# Typed Data and Selection: Fractions, Ratios, Maps, Compositions, and Coordinates Deepen cross-domain question families on seven existing knowledge objects and their prerequisites, keeping mathematical sources unchanged and checking boundaries in exact finite slices. This is a suggested learning sequence, not a universal logical dependency or a formal proof. - Stage 1 · required ## [Rational number](/en/mko/mko-rational-number/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261015%2F) Existing canonical prerequisite for this density pulse. [Related questions](/en/explore/?mko=mko-rational-number) [Concept guides](/en/derivations/?q=mko-rational-number) - Stage 1 · required ## [Arithmetic operations](/en/mko/mko-arithmetic-operations/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261015%2F) Existing canonical prerequisite for this density pulse. [Related questions](/en/explore/?mko=mko-arithmetic-operations) [Concept guides](/en/derivations/?q=mko-arithmetic-operations) - Stage 1 · required ## [Function mapping](/en/mko/mko-function-mapping/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261015%2F) Existing canonical prerequisite for this density pulse. [Related questions](/en/explore/?mko=mko-function-mapping) [Concept guides](/en/derivations/?q=mko-function-mapping) - Stage 1 · required ## [Set](/en/mko/mko-set/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261015%2F) Existing canonical prerequisite for this density pulse. [Related questions](/en/explore/?mko=mko-set) [Concept guides](/en/derivations/?q=mko-set) - Stage 1 · required ## [Natural number](/en/mko/mko-natural-number/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261015%2F) Existing canonical prerequisite for this density pulse. [Related questions](/en/explore/?mko=mko-natural-number) [Concept guides](/en/derivations/?q=mko-natural-number) - Stage 1 · required ## [Real number](/en/mko/mko-real-number/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261015%2F) Existing canonical prerequisite for this density pulse. [Related questions](/en/explore/?mko=mko-real-number) [Concept guides](/en/derivations/?q=mko-real-number) - Stage 1 · required ## [Basic counting principles](/en/mko/mko-counting-principle/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261015%2F) Existing canonical prerequisite for this density pulse. [Related questions](/en/explore/?mko=mko-counting-principle) [Concept guides](/en/derivations/?q=mko-counting-principle) - Stage 1 · required ## [Factorial](/en/mko/mko-factorial/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261015%2F) Existing canonical prerequisite for this density pulse. [Related questions](/en/explore/?mko=mko-factorial) [Concept guides](/en/derivations/?q=mko-factorial) - Stage 1 · required ## [Points, lines, and planes](/en/mko/mko-point-line-plane/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261015%2F) Existing canonical prerequisite for this density pulse. [Related questions](/en/explore/?mko=mko-point-line-plane) [Concept guides](/en/derivations/?q=mko-point-line-plane) - Stage 2 · optional ## [Fraction](/en/mko/mko-fraction/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261015%2F) Keep legal original integer fractions with nonzero denominators, allowing negative denominators. Compare true addition with naive numerator/denominator addition, ordered difference chains, zero-guarded reciprocals, and two raw scaled representations. Zero scaling is illegal; representation/value separation is not a quotient well-definedness formal proof. [Related questions](/en/explore/?mko=mko-fraction) [Concept guides](/en/derivations/?q=mko-fraction) - Stage 2 · optional ## [Domain, codomain, and range](/en/mko/mko-domain-codomain-range/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261015%2F) Use complete Fin(n)→Fin(m) data for domain/codomain/range inventory, two typed-candidate preimages, one-cell replacement with range differences, and one-row protocol audits. Even unused cells must type-check; repeated equal-value rows violate the representation protocol without implying a multivalued function. Data audits are not general function proofs. [Related questions](/en/explore/?mko=mko-domain-codomain-range) [Concept guides](/en/derivations/?q=mko-domain-codomain-range) - Stage 2 · optional ## [Sequence](/en/mko/mko-sequence/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261015%2F) Use explicit finite zero-based indices, declared affine-recurrence windows, two finite extensions of a common four-term prefix, and five prefix sums/ordered differences. Keep order/repetition and genuinely empty windows; equal candidates are not different extensions. Finite prefixes imply neither a unique infinite rule nor convergence or an infinite series. [Related questions](/en/explore/?mko=mko-sequence) [Concept guides](/en/derivations/?q=mko-sequence) - Stage 2 · optional ## [Permutations and combinations](/en/mko/mko-permutation-combination/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261015%2F) Select distinct labeled Fin(n) objects without replacement: ordered/unordered counts, k! and the unique empty selection, three candidate masks, inclusion/exclusion of label 0, and completion counts after a fixed ordered prefix. Do not use repetition, indistinguishable, or circular models, or silently call a counting ratio a probability. [Related questions](/en/explore/?mko=mko-permutation-combination) [Concept guides](/en/derivations/?q=mko-permutation-combination) - Stage 2 · optional ## [Coordinate plane](/en/mko/mko-coordinate-plane/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261015%2F) Use ordered integer pairs in one orthogonal directed xy frame: four-point columns/axis/quadrant masks, component defects/equality, swapped-field candidates, and the first unequal pair. The origin lies on both axes and axis points are not in quadrants. A field swap is not a formal geometric reflection bridge; components are not distances or rendered dimensions. [Related questions](/en/explore/?mko=mko-coordinate-plane) [Concept guides](/en/derivations/?q=mko-coordinate-plane) - Stage 3 · optional ## [Ratio and proportion](/en/mko/mko-ratio-proportion/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261015%2F) Numeric ratios in Q require nonzero second components: forward/reverse guards, a complete three-point common-k law, positive-weight same-unit nonnegative-total partitions, and candidate proportions with both denominator guards. Zero-total 0:0 remains illegal despite zero cross defects. Finite models imply no causality, universal law, or geometric similarity. [Related questions](/en/explore/?mko=mko-ratio-proportion) [Concept guides](/en/derivations/?q=mko-ratio-proportion) - Stage 3 · optional ## [Function composition](/en/mko/mko-composition/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261015%2F) Evaluate f before g and guard the intermediate type/domain first. Check complete paths, g restricted to a declared subdomain while retaining original pairings, both Q affine orders and constant differences, and two parenthesizations of three maps. Affine output pairs are [slope,constant], not ascending coefficients. Outside g’s domain is not evaluated and unused g/h cells remain typed; no universal function-identity formal proof. [Related questions](/en/explore/?mko=mko-composition) [Concept guides](/en/derivations/?q=mko-composition) Path source & statuspath-pulse-foundations-20261015 · candidate [Original paths JSON](/data/architecture/learning-paths.json)