# Wave6 Sixth Layer: Cauchy, Taylor, Compact Closed Subsets, Improper Integrals, Convergence Radii, Legendre and Subset Construction 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 ## [Lagrange theorem for finite groups](/en/mko/mko-lagrange-theorem-finite-groups/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261112%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-lagrange-theorem-finite-groups) [Concept guides](/en/derivations/?q=mko-lagrange-theorem-finite-groups) - Stage 1 · required ## [Prime natural number](/en/mko/mko-prime-number/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261112%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 ## [Finite group](/en/mko/mko-finite-group/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261112%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-finite-group) [Concept guides](/en/derivations/?q=mko-finite-group) - Stage 1 · required ## [Finite group action](/en/mko/mko-finite-group-action/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261112%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 ## [Differentiating polynomial sums, products, and compositions](/en/mko/mko-derivative-rules/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261112%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-derivative-rules) [Concept guides](/en/derivations/?q=mko-derivative-rules) - Stage 1 · required ## [Univariate polynomials over the rationals](/en/mko/mko-polynomial/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261112%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-polynomial) [Concept guides](/en/derivations/?q=mko-polynomial) - Stage 1 · required ## [Factorial](/en/mko/mko-factorial/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261112%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-factorial) [Concept guides](/en/derivations/?q=mko-factorial) - Stage 1 · required ## [Higher-order derivative](/en/mko/mko-higher-order-derivative/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261112%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 ## [Hausdorff space](/en/mko/mko-hausdorff-space/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261112%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-hausdorff-space) [Concept guides](/en/derivations/?q=mko-hausdorff-space) - Stage 1 · required ## [Compact subset](/en/mko/mko-compact-subset/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261112%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-compact-subset) [Concept guides](/en/derivations/?q=mko-compact-subset) - Stage 1 · required ## [Open and closed sets](/en/mko/mko-open-closed-set/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261112%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-open-closed-set) [Concept guides](/en/derivations/?q=mko-open-closed-set) - Stage 1 · required ## [Signed polynomial definite integrals on finite intervals](/en/mko/mko-definite-integral/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261112%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-definite-integral) [Concept guides](/en/derivations/?q=mko-definite-integral) - Stage 1 · required ## [Function limits on punctured real domains](/en/mko/mko-function-limit/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261112%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 limits and exact tail cutoffs](/en/mko/mko-sequence-limit/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261112%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 ## [Absolute convergence of series](/en/mko/mko-absolute-convergence-of-series/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261112%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-absolute-convergence-of-series) [Concept guides](/en/derivations/?q=mko-absolute-convergence-of-series) - Stage 1 · required ## [Infinite series](/en/mko/mko-infinite-series/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261112%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 ## [Supremum and infimum](/en/mko/mko-supremum-and-infimum/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261112%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 ## [Absolute value](/en/mko/mko-absolute-value/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261112%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 ## [Natural-number powers and principal square roots](/en/mko/mko-power-root/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261112%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 ## [Quadratic residue](/en/mko/mko-quadratic-residue/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261112%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-quadratic-residue) [Concept guides](/en/derivations/?q=mko-quadratic-residue) - Stage 1 · required ## [Function mapping](/en/mko/mko-function-mapping/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261112%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 ## [Nondeterministic finite automaton](/en/mko/mko-nondeterministic-finite-automaton/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261112%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-nondeterministic-finite-automaton) [Concept guides](/en/derivations/?q=mko-nondeterministic-finite-automaton) - Stage 1 · required ## [Deterministic finite automaton](/en/mko/mko-finite-automaton/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261112%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-finite-automaton) [Concept guides](/en/derivations/?q=mko-finite-automaton) - Stage 1 · required ## [Subset](/en/mko/mko-subset/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261112%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 2 · optional ## [Cauchy theorem for finite groups](/en/mko/mko-cauchy-theorem-finite-groups/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261112%2F) If a prime p divides the order of a finite group G, G contains an element of exact order p. [Related questions](/en/explore/?mko=mko-cauchy-theorem-finite-groups) [Concept guides](/en/derivations/?q=mko-cauchy-theorem-finite-groups) - Stage 3 · optional ## [Taylor polynomial](/en/mko/mko-taylor-polynomial/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261112%2F) At center a, derivative values of orders0 through n, scaled by their factorials, form the order-n Taylor polynomial. [Related questions](/en/explore/?mko=mko-taylor-polynomial) [Concept guides](/en/derivations/?q=mko-taylor-polynomial) - Stage 4 · optional ## [A compact subset of a Hausdorff space is closed](/en/mko/mko-compact-subset-closed-in-hausdorff/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261112%2F) If K is compact in the Hausdorff space X, then K is closed in X. [Related questions](/en/explore/?mko=mko-compact-subset-closed-in-hausdorff) [Concept guides](/en/derivations/?q=mko-compact-subset-closed-in-hausdorff) - Stage 5 · optional ## [Improper integral](/en/mko/mko-improper-integral/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261112%2F) Integrals over unbounded intervals or with singularities are defined by appropriate one-sided limits, with every required piece converging independently. [Related questions](/en/explore/?mko=mko-improper-integral) [Concept guides](/en/derivations/?q=mko-improper-integral) - Stage 6 · optional ## [Radius of convergence of a power series](/en/mko/mko-power-series-radius-of-convergence/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261112%2F) Around its center c, a power series has absolute convergence inside radius R and divergence outside; endpoints require separate decisions. [Related questions](/en/explore/?mko=mko-power-series-radius-of-convergence) [Concept guides](/en/derivations/?q=mko-power-series-radius-of-convergence) - Stage 7 · optional ## [Legendre symbol](/en/mko/mko-legendre-symbol/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261112%2F) For odd prime p, the Legendre symbol uses0,1,−1 for the zero class, nonzero quadratic residues and nonresidues respectively. [Related questions](/en/explore/?mko=mko-legendre-symbol) [Concept guides](/en/derivations/?q=mko-legendre-symbol) - Stage 8 · optional ## [Subset construction from an NFA to a DFA](/en/mko/mko-subset-construction-nfa-to-dfa/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261112%2F) Use epsilon-NFA state subsets as DFA states, defining start, transitions and acceptance with epsilon closure to preserve the language. [Related questions](/en/explore/?mko=mko-subset-construction-nfa-to-dfa) [Concept guides](/en/derivations/?q=mko-subset-construction-nfa-to-dfa) Path source & statuspath-pulse-foundations-20261112 · candidate [Original paths JSON](/data/architecture/learning-paths.json)