MATH · IN · MODELS

Structure genealogy

Each structure occupies one row, placed at every year a paper reported it; rows are ordered by first appearance (oldest at the bottom). Dot colour marks the object's role, ordered by the year that role first entered the map — so warmer rows arrived later. Dot size = papers that year; a star marks a structure reported by at least two papers that share no author — computed, not hand-assigned. The panel below stacks the same reports into activity per role per year.

role · oldest → newestdynamical objectgeometric objectoperatorconfigurationlabeled data propertymeasurementdistribution propertymetric modelempirical patternhypothesiscombinatorial object
2013201320142014201520152016201620172017201820182019201920202020202120212022202220232023202420242025202520262026Line AttractorLine Attractor · 2013 · 1 paperLine Attractor · 2019 · 1 paperLine Attractor · 2020 · 2 papersLine Attractor · 2025 · 1 paperLine Attractor · 2026 · 1 paper★Linear DirectionLinear Direction · 2013 · 1 paperLinear Direction · 2016 · 2 papersLinear Direction · 2017 · 1 paperLinear Direction · 2018 · 2 papersLinear Direction · 2019 · 2 papersLinear Direction · 2020 · 5 papersLinear Direction · 2021 · 3 papersLinear Direction · 2022 · 8 papersLinear Direction · 2023 · 22 papersLinear Direction · 2024 · 44 papersLinear Direction · 2025 · 75 papersLinear Direction · 2026 · 157 papers★Linear SubspaceLinear Subspace · 2013 · 1 paperLinear Subspace · 2016 · 1 paperLinear Subspace · 2018 · 3 papersLinear Subspace · 2019 · 5 papersLinear Subspace · 2020 · 5 papersLinear Subspace · 2021 · 6 papersLinear Subspace · 2022 · 5 papersLinear Subspace · 2023 · 19 papersLinear Subspace · 2024 · 11 papersLinear Subspace · 2025 · 32 papersLinear Subspace · 2026 · 73 papers★Conceptor (soft ellipsoidal regio…Conceptor (soft ellipsoidal region) · 2014 · 1 paperConceptor (soft ellipsoidal region) · 2024 · 1 paperConceptor (soft ellipsoidal region) · 2026 · 1 paper★Concept Crystals (Parallelogram/T…Concept Crystals (Parallelogram/Trapezoid Structure) · 2016 · 1 paperConcept Crystals (Parallelogram/Trapezoid Structure) · 2019 · 2 papersConcept Crystals (Parallelogram/Trapezoid Structure) · 2020 · 1 paperConcept Crystals (Parallelogram/Trapezoid Structure) · 2024 · 3 papersConcept Crystals (Parallelogram/Trapezoid Structure) · 2025 · 3 papersConcept Crystals (Parallelogram/Trapezoid Structure) · 2026 · 1 paper★Linear SeparabilityLinear Separability · 2016 · 1 paperLinear Separability · 2017 · 1 paperLinear Separability · 2018 · 1 paperLinear Separability · 2020 · 3 papersLinear Separability · 2021 · 2 papersLinear Separability · 2022 · 1 paperLinear Separability · 2023 · 4 papersLinear Separability · 2024 · 7 papersLinear Separability · 2025 · 11 papersLinear Separability · 2026 · 36 papers★SphereSphere · 2017 · 1 paperSphere · 2019 · 1 paperSphere · 2020 · 1 paperSphere · 2025 · 1 paper★Representational-similarity traje…Representational-similarity trajectory across depth · 2017 · 1 paperRepresentational-similarity trajectory across depth · 2018 · 1 paperRepresentational-similarity trajectory across depth · 2019 · 2 papersRepresentational-similarity trajectory across depth · 2024 · 1 paperRepresentational-similarity trajectory across depth · 2025 · 2 papersRepresentational-similarity trajectory across depth · 2026 · 3 papers★CircleCircle · 2018 · 1 paperCircle · 2019 · 1 paperCircle · 2023 · 5 papersCircle · 2024 · 3 papersCircle · 2025 · 8 papersCircle · 2026 · 17 papers★Linear region arrangement (polyhe…Linear region arrangement (polyhedral tessellation of input space) · 2018 · 1 paperLinear region arrangement (polyhedral tessellation of input space) · 2024 · 1 paperLinear region arrangement (polyhedral tessellation of input space) · 2026 · 1 paper★TorusTorus · 2018 · 2 papersTorus · 2020 · 1 paperTorus · 2023 · 1 paperTorus · 2024 · 1 paperTorus · 2025 · 2 papers★Curvature profile of the represen…Curvature profile of the representation manifold · 2018 · 4 papersCurvature profile of the representation manifold · 2021 · 1 paperCurvature profile of the representation manifold · 2023 · 3 papersCurvature profile of the representation manifold · 2024 · 1 paperCurvature profile of the representation manifold · 2025 · 3 papersCurvature profile of the representation manifold · 2026 · 9 papers★AnisotropyAnisotropy · 2018 · 1 paperAnisotropy · 2019 · 2 papersAnisotropy · 2020 · 1 paperAnisotropy · 2021 · 6 papersAnisotropy · 2022 · 1 paperAnisotropy · 2023 · 2 papersAnisotropy · 2024 · 4 papersAnisotropy · 2025 · 6 papersAnisotropy · 2026 · 17 papers★Fixed-point graph topology (popul…Fixed-point graph topology (population-level universality of RNN dynamical structure) · 2019 · 1 paperFixed-point graph topology (population-level universality of RNN dynamical structure) · 2025 · 1 paperFixed-point graph topology (population-level universality of RNN dynamical structure) · 2026 · 1 paper★1D continuum manifold1D continuum manifold · 2019 · 1 paper1D continuum manifold · 2025 · 5 papers1D continuum manifold · 2026 · 8 papers★Polytope (Simplex)Polytope (Simplex) · 2019 · 1 paperPolytope (Simplex) · 2020 · 1 paperPolytope (Simplex) · 2021 · 1 paperPolytope (Simplex) · 2022 · 2 papersPolytope (Simplex) · 2023 · 1 paperPolytope (Simplex) · 2024 · 4 papersPolytope (Simplex) · 2025 · 4 papersPolytope (Simplex) · 2026 · 8 papers★Intrinsic-dimension profile acros…Intrinsic-dimension profile across depth · 2019 · 2 papersIntrinsic-dimension profile across depth · 2021 · 1 paperIntrinsic-dimension profile across depth · 2023 · 4 papersIntrinsic-dimension profile across depth · 2024 · 8 papersIntrinsic-dimension profile across depth · 2025 · 12 papersIntrinsic-dimension profile across depth · 2026 · 23 papers★Tree Metric EmbeddingTree Metric Embedding · 2019 · 2 papersTree Metric Embedding · 2020 · 2 papersTree Metric Embedding · 2021 · 4 papersTree Metric Embedding · 2022 · 2 papersTree Metric Embedding · 2024 · 1 paperTree Metric Embedding · 2025 · 2 papersTree Metric Embedding · 2026 · 2 papers★Decision boundary (as a codimensi…Decision boundary (as a codimension-1 hypersurface) · 2020 · 2 papersDecision boundary (as a codimension-1 hypersurface) · 2025 · 2 papersDecision boundary (as a codimension-1 hypersurface) · 2026 · 3 papers★Pinched Manifold (Singular Quotie…Pinched Manifold (Singular Quotient) · 2020 · 1 paperAffine SubspaceAffine Subspace · 2021 · 1 paperAffine Subspace · 2023 · 1 paperAffine Subspace · 2024 · 2 papersAffine Subspace · 2026 · 2 papers★Persistent-homology / Betti profi…Persistent-homology / Betti profile across depth · 2022 · 1 paperPersistent-homology / Betti profile across depth · 2024 · 1 paperPersistent-homology / Betti profile across depth · 2026 · 1 paper★Dimensional collapseDimensional collapse · 2022 · 1 paperDimensional collapse · 2023 · 1 paperDimensional collapse · 2024 · 2 papersDimensional collapse · 2025 · 7 papersDimensional collapse · 2026 · 11 papers★Limit cycle (stable periodic attr…Limit cycle (stable periodic attractor) · 2023 · 1 paperLimit cycle (stable periodic attractor) · 2026 · 1 paper★Relation-decoding operator (affin…Relation-decoding operator (affine map between representation spaces) · 2023 · 1 paperRelation-decoding operator (affine map between representation spaces) · 2025 · 1 paper★Nonlinear World-Model DecodabilityNonlinear World-Model Decodability · 2023 · 1 paperNonlinear World-Model Decodability · 2026 · 1 paper★ConeCone · 2024 · 3 papersCone · 2025 · 2 papersCone · 2026 · 2 papers★Lissajous CurvesLissajous Curves · 2024 · 1 paperLissajous Curves · 2025 · 4 papersLissajous Curves · 2026 · 1 paper★Representation-Weight-Gradient Al…Representation-Weight-Gradient Alignment (Canonical Representation Hypothesis) · 2024 · 1 paperFeature Lobes (Spatial-Functional…Feature Lobes (Spatial-Functional Modularity) · 2024 · 1 paperBelief State Geometry Hypothesis …Belief State Geometry Hypothesis (Mixed-State Presentation) · 2024 · 1 paperBelief State Geometry Hypothesis (Mixed-State Presentation) · 2025 · 1 paper★Ghost Point (saddle-node bifurcat…Ghost Point (saddle-node bifurcation remnant) · 2025 · 1 paperGeneralized HelixGeneralized Helix · 2025 · 1 paperGeneralized Helix · 2026 · 1 paper★Minkowski Sum of Tile Polytopes (…Minkowski Sum of Tile Polytopes (Minkowski Representation Hypothesis) · 2025 · 1 paperAttribute-Induced Embedding Foldi…Attribute-Induced Embedding Folding · 2025 · 1 paperContinuous causal attention opera…Continuous causal attention operator (CCT) · 2025 · 1 paperSymmetric/skew decomposition of t…Symmetric/skew decomposition of the attention query-key operator · 2025 · 1 paperAttention reference frame (sink-t…Attention reference frame (sink-token anchor configuration) · 2025 · 1 paperAttention-weight frequency-band s…Attention-weight frequency-band specialization (DFT/wavelet decomposition of attention as a position-indexed signal) · 2025 · 1 paperCumulant-expansion profile of the…Cumulant-expansion profile of the next-token softmax entropy · 2025 · 1 paperSemantic information production r…Semantic information production rate along the diffusion trajectory · 2025 · 1 paperScore-Jacobian pullback Riemannia…Score-Jacobian pullback Riemannian metric · 2025 · 1 paperConcept lattice (Formal Concept A…Concept lattice (Formal Concept Analysis half-space model) · 2025 · 1 paperConcept lattice (Formal Concept Analysis half-space model) · 2026 · 2 papers★Hyperbolic ManifoldHyperbolic Manifold · 2026 · 2 papers★Laguerre-Voronoi partition (weigh…Laguerre-Voronoi partition (weighted power diagram of a linear readout layer) · 2026 · 1 paperLocal hyperboloid-type quadric pa…Local hyperboloid-type quadric patch · 2026 · 1 paperParaboloid (circular × pinched co…Paraboloid (circular × pinched continuum) · 2026 · 1 paperStratified Manifold with Continuo…Stratified Manifold with Continuous Fibers · 2026 · 1 paperFinite-Lag Transport TensorFinite-Lag Transport Tensor · 2026 · 1 paperJacobian non-normality (Schur dec…Jacobian non-normality (Schur decomposition of per-layer operators) · 2026 · 1 paperRelation frame (ordered multi-tok…Relation frame (ordered multi-token tuple geometry) · 2026 · 1 paperAttention-graph spectral profile …Attention-graph spectral profile (Fiedler value, HFER, smoothness, spectral entropy) · 2026 · 1 paperSpatial-frequency (spectral) acce…Spatial-frequency (spectral) accessibility profile across depth · 2026 · 1 paperTask-induced symmetry class of a …Task-induced symmetry class of a learned representation · 2026 · 1 paperDiffusion Spacetime Information G…Diffusion Spacetime Information Geometry · 2026 · 1 paperConcept Cluster HeterogeneityConcept Cluster Heterogeneity · 2026 · 3 papers★reports per role per year (collection density, not real-world activity)0200399

This is not a claim that one structure was literally derived from another — the schema doesn't encode structure-to-structure lineage. It shows when each shape entered the literature, how sustained the attention was, and how the mix of mathematical roles shifted over time.

Not yet dated (no paper in data/papers/ lists them) — 1 structure: 2D Grid (Square Lattice).