# Wave5 Third Layer: Matrix Rank, Pointwise Function Limits, Series, Statistical Models, Connectedness, Maximum Matching and Modular Inverses 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 matrices: dimensions, indexing, and transpose](/en/mko/mko-matrix/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261103%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-matrix) [Concept guides](/en/derivations/?q=mko-matrix) - Stage 1 · required ## [Linear combinations and span](/en/mko/mko-linear-combination-span/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261103%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-linear-combination-span) [Concept guides](/en/derivations/?q=mko-linear-combination-span) - Stage 1 · required ## [Dimension of a finite-dimensional vector space](/en/mko/mko-vector-space-dimension/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261103%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 1 · required ## [Sequence](/en/mko/mko-sequence/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261103%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 ## [Function mapping](/en/mko/mko-function-mapping/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261103%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-20261103%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 ## [Quantifiers and explicit domains](/en/mko/mko-quantifier/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261103%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 ## [Arithmetic operations](/en/mko/mko-arithmetic-operations/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261103%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-arithmetic-operations) [Concept guides](/en/derivations/?q=mko-arithmetic-operations) - Stage 1 · required ## [Finite joint distribution](/en/mko/mko-finite-joint-distribution/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261103%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-finite-joint-distribution) [Concept guides](/en/derivations/?q=mko-finite-joint-distribution) - Stage 1 · required ## [Probability on finite sample spaces](/en/mko/mko-probability/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261103%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-probability) [Concept guides](/en/derivations/?q=mko-probability) - Stage 1 · required ## [Set](/en/mko/mko-set/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261103%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-set) [Concept guides](/en/derivations/?q=mko-set) - Stage 1 · required ## [Topological space](/en/mko/mko-topological-space/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261103%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 ## [Open and closed sets](/en/mko/mko-open-closed-set/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261103%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 ## [Set operations](/en/mko/mko-set-operations/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261103%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 ## [Graph matching](/en/mko/mko-graph-matching/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261103%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-graph-matching) [Concept guides](/en/derivations/?q=mko-graph-matching) - Stage 1 · required ## [Optimization problem](/en/mko/mko-optimization-problem/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261103%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 ## [Greatest common divisor and coprimality](/en/mko/mko-gcd-coprime/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261103%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 congruence modulo a positive modulus](/en/mko/mko-modular-congruence/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261103%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 ## [Equation](/en/mko/mko-equation/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261103%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-equation) [Concept guides](/en/derivations/?q=mko-equation) - Stage 2 · optional ## [Matrix rank](/en/mko/mko-matrix-rank/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261103%2F) Matrix rank is the dimension of the row or column space and can be computed as the number of pivots after exact rank-preserving elimination. [Related questions](/en/explore/?mko=mko-matrix-rank) [Concept guides](/en/derivations/?q=mko-matrix-rank) - Stage 3 · optional ## [Pointwise convergence of functions](/en/mko/mko-pointwise-convergence-of-functions/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261103%2F) A function sequence on one domain converges pointwise when its value sequence converges at each fixed x; the witness N may depend on x and epsilon. [Related questions](/en/explore/?mko=mko-pointwise-convergence-of-functions) [Concept guides](/en/derivations/?q=mko-pointwise-convergence-of-functions) - Stage 4 · optional ## [Infinite series](/en/mko/mko-infinite-series/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261103%2F) An infinite series is defined through the limit of its partial-sum sequence; terms tending to zero are necessary but not sufficient for convergence. [Related questions](/en/explore/?mko=mko-infinite-series) [Concept guides](/en/derivations/?q=mko-infinite-series) - Stage 5 · optional ## [Parametric statistical model](/en/mko/mko-parametric-statistical-model/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261103%2F) A parametric statistical model assigns each parameter theta a complete probability law P_theta on the same finite measurable sample space. [Related questions](/en/explore/?mko=mko-parametric-statistical-model) [Concept guides](/en/derivations/?q=mko-parametric-statistical-model) - Stage 6 · optional ## [Connected topological space](/en/mko/mko-connected-topological-space/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261103%2F) A space is connected when it has no separation into two disjoint nonempty open sets, equivalently no nontrivial clopen subset. [Related questions](/en/explore/?mko=mko-connected-topological-space) [Concept guides](/en/derivations/?q=mko-connected-topological-space) - Stage 7 · optional ## [Maximum matching](/en/mko/mko-maximum-matching/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261103%2F) A maximum matching has the greatest edge count among all valid matchings of the complete finite graph; it may be nonunique and need not be perfect. [Related questions](/en/explore/?mko=mko-maximum-matching) [Concept guides](/en/derivations/?q=mko-maximum-matching) - Stage 8 · optional ## [Modular inverse](/en/mko/mko-modular-inverse/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261103%2F) For n≥2, a has a multiplicative inverse modulo n exactly when gcd(a,n)=1; the inverse is unique as a residue class modulo n. [Related questions](/en/explore/?mko=mko-modular-inverse) [Concept guides](/en/derivations/?q=mko-modular-inverse) Path source & statuspath-pulse-foundations-20261103 · candidate [Original paths JSON](/data/architecture/learning-paths.json)