# Wave4 Closure: Countability, Bipartite Graphs, Distributional Convergence, Quotient Rings, Coloring and Two Depth Extensions 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 ## [Cardinality comparison of sets](/en/mko/mko-cardinality-comparison/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261031%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-cardinality-comparison) [Concept guides](/en/derivations/?q=mko-cardinality-comparison) - Stage 1 · required ## [Natural number](/en/mko/mko-natural-number/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261031%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-natural-number) [Concept guides](/en/derivations/?q=mko-natural-number) - Stage 1 · required ## [Injectivity and distinct inputs](/en/mko/mko-injective/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261031%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-injective) [Concept guides](/en/derivations/?q=mko-injective) - Stage 1 · required ## [Surjectivity and codomain coverage](/en/mko/mko-surjective/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261031%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-surjective) [Concept guides](/en/derivations/?q=mko-surjective) - Stage 1 · required ## [Finite simple undirected graphs](/en/mko/mko-graph/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261031%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 ## [Subset](/en/mko/mko-subset/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261031%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-subset) [Concept guides](/en/derivations/?q=mko-subset) - Stage 1 · required ## [Function mapping](/en/mko/mko-function-mapping/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261031%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-function-mapping) [Concept guides](/en/derivations/?q=mko-function-mapping) - Stage 1 · required ## [Cumulative distribution function](/en/mko/mko-cumulative-distribution-function/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261031%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-cumulative-distribution-function) [Concept guides](/en/derivations/?q=mko-cumulative-distribution-function) - Stage 1 · required ## [Function limits on punctured real domains](/en/mko/mko-function-limit/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261031%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-function-limit) [Concept guides](/en/derivations/?q=mko-function-limit) - Stage 1 · required ## [Sequence](/en/mko/mko-sequence/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261031%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 joint distribution](/en/mko/mko-finite-joint-distribution/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261031%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-finite-joint-distribution) [Concept guides](/en/derivations/?q=mko-finite-joint-distribution) - Stage 1 · required ## [Finite unital ring](/en/mko/mko-finite-unital-ring/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261031%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-finite-unital-ring) [Concept guides](/en/derivations/?q=mko-finite-unital-ring) - Stage 1 · required ## [Finite two-sided ideal](/en/mko/mko-finite-two-sided-ideal/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261031%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-finite-two-sided-ideal) [Concept guides](/en/derivations/?q=mko-finite-two-sided-ideal) - Stage 1 · required ## [Equivalence class](/en/mko/mko-equivalence-class/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261031%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-equivalence-class) [Concept guides](/en/derivations/?q=mko-equivalence-class) - Stage 1 · required ## [Quotient set](/en/mko/mko-quotient-set/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261031%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-quotient-set) [Concept guides](/en/derivations/?q=mko-quotient-set) - Stage 1 · required ## [Usual order on the real numbers](/en/mko/mko-order-relation/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261031%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 ## [Finite integral domain](/en/mko/mko-finite-integral-domain/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261031%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-finite-integral-domain) [Concept guides](/en/derivations/?q=mko-finite-integral-domain) - Stage 1 · required ## [Finite abelian group](/en/mko/mko-finite-abelian-group/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261031%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-finite-abelian-group) [Concept guides](/en/derivations/?q=mko-finite-abelian-group) - Stage 1 · required ## [Finite filtration](/en/mko/mko-finite-filtration/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261031%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-finite-filtration) [Concept guides](/en/derivations/?q=mko-finite-filtration) - Stage 1 · required ## [Finite conditional expectation](/en/mko/mko-conditional-expectation/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261031%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-conditional-expectation) [Concept guides](/en/derivations/?q=mko-conditional-expectation) - Stage 1 · required ## [Discrete-time stochastic process](/en/mko/mko-discrete-stochastic-process/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261031%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-discrete-stochastic-process) [Concept guides](/en/derivations/?q=mko-discrete-stochastic-process) - Stage 1 · required ## [Expectation in finite models](/en/mko/mko-expectation/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261031%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 ## [Measurable function](/en/mko/mko-measurable-function/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261031%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-measurable-function) [Concept guides](/en/derivations/?q=mko-measurable-function) - Stage 2 · optional ## [At-most-countable set](/en/mko/mko-at-most-countable-set/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261031%2F) A set is at most countable if it is finite or has an explicit injection into the natural numbers; this is not the same as countably infinite. [Related questions](/en/explore/?mko=mko-at-most-countable-set) [Concept guides](/en/derivations/?q=mko-at-most-countable-set) - Stage 2 · optional ## [Bipartite graph](/en/mko/mko-bipartite-graph/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261031%2F) A finite undirected simple graph is bipartite when all vertices split into two disjoint parts and every edge crosses between the parts. [Related questions](/en/explore/?mko=mko-bipartite-graph) [Concept guides](/en/derivations/?q=mko-bipartite-graph) - Stage 2 · optional ## [Convergence in distribution](/en/mko/mko-convergence-in-distribution/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261031%2F) Convergence in distribution means that F_n tends pointwise to F at every continuity point of the limiting CDF. [Related questions](/en/explore/?mko=mko-convergence-in-distribution) [Concept guides](/en/derivations/?q=mko-convergence-in-distribution) - Stage 2 · optional ## [Quotient ring of a finite ring](/en/mko/mko-finite-quotient-ring/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261031%2F) Additive cosets of a proper two-sided ideal in a finite ring form a quotient ring when its operations are independent of representative choice. [Related questions](/en/explore/?mko=mko-finite-quotient-ring) [Concept guides](/en/derivations/?q=mko-finite-quotient-ring) - Stage 2 · optional ## [Graph coloring](/en/mko/mko-graph-coloring/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261031%2F) A proper k-coloring uses at most k colors and gives different colors to the endpoints of every edge. [Related questions](/en/explore/?mko=mko-graph-coloring) [Concept guides](/en/derivations/?q=mko-graph-coloring) - Stage 2 · optional ## [Finite field](/en/mko/mko-finite-field/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261031%2F) Deepen characteristic, nonzero multiplicative groups, affine equations and Frobenius identities in the complete finite-field model. [Related questions](/en/explore/?mko=mko-finite-field) [Concept guides](/en/derivations/?q=mko-finite-field) - Stage 2 · optional ## [Finite martingale](/en/mko/mko-martingale/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261031%2F) Deepen martingale differences, affine stability, fair partial sums and constant-expectation counterexamples under finite filtrations. [Related questions](/en/explore/?mko=mko-martingale) [Concept guides](/en/derivations/?q=mko-martingale) Path source & statuspath-pulse-foundations-20261031 · candidate [Original paths JSON](/data/architecture/learning-paths.json)