# Wave5 Closure: Chromatic Minima, Cliques, Estimator Bias and Variance, Fermat, Diophantine Equations and Quotients 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 ## [Graph coloring](/en/mko/mko-graph-coloring/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261106%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-graph-coloring) [Concept guides](/en/derivations/?q=mko-graph-coloring) - Stage 1 · required ## [Natural number](/en/mko/mko-natural-number/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261106%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 ## [Optimization problem](/en/mko/mko-optimization-problem/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261106%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-optimization-problem) [Concept guides](/en/derivations/?q=mko-optimization-problem) - Stage 1 · required ## [Finite simple undirected graphs](/en/mko/mko-graph/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261106%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-20261106%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 ## [Sampling distribution](/en/mko/mko-sampling-distribution/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261106%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-sampling-distribution) [Concept guides](/en/derivations/?q=mko-sampling-distribution) - Stage 1 · required ## [Expectation in finite models](/en/mko/mko-expectation/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261106%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 ## [Parametric statistical model](/en/mko/mko-parametric-statistical-model/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261106%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 ## [Variance in finite probability models](/en/mko/mko-variance/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261106%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 ## [Prime natural number](/en/mko/mko-prime-number/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261106%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 congruence modulo a positive modulus](/en/mko/mko-modular-congruence/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261106%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-modular-congruence) [Concept guides](/en/derivations/?q=mko-modular-congruence) - Stage 1 · required ## [Natural-number powers and principal square roots](/en/mko/mko-power-root/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261106%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-power-root) [Concept guides](/en/derivations/?q=mko-power-root) - Stage 1 · required ## [Integer](/en/mko/mko-integer/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261106%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-integer) [Concept guides](/en/derivations/?q=mko-integer) - Stage 1 · required ## [Greatest common divisor and coprimality](/en/mko/mko-gcd-coprime/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261106%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 ## [Linear equation in one variable](/en/mko/mko-linear-equation/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261106%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-linear-equation) [Concept guides](/en/derivations/?q=mko-linear-equation) - Stage 1 · required ## [Euclidean algorithm](/en/mko/mko-euclidean-algorithm/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261106%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 ## [Real vector subspace](/en/mko/mko-vector-subspace/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261106%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 ## [Quotient set](/en/mko/mko-quotient-set/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261106%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 ## [Real vector space](/en/mko/mko-real-vector-space/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261106%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 ## [Equivalence class](/en/mko/mko-equivalence-class/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261106%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 ## [Dimension of a finite-dimensional vector space](/en/mko/mko-vector-space-dimension/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261106%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-vector-space-dimension) [Concept guides](/en/derivations/?q=mko-vector-space-dimension) - Stage 2 · optional ## [Chromatic number](/en/mko/mko-chromatic-number/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261106%2F) The chromatic number of a finite simple graph is the minimum number of colors in a proper vertex coloring; equality needs both upper and lower certificates. [Related questions](/en/explore/?mko=mko-chromatic-number) [Concept guides](/en/derivations/?q=mko-chromatic-number) - Stage 3 · optional ## [Cliques and independent sets](/en/mko/mko-clique-and-independent-set/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261106%2F) Every pair of distinct vertices of a clique is adjacent; every pair of an independent set is nonadjacent. Inclusion-maximal and maximum-cardinality are different. [Related questions](/en/explore/?mko=mko-clique-and-independent-set) [Concept guides](/en/derivations/?q=mko-clique-and-independent-set) - Stage 4 · optional ## [Estimator bias](/en/mko/mko-estimator-bias/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261106%2F) Bias at theta is the estimator expectation under its theta sampling law minus the target parameter; unbiasedness requires zero bias at every theta. [Related questions](/en/explore/?mko=mko-estimator-bias) [Concept guides](/en/derivations/?q=mko-estimator-bias) - Stage 5 · optional ## [Estimator variance](/en/mko/mko-estimator-variance/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261106%2F) At each theta, estimator variance is the second central moment of its sampling law around its own mean; it is not bias or MSE by itself. [Related questions](/en/explore/?mko=mko-estimator-variance) [Concept guides](/en/derivations/?q=mko-estimator-variance) - Stage 6 · optional ## [Fermat's little theorem](/en/mko/mko-fermat-little-theorem/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261106%2F) For a prime p, a^p is congruent to a modulo p; if p does not divide a, a^(p−1) is congruent to1 modulo p. The converse fails. [Related questions](/en/explore/?mko=mko-fermat-little-theorem) [Concept guides](/en/derivations/?q=mko-fermat-little-theorem) - Stage 7 · optional ## [Linear Diophantine equation](/en/mko/mko-linear-diophantine-equation/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261106%2F) A nondegenerate integer equation a*x+b*y=c has a solution exactly when gcd(a,b) divides c; all solutions have one integer parameter. [Related questions](/en/explore/?mko=mko-linear-diophantine-equation) [Concept guides](/en/derivations/?q=mko-linear-diophantine-equation) - Stage 8 · optional ## [Quotient vector space](/en/mko/mko-quotient-vector-space/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261106%2F) Cosets of a subspace W form the quotient vector space V/W, whose operations must be independent of representative choices. [Related questions](/en/explore/?mko=mko-quotient-vector-space) [Concept guides](/en/derivations/?q=mko-quotient-vector-space) Path source & statuspath-pulse-foundations-20261106 · candidate [Original paths JSON](/data/architecture/learning-paths.json)