# Wave5 Fifth Layer: Direct Sums, Duals, Euclid, Euler Trails, Homeomorphisms, Sampling Laws and the WLLN 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 ## [Real vector subspace](/en/mko/mko-vector-subspace/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261105%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-vector-subspace) [Concept guides](/en/derivations/?q=mko-vector-subspace) - Stage 1 · required ## [Set operations](/en/mko/mko-set-operations/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261105%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 ## [Linear independence and bases](/en/mko/mko-linear-independence-basis/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261105%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-linear-independence-basis) [Concept guides](/en/derivations/?q=mko-linear-independence-basis) - Stage 1 · required ## [Linear map between real vector spaces](/en/mko/mko-general-linear-map/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261105%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-general-linear-map) [Concept guides](/en/derivations/?q=mko-general-linear-map) - Stage 1 · required ## [Real vector space](/en/mko/mko-real-vector-space/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261105%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 ## [Real number](/en/mko/mko-real-number/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261105%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-real-number) [Concept guides](/en/derivations/?q=mko-real-number) - Stage 1 · required ## [Greatest common divisor and coprimality](/en/mko/mko-gcd-coprime/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261105%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 ## [Natural-number divisibility](/en/mko/mko-natural-divisibility/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261105%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 ## [Algorithm correctness](/en/mko/mko-algorithm-correctness/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261105%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-algorithm-correctness) [Concept guides](/en/derivations/?q=mko-algorithm-correctness) - Stage 1 · required ## [Graph walks and simple paths](/en/mko/mko-graph-walk-simple-path/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261105%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-graph-walk-simple-path) [Concept guides](/en/derivations/?q=mko-graph-walk-simple-path) - Stage 1 · required ## [Finite simple undirected graphs](/en/mko/mko-graph/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261105%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 ## [Connected component](/en/mko/mko-connected-component/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261105%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-connected-component) [Concept guides](/en/derivations/?q=mko-connected-component) - Stage 1 · required ## [Continuous maps between topological spaces](/en/mko/mko-topological-continuity/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261105%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-topological-continuity) [Concept guides](/en/derivations/?q=mko-topological-continuity) - Stage 1 · required ## [Bijectivity and unique preimages](/en/mko/mko-bijective/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261105%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 ## [Inverse functions of bijections](/en/mko/mko-inverse-function/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261105%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-inverse-function) [Concept guides](/en/derivations/?q=mko-inverse-function) - Stage 1 · required ## [Topological space](/en/mko/mko-topological-space/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261105%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 ## [Parametric statistical model](/en/mko/mko-parametric-statistical-model/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261105%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-parametric-statistical-model) [Concept guides](/en/derivations/?q=mko-parametric-statistical-model) - Stage 1 · required ## [Statistical estimator](/en/mko/mko-statistical-estimator/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261105%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-statistical-estimator) [Concept guides](/en/derivations/?q=mko-statistical-estimator) - Stage 1 · required ## [Finite discrete probability distribution](/en/mko/mko-finite-discrete-probability-distribution/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261105%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 ## [Independent identically distributed sequence](/en/mko/mko-iid-sequence/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261105%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-iid-sequence) [Concept guides](/en/derivations/?q=mko-iid-sequence) - Stage 1 · required ## [Expectation in finite models](/en/mko/mko-expectation/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261105%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-20261105%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 ## [Convergence in probability](/en/mko/mko-convergence-in-probability/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261105%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-convergence-in-probability) [Concept guides](/en/derivations/?q=mko-convergence-in-probability) - Stage 2 · optional ## [Direct sum of subspaces](/en/mko/mko-direct-sum/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261105%2F) The sum of two subspaces is an internal direct sum when every vector in that sum has a unique decomposition; covering an ambient space is a separate spanning obligation. [Related questions](/en/explore/?mko=mko-direct-sum) [Concept guides](/en/derivations/?q=mko-direct-sum) - Stage 3 · optional ## [Dual vector space](/en/mko/mko-dual-vector-space/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261105%2F) The dual of a real vector space consists of all real-valued linear functionals; a covector coordinate row depends on a chosen basis. [Related questions](/en/explore/?mko=mko-dual-vector-space) [Concept guides](/en/derivations/?q=mko-dual-vector-space) - Stage 4 · optional ## [Euclidean algorithm](/en/mko/mko-euclidean-algorithm/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261105%2F) The Euclidean algorithm computes the gcd using exact quotient-remainder divisions, gcd invariance and strictly decreasing nonnegative remainders. [Related questions](/en/explore/?mko=mko-euclidean-algorithm) [Concept guides](/en/derivations/?q=mko-euclidean-algorithm) - Stage 5 · optional ## [Euler trails and circuits](/en/mko/mko-eulerian-trail/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261105%2F) An Euler trail uses each edge once; on connected nonzero-edge support, zero odd-degree vertices give a circuit and two give an open Euler trail. [Related questions](/en/explore/?mko=mko-eulerian-trail) [Concept guides](/en/derivations/?q=mko-eulerian-trail) - Stage 6 · optional ## [Homeomorphism](/en/mko/mko-homeomorphism/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261105%2F) A homeomorphism is a bijection that is continuous in both directions; a continuous bijection alone need not be a homeomorphism. [Related questions](/en/explore/?mko=mko-homeomorphism) [Concept guides](/en/derivations/?q=mko-homeomorphism) - Stage 7 · optional ## [Sampling distribution](/en/mko/mko-sampling-distribution/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261105%2F) At a fixed parameter, an estimator pushes the sample law forward to its sampling distribution; all parameters give a family of such laws. [Related questions](/en/explore/?mko=mko-sampling-distribution) [Concept guides](/en/derivations/?q=mko-sampling-distribution) - Stage 8 · optional ## [Weak law of large numbers](/en/mko/mko-weak-law-large-numbers/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261105%2F) For IID variables with finite variance, a Chebyshev bound shows the sample mean converges in probability to the common mean. [Related questions](/en/explore/?mko=mko-weak-law-large-numbers) [Concept guides](/en/derivations/?q=mko-weak-law-large-numbers) Path source & statuspath-pulse-foundations-20261105 · candidate [Original paths JSON](/data/architecture/learning-paths.json)