XCT.QLT
- Other names and narrower kinds
- vertical loop, Pochhammer contour
What it is Filled
In a geometric or topological sense, a loop is a continuous path whose endpoint coincides with its starting point.
In scope
- Continuous closed paths, including constant loops and paths with repeated points
- Ambient topology, excluded regions and endpoint identification
- Parameterisation, traversal direction and distinctions between a map and its image
- Based homotopy, null-homotopy and the boundary with free-loop conventions
- Loop concatenation, reversal and applicable invariants
Out of scope
- Programming loops, feedback loops and recurring processes
- Algebraic loops defined by a binary operation
- Graph self-edges and closed walks except through an explicit topological realisation
- Knots and links as embedded objects with isotopy constraints
- The full structure of fundamental groups and loop spaces
- Physical circuits, ropes and tracks independently of their topological abstraction
Why it exists Filled
Enable an agent to recognise a topological loop, assess evidence about its continuity and deformation class, and choose valid transformations within its ambient space.
Distinguishing features Filled
- The candidate supplies a continuous path whose endpoints coincide in X; visual proximity alone does not establish closure.
- Self-intersections, retracing and constant maps are permitted, so being a simple closed curve is not required.
- Traversal data matters: two maps can have the same image while traversing it in different directions or different numbers of times.
- A fixed basepoint must be distinguished from a convention allowing that point to move during deformation.
- Closure does not establish null-homotopy; deformability must be assessed in the specified ambient space.
What robots and AI may and may not do Filled
Must not
- Confuse a loop with a closed curve that is not continuous or with a cycle in a graph without saying so.
- Present incorrect proofs or definitions as established.
- Misattribute results in topology to the wrong mathematicians.
- Present homework answers as a student's own work for assessment.
Only with a human decision
- Certifying a proof for publication or assessment.
May
- Explain loops in geometry and topology from mathematics sources.
- Compute or visualize loops and their properties for learners.
Moral aspects Filled
- Accurate definitions matter in teaching and in fields that build on topology.
- Credit for mathematical discoveries matters to researchers.
Who is affected
- Students
- Mathematicians
Owners Filled
Steward
Nobody owns the concept; it belongs to the mathematical commons.
Links to other meta-models Filled
parent
- Q1366002 - registry parent class
- Q3001206 - registry parent class
What else AI and robots need to interact with it Filled
Identity and identifiers required Filled
- Vercy registry: vr.tr.loop-q4558786
- Wikidata: Q4558786 (https://www.wikidata.org/wiki/Q4558786)
Direct properties not applicable Not applicable
- Ambient space: Space X with topology, identifications and excluded subsets
- Loop representation: Formula; piecewise map; circle-domain map; certified polygonal path; sampled candidate
- Closure and continuity evidence: Established; conditional; unverified; refuted, recorded separately
- Basepoint convention: Distinguished fixed basepoint; basepoint forgotten for comparison; unspecified
- Basepoint: x0 = γ(0) = γ(1) in X
- Traversal structure: Constant; injective except at endpoints; repeated points; retracing; undetermined
- Comparison relation: Map equality; image equality; specified reparameterisation; based homotopy; free homotopy
- Null-homotopy status: Established; obstructed; unresolved, relative to X and the chosen convention
- Winding number: Integer, only with a suitable target, orientation and reference point
Plane XCT: no invented physical properties.
Recognition not applicable Not applicable
Not applicable
A topological loop is an abstract mathematical object whose defining properties cannot be established by visual appearance alone.
Capabilities and actions required Filled
- Validate a proposed loop against continuity, closure and ambient-space requirements.
- Compare loop representations using an explicitly selected equivalence relation.
- Concatenate loops after checking their common basepoint and traversal convention.
- Reverse or reparameterise a loop while recording the resulting map.
- Seek a contraction or another homotopy and verify its constraints.
- Compute applicable invariants and report what they establish or leave unresolved.
Hazards and failure modes required Filled
- Confusing a closed path with a simple closed curve can invalidate arguments because a loop may intersect itself.
- Assuming every loop can contract to a point can produce incorrect conclusions about the topology of its ambient space.
Standards and interfaces not applicable Not applicable
Not applicable
No generally applicable official standard, law or classification governs a loop as a mathematical object.
Context of use required Filled
- Loops are studied in topology and geometry to describe continuous paths that return to their starting point.
- They are used to investigate holes, fundamental groups and whether paths can be continuously deformed within a specified space.
Sources Filled
- Wikidata item Q4558786: loop - identity and sense of the item
- Wikipedia: Loop (topology) - general description of the item
Open questions
- Does Vercy intend this entry to require a distinguished basepoint, given that the linked registry item distinguishes free loops while also treating closed curve as an equivalent name?
- Does an existing world-model publication already own topological loops or closed paths, requiring this entry to link to it?
- Which representation families must the model support first: symbolic maps, combinatorial realisations, geometric paths or sampled observations?
- What evidence standard should distinguish certified continuity and homotopy from numerical or visual indications?
- Which ambient-space families warrant specialised invariant extensions, without implying that winding number classifies arbitrary loops?
Machine files
Provenance
thing registry research (pass 2) · unreviewed
Built from: models/things/publications/thing-q4558786/spec.json