Classification — the road of classes — Math in LLMs Research Map

Classification — the road of classes

Built live from the theory collection. An object travels root → ontological sort → role → facets → leaf; every level asks one criterion. Leaves are clickable. The invariants doing the splitting are real: curvature $K$ , homology $H_1$ , spectrum $0\preceq C\preceq I$ .

invariant basis · role + facetssingle basis · one criterion / levelmutually exclusive · one class / leveldata-driven · derived from theory nodes

An object tracked in the map

ontological sort — object / tool / property / claim

OBJECT — a thing in the space

role

geometric-object

regularity

curvature

K

→ decision-boundary manifolds-helix manifolds-hyperboloid manifolds-paraboloid manifolds-stratified-fiber

topology / embedding

→ laguerre-voronoi-partition linear-region-arrangement

→ linear-direction linear-subspace manifolds-affine

→ manifolds-1d-continuum

→ manifolds-circle manifolds-grid manifolds-torus

→ manifolds-hyperbolic

→ manifolds-sphere

→ lissajous-curves manifolds-cone pinched-manifold

→ manifolds-polytope minkowski-sum-polytope

configuration

→ attention-reference-frame concept-crystals relation-frame representation-weight-gradient-alignment

combinatorial-object

→ concept-lattice

dynamical-object

→ fixed-point-topology ghost-point limit-cycle line-attractor

TOOL — acts on the cloud

role

operator

→ attention-qk-symmetry attribute-induced-embedding-folding conceptor-operator continuous-causal-attention-operator finite-lag-transport-tensor jacobian-non-normality relation-decoding-operator

metric-model

→ diffusion-spacetime-information-geometry score-jacobian-pullback-metric tree-metric-probe

PROPERTY — a measured characteristic

source of the quantity

distribution-property

→ anisotropy dimensional-collapse task-induced-symmetry-class

labeled-data-property

→ linear-separability nonlinear-world-model-decodability

measurement

→ attention-frequency-specialization attention-graph-spectral-profile cumulant-entropy-profile diffusion-semantic-information-rate intrinsic-dimension-profile manifold-curvature-profile persistent-homology-profile representational-similarity-trajectory spectral-accessibility-profile

empirical-pattern

→ concept-cluster-heterogeneity feature-lobes

CLAIM — a general assertion

role

hypothesis

→ aristotelian-representation-hypothesis belief-state-simplex conceptual-belief-space constraint-algebra-basis constructive-interference efficient-coding-manifolds linear-centroids-hypothesis lrh minkowski-representation platonic-representation spatial-functional-modularity sufficiency-staging tangent-aligned-anisotropy translation-symmetry

Root → sort → role → leaf

The same road, drawn as a schematic: every object in the map enters at the root, is placed in an ontological sort, then in a role, and lands on a leaf. Leaves are clickable.

root

an object in the map

any shape found in a trained model

ontological sort

OBJECT

lives in the space

TOOL

acts on the cloud

PROPERTY

a measured trait

role · type

geometric-object