# Structural Depth Wave4, First Slice: Abelian Groups, Norms, Cauchy Sequences, Joint Laws, Covariance, Graph Isomorphisms and Equivalence Relations Complete seven traceable knowledge objects in dependency order on the existing number-system and function foundations. This is a suggested learning sequence, not a universal logical dependency or a formal proof. - Stage 1 · required ## [Finite group](/en/mko/mko-finite-group/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261026%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-finite-group) [Concept guides](/en/derivations/?q=mko-finite-group) - Stage 1 · required ## [Quantifiers and explicit domains](/en/mko/mko-quantifier/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261026%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-quantifier) [Concept guides](/en/derivations/?q=mko-quantifier) - Stage 1 · required ## [Real vector space](/en/mko/mko-real-vector-space/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261026%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-real-vector-space) [Concept guides](/en/derivations/?q=mko-real-vector-space) - Stage 1 · required ## [Absolute value](/en/mko/mko-absolute-value/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261026%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-absolute-value) [Concept guides](/en/derivations/?q=mko-absolute-value) - Stage 1 · required ## [Usual order on the real numbers](/en/mko/mko-order-relation/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261026%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-order-relation) [Concept guides](/en/derivations/?q=mko-order-relation) - Stage 1 · required ## [Metric space](/en/mko/mko-metric-space/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261026%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-metric-space) [Concept guides](/en/derivations/?q=mko-metric-space) - Stage 1 · required ## [Sequence](/en/mko/mko-sequence/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261026%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-sequence) [Concept guides](/en/derivations/?q=mko-sequence) - Stage 1 · required ## [Finite discrete probability distribution](/en/mko/mko-finite-discrete-probability-distribution/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261026%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-finite-discrete-probability-distribution) [Concept guides](/en/derivations/?q=mko-finite-discrete-probability-distribution) - Stage 1 · required ## [Real random variables on finite sample spaces](/en/mko/mko-random-variable/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261026%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-random-variable) [Concept guides](/en/derivations/?q=mko-random-variable) - Stage 1 · required ## [Probability on finite sample spaces](/en/mko/mko-probability/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261026%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-probability) [Concept guides](/en/derivations/?q=mko-probability) - Stage 1 · required ## [Sample space](/en/mko/mko-sample-space/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261026%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-sample-space) [Concept guides](/en/derivations/?q=mko-sample-space) - Stage 1 · required ## [Expectation in finite models](/en/mko/mko-expectation/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261026%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-expectation) [Concept guides](/en/derivations/?q=mko-expectation) - Stage 1 · required ## [Variance in finite probability models](/en/mko/mko-variance/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261026%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-variance) [Concept guides](/en/derivations/?q=mko-variance) - Stage 1 · required ## [Finite simple undirected graphs](/en/mko/mko-graph/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261026%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-graph) [Concept guides](/en/derivations/?q=mko-graph) - Stage 1 · required ## [Bijectivity and unique preimages](/en/mko/mko-bijective/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261026%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-bijective) [Concept guides](/en/derivations/?q=mko-bijective) - Stage 1 · required ## [Binary relations and set projections](/en/mko/mko-relation/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261026%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-relation) [Concept guides](/en/derivations/?q=mko-relation) - Stage 1 · required ## [Logical connectives](/en/mko/mko-logical-connectives/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261026%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-logical-connectives) [Concept guides](/en/derivations/?q=mko-logical-connectives) - Stage 2 · optional ## [Finite abelian group](/en/mko/mko-finite-abelian-group/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261026%2F) A finite group is abelian when its complete Cayley table satisfies commutativity for every pair of elements. [Related questions](/en/explore/?mko=mko-finite-abelian-group) [Concept guides](/en/derivations/?q=mko-finite-abelian-group) - Stage 3 · optional ## [Real normed vector space](/en/mko/mko-real-normed-vector-space/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261026%2F) A real vector space equipped with a positive-definite, absolutely homogeneous real-valued norm satisfying the triangle inequality. [Related questions](/en/explore/?mko=mko-real-normed-vector-space) [Concept guides](/en/derivations/?q=mko-real-normed-vector-space) - Stage 4 · optional ## [Cauchy sequence](/en/mko/mko-cauchy-sequence/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261026%2F) A sequence in a metric space is Cauchy if, for every positive epsilon, all pairs of sufficiently late terms are less than epsilon apart. [Related questions](/en/explore/?mko=mko-cauchy-sequence) [Concept guides](/en/derivations/?q=mko-cauchy-sequence) - Stage 5 · optional ## [Finite joint distribution](/en/mko/mko-finite-joint-distribution/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261026%2F) Value tuples of random variables on one common finite probability space, together with exact masses, specify a finite joint distribution. [Related questions](/en/explore/?mko=mko-finite-joint-distribution) [Concept guides](/en/derivations/?q=mko-finite-joint-distribution) - Stage 6 · optional ## [Covariance](/en/mko/mko-covariance/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261026%2F) Under one finite joint distribution, covariance is E[XY]−E[X]E[Y]; zero covariance does not in general imply independence. [Related questions](/en/explore/?mko=mko-covariance) [Concept guides](/en/derivations/?q=mko-covariance) - Stage 7 · optional ## [Graph isomorphism](/en/mko/mko-graph-isomorphism/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261026%2F) An isomorphism of finite simple graphs is a vertex bijection preserving both adjacency and nonadjacency for every pair. [Related questions](/en/explore/?mko=mko-graph-isomorphism) [Concept guides](/en/derivations/?q=mko-graph-isomorphism) - Stage 8 · optional ## [Equivalence relation](/en/mko/mko-equivalence-relation/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261026%2F) A relation on a fixed carrier is an equivalence relation when it fully satisfies reflexivity, symmetry and transitivity. [Related questions](/en/explore/?mko=mko-equivalence-relation) [Concept guides](/en/derivations/?q=mko-equivalence-relation) Path source & statuspath-pulse-foundations-20261026 · candidate [Original paths JSON](/data/architecture/learning-paths.json)