# Wave6 Seventh Layer: Normal Subgroups, Real Exponential, Analytic OGFs, Euler Criterion, Turing Decisions, Gradients and Flow Networks 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](/en/mko/mko-finite-group/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261113%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 ## [Subgroups and group homomorphisms](/en/mko/mko-subgroup-homomorphism/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261113%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-subgroup-homomorphism) [Concept guides](/en/derivations/?q=mko-subgroup-homomorphism) - Stage 1 · required ## [Quantifiers and explicit domains](/en/mko/mko-quantifier/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261113%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-quantifier) [Concept guides](/en/derivations/?q=mko-quantifier) - Stage 1 · required ## [Infinite series](/en/mko/mko-infinite-series/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261113%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 ## [Real number](/en/mko/mko-real-number/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261113%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 ## [Function mapping](/en/mko/mko-function-mapping/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261113%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 ## [Sequence limits and exact tail cutoffs](/en/mko/mko-sequence-limit/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261113%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 ## [Radius of convergence of a power series](/en/mko/mko-power-series-radius-of-convergence/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261113%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-power-series-radius-of-convergence) [Concept guides](/en/derivations/?q=mko-power-series-radius-of-convergence) - Stage 1 · required ## [Factorial](/en/mko/mko-factorial/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261113%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 ## [Absolute convergence of series](/en/mko/mko-absolute-convergence-of-series/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261113%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 ## [Binomial theorem](/en/mko/mko-binomial-theorem/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261113%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-binomial-theorem) [Concept guides](/en/derivations/?q=mko-binomial-theorem) - Stage 1 · required ## [Recurrence relation](/en/mko/mko-recurrence-relation/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261113%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-recurrence-relation) [Concept guides](/en/derivations/?q=mko-recurrence-relation) - Stage 1 · required ## [Sequence](/en/mko/mko-sequence/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261113%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-sequence) [Concept guides](/en/derivations/?q=mko-sequence) - Stage 1 · required ## [Euler's theorem in number theory](/en/mko/mko-euler-theorem-number-theory/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261113%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 1 · required ## [Quadratic residue](/en/mko/mko-quadratic-residue/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261113%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 ## [Legendre symbol](/en/mko/mko-legendre-symbol/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261113%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-legendre-symbol) [Concept guides](/en/derivations/?q=mko-legendre-symbol) - Stage 1 · required ## [Abstract computational model](/en/mko/mko-computational-model/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261113%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-computational-model) [Concept guides](/en/derivations/?q=mko-computational-model) - Stage 1 · required ## [Decision problem](/en/mko/mko-decision-problem/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261113%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-decision-problem) [Concept guides](/en/derivations/?q=mko-decision-problem) - Stage 1 · required ## [Deterministic finite automaton](/en/mko/mko-finite-automaton/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261113%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 ## [Algorithm correctness](/en/mko/mko-algorithm-correctness/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261113%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 ## [Partial derivative](/en/mko/mko-partial-derivative/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261113%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-partial-derivative) [Concept guides](/en/derivations/?q=mko-partial-derivative) - Stage 1 · required ## [Plane vectors in a fixed basis](/en/mko/mko-vector/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261113%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-vector) [Concept guides](/en/derivations/?q=mko-vector) - Stage 1 · required ## [Multivariable real-valued function](/en/mko/mko-multivariable-real-function/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261113%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-multivariable-real-function) [Concept guides](/en/derivations/?q=mko-multivariable-real-function) - Stage 1 · required ## [Finite directed graph](/en/mko/mko-finite-directed-graph/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261113%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 ## [Linear programming](/en/mko/mko-linear-programming/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261113%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-linear-programming) [Concept guides](/en/derivations/?q=mko-linear-programming) - Stage 2 · optional ## [Finite normal subgroup](/en/mko/mko-finite-normal-subgroup/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261113%2F) A subgroup H of a finite group G is normal when gHg^(-1)=H for every g, equivalently when each left coset equals the corresponding right coset. [Related questions](/en/explore/?mko=mko-finite-normal-subgroup) [Concept guides](/en/derivations/?q=mko-finite-normal-subgroup) - Stage 3 · optional ## [Real exponential function](/en/mko/mko-real-exponential-function/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261113%2F) Define E(x)=sum over n≥0 of x^n/n!, absolutely convergent for every real x, with E(x+y)=E(x)E(y) and E(x)>0. [Related questions](/en/explore/?mko=mko-real-exponential-function) [Concept guides](/en/derivations/?q=mko-real-exponential-function) - Stage 4 · optional ## [Analytic ordinary generating function](/en/mko/mko-analytic-ordinary-generating-function/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261113%2F) An ordinary power series A(x)=sum over n≥0 of a_n*x^n analytically encodes a sequence on an explicitly declared convergence domain. [Related questions](/en/explore/?mko=mko-analytic-ordinary-generating-function) [Concept guides](/en/derivations/?q=mko-analytic-ordinary-generating-function) - Stage 5 · optional ## [Euler criterion for quadratic residues](/en/mko/mko-euler-criterion-quadratic-residue/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261113%2F) For odd prime p and a not divisible by p, a^((p−1)/2) modulo p is the modulo-p representative of the Legendre symbol (a/p). [Related questions](/en/explore/?mko=mko-euler-criterion-quadratic-residue) [Concept guides](/en/derivations/?q=mko-euler-criterion-quadratic-residue) - Stage 6 · optional ## [Turing-machine decision model](/en/mko/mko-turing-machine-decision-model/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261113%2F) Under fixed deterministic single-tape semantics, a decider must halt on every legal input and return the correct accept/reject answer. [Related questions](/en/explore/?mko=mko-turing-machine-decision-model) [Concept guides](/en/derivations/?q=mko-turing-machine-decision-model) - Stage 7 · optional ## [Gradient vector](/en/mko/mko-gradient-vector/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261113%2F) In fixed standard Euclidean coordinates, the first partial derivatives of a differentiable real-valued function form an ordered gradient vector. [Related questions](/en/explore/?mko=mko-gradient-vector) [Concept guides](/en/derivations/?q=mko-gradient-vector) - Stage 8 · optional ## [Finite flow network](/en/mko/mko-finite-flow-network/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261113%2F) A nonnegative flow in a finite directed network must respect every edge capacity and conserve flow at each vertex other than source and sink. [Related questions](/en/explore/?mko=mko-finite-flow-network) [Concept guides](/en/derivations/?q=mko-finite-flow-network) Path source & statuspath-pulse-foundations-20261113 · candidate [Original paths JSON](/data/architecture/learning-paths.json)