{"schema":"https://ver.cy/schemas/card/1.0.0","id":"vr.tr.foliation-q684220","code":"thing-q684220","url":"https://ver.cy/models/thing/q684220/","name":"foliation","alternateNames":["foliation of codimension 1","Lagrangian foliation","Taut foliation"],"kind":"thing","status":"published","version":"0.1.0-research.1","language":"en","classifiers":{"family":"Thing Registry","category":"Activities and processes","entryKind":"thing","plane":"ACT","domain":["ACT.ACT"],"industry":[],"navPath":"","tags":[],"facets":{}},"whatItIs":"In differential geometry, a regular smooth foliation of an n-dimensional manifold is a partition into connected immersed submanifolds, called leaves, of a fixed dimension p, with local coordinates in which the connected pieces of leaves are slices obtained by fixing n−p transverse coordinates.","purpose":"Enable an agent to recognise a mathematical foliation, assess evidence for its local and global structure, and determine which constructions and deductions its hypotheses permit.","scope":{"in":["Ambient manifold, leaf dimension, codimension and differentiability assumptions","Foliated charts, plaques and their assembly into maximal connected leaves","Tangent distributions and evidence establishing their integrability","Leaf topology, closures, transverse structure and holonomy","Leaf-space properties and conditions supporting constructions or equivalences","Explicit boundaries between regular foliations and singular or other extended conventions"],"out":["Geological foliation in rocks and foliation as manuscript leaf numbering","General manifold geometry unrelated to the specified leaf decomposition","Arbitrary partitions or stratifications without the required foliation structure","Nonintegrable distributions considered independently of a proposed foliation","Flow timing and vector-field magnitudes beyond the leaves they generate","General fibre-bundle theory beyond its relationship to the specified foliation"],"boundaries":[]},"distinguishingFeatures":["A regular foliation requires compatible local product charts whose plaques describe the decomposition; an arbitrary partition into submanifolds is insufficient.","Its leaves partition the entire ambient manifold, whereas a lamination may occupy only a proper subset.","In the smooth regular setting, its tangent planes form a constant-rank involutive distribution; a plane field lacking involutivity does not qualify. See [Moerdijk and Mrčun, Foliations](https://resolve.cambridge.org/core/services/aop-cambridge-core/content/view/A7953BF5BF52D9DA6CB9518D98AF20D5/9780511615450c1_p4-18_CBO.pdf/foliations.pdf).","A foliation does not automatically supply a global base manifold or locally trivial bundle projection.","A one-dimensional foliation specifies leaves without necessarily specifying an orientation, speed or preferred generating vector field."],"structure":{"bundles":[{"id":"foliation-convention","name":"Foliation convention","description":"Fix the mathematical meaning and ambient assumptions before interpreting a proposed instance.","layers":[{"id":"regularity-and-definition","name":"Regularity and definition","description":"Identify the adopted definition and the category in which it operates.","findings":[{"id":"adopted-foliation-framework","name":"Adopted foliation framework","description":"Record the author's definition and whether the proposal uses a regular atlas, an integrable distribution or a named extension.","questions":[{"text":"Which definition and differentiability category does the author use?","id":"adopted-foliation-framework-q01","kind":"definition"},{"text":"Which source explicitly supplies that definition and its hypotheses?","id":"adopted-foliation-framework-q02","kind":"provenance"},{"text":"Are varying leaf dimensions admitted, and under which singular-foliation framework?","id":"adopted-foliation-framework-q03","kind":"boundary"}]}]},{"id":"ambient-domain","name":"Ambient domain","description":"Specify where the leaf decomposition is asserted to exist.","findings":[{"id":"manifold-and-dimensional-assumptions","name":"Manifold and dimensional assumptions","description":"Record the ambient manifold, dimensions and any boundary conditions or excluded locus.","questions":[{"text":"What are the ambient dimension n, leaf dimension p and codimension n − p?","id":"manifold-and-dimensional-assumptions-q01","kind":"measurement"},{"text":"Does the structure cover all of M, or only a specified open subset or regular locus?","id":"manifold-and-dimensional-assumptions-q02","kind":"boundary"},{"text":"If M has boundary, what tangency or transversality convention is imposed there?","id":"manifold-and-dimensional-assumptions-q03","kind":"definition"}]}]}]},{"id":"local-product-and-integrability","name":"Local product and integrability","description":"Establish the local evidence that distinguishes a foliation from a proposed decomposition or plane field.","layers":[{"id":"foliated-atlas","name":"Foliated atlas","description":"Inspect local product coordinates and compatibility on overlaps.","findings":[{"id":"plaque-preserving-charts","name":"Plaque-preserving charts","description":"Record charts with coordinates (x, y), plaques at fixed y, and transition maps preserving the local plaque structure.","questions":[{"text":"Which charts cover the claimed domain, and which coordinates are leafwise and transverse?","id":"plaque-preserving-charts-q01","kind":"definition"},{"text":"What establishes that transverse transition coordinates are locally independent of leafwise coordinates?","id":"plaque-preserving-charts-q02","kind":"provenance"}]}]},{"id":"distribution-test","name":"Distribution test","description":"Evaluate a differential presentation under its stated regularity assumptions.","findings":[{"id":"constant-rank-and-involutivity","name":"Constant rank and involutivity","description":"For a smooth distribution presentation, record rank checks and closure under Lie brackets as the Frobenius test.","questions":[{"text":"Do the proposed tangent generators span a smooth subbundle of constant rank throughout the domain?","id":"constant-rank-and-involutivity-q01","kind":"measurement"},{"text":"Which bracket computations or theorem establish involutivity?","id":"constant-rank-and-involutivity-q02","kind":"provenance"},{"text":"Where rank or involutivity fails, must the domain be restricted or the claimed framework changed?","id":"constant-rank-and-involutivity-q03","kind":"action"}]}]}]},{"id":"leaves-and-global-assembly","name":"Leaves and global assembly","description":"Track how local plaques form leaves and how those leaves occupy the manifold.","layers":[{"id":"leaf-identification","name":"Leaf identification","description":"Resolve global leaves from the chosen local or differential presentation.","findings":[{"id":"maximal-connected-leaves","name":"Maximal connected leaves","description":"Record how connected chains of plaques or maximal integral manifolds identify individual leaves.","questions":[{"text":"What criterion determines whether two specified points lie on the same leaf?","id":"maximal-connected-leaves-q01","kind":"definition"},{"text":"What establishes that a proposed leaf is maximal rather than a proper local plaque or integral patch?","id":"maximal-connected-leaves-q02","kind":"provenance"}]}]},{"id":"leaf-topology-and-closure","name":"Leaf topology and closure","description":"Keep intrinsic properties distinct from properties of the inclusion into the ambient manifold.","findings":[{"id":"intrinsic-versus-ambient-behaviour","name":"Intrinsic versus ambient behaviour","description":"Record topology, compactness, embeddedness, properness and closure for identified leaves, marking unresolved properties explicitly.","questions":[{"text":"What is known about the leaf's intrinsic topology, and is its inclusion embedded or proper?","id":"intrinsic-versus-ambient-behaviour-q01","kind":"measurement"},{"text":"What is its closure in M, and is the leaf closed or dense in a specified region?","id":"intrinsic-versus-ambient-behaviour-q02","kind":"measurement"},{"text":"Which claims hold for every leaf, and which hold only for selected or exceptional leaves?","id":"intrinsic-versus-ambient-behaviour-q03","kind":"boundary"}]}]}]},{"id":"transverse-behaviour","name":"Transverse behaviour","description":"Record transverse transport and the conditions under which neighbouring leaves can be compared.","layers":[{"id":"holonomy-data","name":"Holonomy data","description":"Specify transverse return maps with their domains and choices.","findings":[{"id":"loop-induced-return-germs","name":"Loop-induced return germs","description":"Record the germ induced by a leafwise loop on a local transversal, retaining basepoint and equivalence conventions.","questions":[{"text":"Which leaf, basepoint, loop and transversal define the proposed holonomy germ?","id":"loop-induced-return-germs-q01","kind":"definition"},{"text":"What calculation or proof establishes whether the relevant holonomy is trivial, finite or otherwise?","id":"loop-induced-return-germs-q02","kind":"provenance"},{"text":"How are results compared after changing the transversal or basepoint?","id":"loop-induced-return-germs-q03","kind":"boundary"}]}]},{"id":"neighbourhood-conclusions","name":"Neighbourhood conclusions","description":"Assess claims about neighbourhoods of leaves and additional transverse structure.","findings":[{"id":"stability-hypotheses","name":"Stability hypotheses","description":"Record the exact stability result proposed and evidence for every required assumption.","questions":[{"text":"Which stability theorem is being invoked, and are its compactness, holonomy and regularity hypotheses established?","id":"stability-hypotheses-q01","kind":"provenance"},{"text":"What neighbourhood conclusion does that theorem permit for this particular leaf?","id":"stability-hypotheses-q02","kind":"action"}]},{"id":"additional-transverse-structure","name":"Additional transverse structure","description":"Record transverse orientation, invariant measures or transverse geometric structures only when supplied or established.","questions":[{"text":"Which transverse structures are asserted, and how are they defined?","id":"additional-transverse-structure-q01","kind":"definition"},{"text":"What verifies their compatibility with the transverse transition or holonomy maps?","id":"additional-transverse-structure-q02","kind":"provenance"}]}]}]},{"id":"quotients-and-permitted-constructions","name":"Quotients and permitted constructions","description":"Determine how the foliation can be represented globally, transformed or compared.","layers":[{"id":"leaf-space","name":"Leaf space","description":"Assess the quotient obtained by identifying points on the same leaf.","findings":[{"id":"quotient-and-fibration-evidence","name":"Quotient and fibration evidence","description":"Record quotient topology and separate evidence for any manifold, submersion or fibre-bundle presentation.","questions":[{"text":"What is established about the separation properties and manifold status of M modulo its leaves?","id":"quotient-and-fibration-evidence-q01","kind":"measurement"},{"text":"If a map to a base is supplied, are its fibres individual leaves or unions of leaves?","id":"quotient-and-fibration-evidence-q02","kind":"boundary"},{"text":"What additional evidence supports a submersion or locally trivial bundle description?","id":"quotient-and-fibration-evidence-q03","kind":"provenance"}]}]},{"id":"construction-and-comparison","name":"Construction and comparison","description":"Specify admissible transformations and the equivalence relation used to compare outputs.","findings":[{"id":"restriction-and-pullback","name":"Restriction and pullback","description":"Record the subset or map involved and verify the conditions needed for an induced foliation.","questions":[{"text":"Is the proposed restriction to an open set, a saturated subset or a transverse submanifold?","id":"restriction-and-pullback-q01","kind":"boundary"},{"text":"For a pullback along a smooth map, what establishes transversality to the foliation?","id":"restriction-and-pullback-q02","kind":"provenance"},{"text":"What are the resulting domain and leaf dimension, and which connected components become leaves?","id":"restriction-and-pullback-q03","kind":"action"}]},{"id":"foliated-equivalence","name":"Foliated equivalence","description":"Record whether comparison means the same foliation on M, equivalence by a leaf-preserving diffeomorphism or another explicitly defined relation.","questions":[{"text":"Which equivalence relation is required for this comparison?","id":"foliated-equivalence-q01","kind":"definition"},{"text":"What map or atlas-compatibility argument establishes the claimed equivalence?","id":"foliated-equivalence-q02","kind":"provenance"},{"text":"Must the comparison preserve leaf orientation, transverse orientation or additional transverse structure?","id":"foliated-equivalence-q03","kind":"boundary"}]}]}]},{"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-confidence-notes","name":"Check these first","description":"Recalled without web access and unsourced; every item is a lead to verify.","questions":[{"text":"Which of these check these first hold for the sense of foliation this model covers, and on what evidence?","id":"evidence-confidence-notes-q01","kind":"provenance"}]},{"id":"evidence-kinds","name":"Kinds and varieties","description":"Recalled without web access and unsourced; every item is a lead to verify.","questions":[{"text":"Which of these kinds and varieties hold for the sense of foliation this model covers, and on what evidence?","id":"evidence-kinds-q01","kind":"provenance"}]},{"id":"evidence-real-world-use","name":"Real-world use","description":"Recalled without web access and unsourced; every item is a lead to verify.","questions":[{"text":"Which of these real-world use hold for the sense of foliation this model covers, and on what evidence?","id":"evidence-real-world-use-q01","kind":"provenance"}]},{"id":"evidence-measurements","name":"Typical measurements","description":"Recalled without web access and unsourced; every item is a lead to verify.","questions":[{"text":"Which of these typical measurements hold for the sense of foliation 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":"Recalled without web access and unsourced; every item is a lead to verify.","questions":[{"text":"Which of these failure modes and hazards hold for the sense of foliation this model covers, and on what evidence?","id":"evidence-failure-modes-and-hazards-q01","kind":"provenance"}]},{"id":"evidence-neighbours","name":"Neighbouring kinds and how to tell them apart","description":"Recalled without web access and unsourced; every item is a lead to verify.","questions":[{"text":"Which of these neighbouring kinds and how to tell them apart hold for the sense of foliation this model covers, and on what evidence?","id":"evidence-neighbours-q01","kind":"provenance"}]}]}]}]},"agentConduct":{"may":["Explain foliations, their leaves and standard examples such as the Reeb foliation.","Check stated definitions and conditions when helping with proofs."],"mustNot":["Present an unproved statement about foliations as a theorem.","Confuse regular foliations with singular ones in results others rely on.","Copy proofs or figures from published works without credit."],"requiresHuman":["Publishing new claims about foliations under a person's name."]},"ethics":{"considerations":["Accurate attribution and correct proofs matter in mathematical research and teaching."],"affectedParties":["Students","Mathematicians","Physicists using the theory"]},"owners":{"steward":"Nobody owns it; it is a concept of differential geometry.","roles":[],"masterSystems":[]},"relations":[{"target":"Q381060","type":"parent","note":"registry parent class"},{"target":"Q5638159","type":"parent","note":"registry parent class"}],"interaction":{"identity":{"applicability":"required","items":["Vercy registry: vr.tr.foliation-q684220","Wikidata: Q684220 (https://www.wikidata.org/wiki/Q684220)"]},"properties":{"applicability":"not-applicable","items":["Ambient and leaf dimensions: Integers n and p with 0 ≤ p ≤ n; codimension q = n − p","Regularity convention: Topological, C^r with r specified, smooth, real analytic or holomorphic","Definition framework: Regular foliated atlas; regular integrable distribution; explicitly named singular framework","Integrability assessment: Established, disproved, unresolved or not applicable to this presentation","Leaf inclusion and closure: For each specified leaf: inclusion into M, intrinsic topology and closure in M","Leaf holonomy: Based leaf loops mapped to germs of transverse return maps, with transversal and basepoint recorded","Orientation data: Orientability and chosen orientation recorded separately for tangent and normal bundles; unknown allowed","Leaf-space assessment: Separation properties, manifold status and quotient-map regularity, each established or unresolved"]},"recognition":{"applicability":"not-applicable","items":[]},"capabilities":{"applicability":"required","items":["Check whether proposed charts, equations or tangent planes define a foliation under stated assumptions.","Recover local plaques and identify maximal leaves where the available presentation permits.","Assess leaf closures and transverse return behaviour using explicit examples or proofs.","Determine whether restriction, transverse pullback or transport by diffeomorphism preserves the required structure.","Compare two presentations using a specified notion of foliated equivalence.","Gate theorem applications on verified regularity, compactness, holonomy and boundary hypotheses."]},"hazards":{"applicability":"required","items":["Mixing up local and global conditions leads to false proofs."]},"interfaces":{"applicability":"not-applicable","items":[]},"context":{"applicability":"required","items":["Foliations are used in differential geometry and topology to study manifolds through families of connected leaves with a consistent local structure.","They also arise in dynamical systems and the study of integrable distributions, where local solutions organize into geometric subspaces."]}},"sources":[{"title":"Wikidata item Q684220: foliation","url":"https://www.wikidata.org/wiki/Q684220","note":"identity and sense of the item"},{"title":"Wikipedia: Foliation","url":"https://en.wikipedia.org/wiki/Foliation","note":"general description of the item"}],"openQuestions":["Should this registry entry's eventual researched model encompass named singular-foliation frameworks, or describe them only as explicit extensions of the regular baseline?","Which source-backed conventions should govern topological regularity and manifolds with boundary?","Does an existing Vercy world model already own this mathematical concept, requiring a registry link instead of a second publication?","Which worked examples should validate recognition of compact leaves, dense leaves and nontrivial holonomy?","Which equivalence notion and level of proof should an agent require before treating two computational or symbolic presentations as the same foliation?"],"resources":{"spec":"/models/things/publications/thing-q684220/spec.json"},"provenance":{"origin":"thing registry research (pass 2)","builtFrom":["models/things/publications/thing-q684220/spec.json"],"providers":["Codex"],"researchStatus":"unreviewed","generatedAt":"2026-09-12T09:48:47Z","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":"not-applicable","interaction.context":"filled","sources":"filled"},"notes":{"_gate":"Published through the thing publication gate 1.0 on 2026-10-07T10:10:32+00:00: completeness 1.00; sense: Wikidata description; 2 live sources; review by codex: pass","interaction.properties":"Plane ACT: no invented physical properties.","interaction.interfaces":"A mathematical foliation is governed by its stated definitions and hypotheses rather than an applicable regulatory or technical standard.","interaction.recognition":"A foliation is an abstract mathematical structure whose defining properties cannot be established by visual inspection of a diagram."},"score":1.0}}