OCME · Bilingual mathematics & derivations
Typed Data and Selection: Fractions, Ratios, Maps, Compositions, and Coordinates
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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.
Rational number
Existing canonical prerequisite for this density pulse.
Arithmetic operations
Existing canonical prerequisite for this density pulse.
Function mapping
Existing canonical prerequisite for this density pulse.
Set
Existing canonical prerequisite for this density pulse.
Natural number
Existing canonical prerequisite for this density pulse.
Real number
Existing canonical prerequisite for this density pulse.
Basic counting principles
Existing canonical prerequisite for this density pulse.
Factorial
Existing canonical prerequisite for this density pulse.
Points, lines, and planes
Existing canonical prerequisite for this density pulse.
Fraction
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.
Domain, codomain, and range
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.
Sequence
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.
Permutations and combinations
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.
Coordinate plane
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.
Ratio and proportion
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.
Function composition
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.
Path source & status
path-pulse-foundations-20261015 · candidate