# Wave6 Second Layer: Orbits, Monotone Limits, Subspace Topology, Chebyshev, Inclusion-Exclusion, Prime Factorization and Digraphs 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 action](/en/mko/mko-finite-group-action/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261108%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-finite-group-action) [Concept guides](/en/derivations/?q=mko-finite-group-action) - Stage 1 · required ## [Equivalence class](/en/mko/mko-equivalence-class/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261108%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 ## [Supremum and infimum](/en/mko/mko-supremum-and-infimum/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261108%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-supremum-and-infimum) [Concept guides](/en/derivations/?q=mko-supremum-and-infimum) - Stage 1 · required ## [Sequence limits and exact tail cutoffs](/en/mko/mko-sequence-limit/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261108%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-sequence-limit) [Concept guides](/en/derivations/?q=mko-sequence-limit) - Stage 1 · required ## [Usual order on the real numbers](/en/mko/mko-order-relation/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261108%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 ## [Topological space](/en/mko/mko-topological-space/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261108%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-topological-space) [Concept guides](/en/derivations/?q=mko-topological-space) - Stage 1 · required ## [Subset](/en/mko/mko-subset/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261108%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 ## [Topological basis](/en/mko/mko-topological-basis/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261108%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-topological-basis) [Concept guides](/en/derivations/?q=mko-topological-basis) - Stage 1 · required ## [Markov inequality](/en/mko/mko-markov-inequality/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261108%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-markov-inequality) [Concept guides](/en/derivations/?q=mko-markov-inequality) - Stage 1 · required ## [Variance in finite probability models](/en/mko/mko-variance/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261108%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 ## [Absolute value](/en/mko/mko-absolute-value/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261108%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 ## [Set operations](/en/mko/mko-set-operations/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261108%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-set-operations) [Concept guides](/en/derivations/?q=mko-set-operations) - Stage 1 · required ## [Basic counting principles](/en/mko/mko-counting-principle/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261108%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-counting-principle) [Concept guides](/en/derivations/?q=mko-counting-principle) - Stage 1 · required ## [Cardinality comparison of sets](/en/mko/mko-cardinality-comparison/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261108%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 ## [Mathematical induction](/en/mko/mko-mathematical-induction/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261108%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-mathematical-induction) [Concept guides](/en/derivations/?q=mko-mathematical-induction) - Stage 1 · required ## [Prime natural number](/en/mko/mko-prime-number/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261108%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-prime-number) [Concept guides](/en/derivations/?q=mko-prime-number) - Stage 1 · required ## [Natural-number divisibility](/en/mko/mko-natural-divisibility/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261108%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-natural-divisibility) [Concept guides](/en/derivations/?q=mko-natural-divisibility) - Stage 1 · required ## [Euclidean algorithm](/en/mko/mko-euclidean-algorithm/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261108%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-euclidean-algorithm) [Concept guides](/en/derivations/?q=mko-euclidean-algorithm) - Stage 1 · required ## [Greatest common divisor and coprimality](/en/mko/mko-gcd-coprime/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261108%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-gcd-coprime) [Concept guides](/en/derivations/?q=mko-gcd-coprime) - Stage 1 · required ## [Binary relations and set projections](/en/mko/mko-relation/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261108%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 ## [Set](/en/mko/mko-set/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261108%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-set) [Concept guides](/en/derivations/?q=mko-set) - Stage 2 · optional ## [Orbit of a finite group action](/en/mko/mko-finite-group-orbit/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261108%2F) The images of one point under all group elements form its orbit; all orbits partition the point carrier. [Related questions](/en/explore/?mko=mko-finite-group-orbit) [Concept guides](/en/derivations/?q=mko-finite-group-orbit) - Stage 3 · optional ## [Monotone bounded sequence convergence theorem](/en/mko/mko-monotone-sequence-convergence-theorem/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261108%2F) A nondecreasing real sequence bounded above converges to the supremum of its terms; the nonincreasing bounded-below version converges to the infimum. [Related questions](/en/explore/?mko=mko-monotone-sequence-convergence-theorem) [Concept guides](/en/derivations/?q=mko-monotone-sequence-convergence-theorem) - Stage 4 · optional ## [Subspace topology](/en/mko/mko-subspace-topology/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261108%2F) For Y⊆X, the relative topology consists of every intersection of an ambient open set with Y. [Related questions](/en/explore/?mko=mko-subspace-topology) [Concept guides](/en/derivations/?q=mko-subspace-topology) - Stage 5 · optional ## [Chebyshev inequality](/en/mko/mko-chebyshev-inequality/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261108%2F) For a variable with finite mean and variance, the probability of absolute deviation at least epsilon is at most variance/epsilon^2. [Related questions](/en/explore/?mko=mko-chebyshev-inequality) [Concept guides](/en/derivations/?q=mko-chebyshev-inequality) - Stage 6 · optional ## [Inclusion-exclusion principle](/en/mko/mko-inclusion-exclusion-principle/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261108%2F) The union cardinality of a finite indexed set family is the alternating sum of intersection sizes over every nonempty index subset. [Related questions](/en/explore/?mko=mko-inclusion-exclusion-principle) [Concept guides](/en/derivations/?q=mko-inclusion-exclusion-principle) - Stage 7 · optional ## [Unique prime factorization of integers](/en/mko/mko-unique-prime-factorization/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261108%2F) Every positive integer n>1 has a prime factorization whose prime multiset is unique up to factor order. [Related questions](/en/explore/?mko=mko-unique-prime-factorization) [Concept guides](/en/derivations/?q=mko-unique-prime-factorization) - Stage 8 · optional ## [Finite directed graph](/en/mko/mko-finite-directed-graph/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261108%2F) Under the loop-free simple convention, a finite digraph has a finite vertex carrier and an ordered-pair edge set without loops or duplicate same-direction edges. [Related questions](/en/explore/?mko=mko-finite-directed-graph) [Concept guides](/en/derivations/?q=mko-finite-directed-graph) Path source & statuspath-pulse-foundations-20261108 · candidate [Original paths JSON](/data/architecture/learning-paths.json)