# Wave6 Tenth Layer: Contraction, IVT, Taylor Remainders, Finite MLE, Sufficiency, Topological Orders and Modular Order 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 ## [Orbits, fixed points and periodic points](/en/mko/mko-orbit-fixed-periodic/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261116%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-orbit-fixed-periodic) [Concept guides](/en/derivations/?q=mko-orbit-fixed-periodic) - Stage 1 · required ## [Complete metric space](/en/mko/mko-complete-metric-space/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261116%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-complete-metric-space) [Concept guides](/en/derivations/?q=mko-complete-metric-space) - Stage 1 · required ## [Convergence in a metric space](/en/mko/mko-metric-convergence/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261116%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-metric-convergence) [Concept guides](/en/derivations/?q=mko-metric-convergence) - Stage 1 · required ## [Infinite series](/en/mko/mko-infinite-series/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261116%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-infinite-series) [Concept guides](/en/derivations/?q=mko-infinite-series) - Stage 1 · required ## [Point values and continuity of real functions](/en/mko/mko-continuity/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261116%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-continuity) [Concept guides](/en/derivations/?q=mko-continuity) - Stage 1 · required ## [Real number](/en/mko/mko-real-number/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261116%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 ## [Usual order on the real numbers](/en/mko/mko-order-relation/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261116%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 ## [Function mapping](/en/mko/mko-function-mapping/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261116%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 ## [Supremum and infimum](/en/mko/mko-supremum-and-infimum/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261116%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 ## [Taylor polynomial](/en/mko/mko-taylor-polynomial/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261116%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-taylor-polynomial) [Concept guides](/en/derivations/?q=mko-taylor-polynomial) - Stage 1 · required ## [Higher-order derivative](/en/mko/mko-higher-order-derivative/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261116%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-higher-order-derivative) [Concept guides](/en/derivations/?q=mko-higher-order-derivative) - Stage 1 · required ## [Mean value theorem](/en/mko/mko-mean-value-theorem/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261116%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-mean-value-theorem) [Concept guides](/en/derivations/?q=mko-mean-value-theorem) - Stage 1 · required ## [Absolute value](/en/mko/mko-absolute-value/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261116%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 ## [Parametric statistical model](/en/mko/mko-parametric-statistical-model/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261116%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-20261116%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 ## [Optimization problem](/en/mko/mko-optimization-problem/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261116%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 ## [Sampling distribution](/en/mko/mko-sampling-distribution/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261116%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 ## [Conditional probability in finite models](/en/mko/mko-conditional-probability/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261116%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-conditional-probability) [Concept guides](/en/derivations/?q=mko-conditional-probability) - Stage 1 · required ## [Finite directed graph](/en/mko/mko-finite-directed-graph/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261116%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-finite-directed-graph) [Concept guides](/en/derivations/?q=mko-finite-directed-graph) - Stage 1 · required ## [Algorithm correctness](/en/mko/mko-algorithm-correctness/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261116%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 ## [Mathematical induction](/en/mko/mko-mathematical-induction/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261116%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 ## [Modular inverse](/en/mko/mko-modular-inverse/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261116%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-modular-inverse) [Concept guides](/en/derivations/?q=mko-modular-inverse) - Stage 1 · required ## [Greatest common divisor and coprimality](/en/mko/mko-gcd-coprime/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261116%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-20261116%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 ## [Euler's theorem in number theory](/en/mko/mko-euler-theorem-number-theory/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261116%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-euler-theorem-number-theory) [Concept guides](/en/derivations/?q=mko-euler-theorem-number-theory) - Stage 2 · optional ## [Contraction mapping theorem](/en/mko/mko-contraction-mapping-theorem/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261116%2F) A self-map on a nonempty complete metric space with one global contraction constant q<1 has a unique fixed point and convergent Picard iterates. [Related questions](/en/explore/?mko=mko-contraction-mapping-theorem) [Concept guides](/en/derivations/?q=mko-contraction-mapping-theorem) - Stage 3 · optional ## [Intermediate value theorem](/en/mko/mko-intermediate-value-theorem/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261116%2F) A continuous real function on a closed interval attains each value between its endpoint values. [Related questions](/en/explore/?mko=mko-intermediate-value-theorem) [Concept guides](/en/derivations/?q=mko-intermediate-value-theorem) - Stage 4 · optional ## [Taylor remainder theorem](/en/mko/mko-taylor-remainder-theorem/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261116%2F) Under sufficient differentiability, a Lagrange remainder and derivative bound control the Taylor approximation error. [Related questions](/en/explore/?mko=mko-taylor-remainder-theorem) [Concept guides](/en/derivations/?q=mko-taylor-remainder-theorem) - Stage 5 · optional ## [Maximum-likelihood estimator in a finite model](/en/mko/mko-finite-maximum-likelihood-estimator/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261116%2F) A finite-model MLE chooses a parameter maximizing the observed likelihood; ties use one fixed rule independent of the unknown parameter. [Related questions](/en/explore/?mko=mko-finite-maximum-likelihood-estimator) [Concept guides](/en/derivations/?q=mko-finite-maximum-likelihood-estimator) - Stage 6 · optional ## [Sufficient statistic in a finite model](/en/mko/mko-finite-sufficient-statistic/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261116%2F) A statistic is sufficient when the conditional sample law given its value does not depend on theta wherever that condition has positive mass. [Related questions](/en/explore/?mko=mko-finite-sufficient-statistic) [Concept guides](/en/derivations/?q=mko-finite-sufficient-statistic) - Stage 7 · optional ## [Topological ordering of a directed acyclic graph](/en/mko/mko-topological-ordering-dag/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261116%2F) A finite directed graph has a topological ordering exactly when it has no directed cycle. [Related questions](/en/explore/?mko=mko-topological-ordering-dag) [Concept guides](/en/derivations/?q=mko-topological-ordering-dag) - Stage 8 · optional ## [Multiplicative order modulo n](/en/mko/mko-multiplicative-order-mod-n/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261116%2F) For gcd(a,n)=1, the multiplicative order is the smallest positive k with a^k congruent to1 modulo n. [Related questions](/en/explore/?mko=mko-multiplicative-order-mod-n) [Concept guides](/en/derivations/?q=mko-multiplicative-order-mod-n) Path source & statuspath-pulse-foundations-20261116 · candidate [Original paths JSON](/data/architecture/learning-paths.json)