OCME · Bilingual mathematics & derivations
Structural Depth Wave4, First Slice: Abelian Groups, Norms, Cauchy Sequences, Joint Laws, Covariance, Graph Isomorphisms and Equivalence Relations
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- Finite group
- Quantifiers and explicit domains
- Real vector space
- Absolute value
- Usual order on the real numbers
- Metric space
- Sequence
- Finite discrete probability distribution
- Real random variables on finite sample spaces
- Probability on finite sample spaces
- Sample space
- Expectation in finite models
- Variance in finite probability models
- Finite simple undirected graphs
- Bijectivity and unique preimages
- Binary relations and set projections
- Logical connectives
- Finite abelian group
- Real normed vector space
- Cauchy sequence
- Finite joint distribution
- Covariance
- Graph isomorphism
- Equivalence relation
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.
Finite group
Existing canonical prerequisite for the new objects in this pulse.
Quantifiers and explicit domains
Existing canonical prerequisite for the new objects in this pulse.
Real vector space
Existing canonical prerequisite for the new objects in this pulse.
Absolute value
Existing canonical prerequisite for the new objects in this pulse.
Usual order on the real numbers
Existing canonical prerequisite for the new objects in this pulse.
Metric space
Existing canonical prerequisite for the new objects in this pulse.
Sequence
Existing canonical prerequisite for the new objects in this pulse.
Finite discrete probability distribution
Existing canonical prerequisite for the new objects in this pulse.
Real random variables on finite sample spaces
Existing canonical prerequisite for the new objects in this pulse.
Probability on finite sample spaces
Existing canonical prerequisite for the new objects in this pulse.
Sample space
Existing canonical prerequisite for the new objects in this pulse.
Expectation in finite models
Existing canonical prerequisite for the new objects in this pulse.
Variance in finite probability models
Existing canonical prerequisite for the new objects in this pulse.
Finite simple undirected graphs
Existing canonical prerequisite for the new objects in this pulse.
Bijectivity and unique preimages
Existing canonical prerequisite for the new objects in this pulse.
Binary relations and set projections
Existing canonical prerequisite for the new objects in this pulse.
Logical connectives
Existing canonical prerequisite for the new objects in this pulse.
Finite abelian group
A finite group is abelian when its complete Cayley table satisfies commutativity for every pair of elements.
Real normed vector space
A real vector space equipped with a positive-definite, absolutely homogeneous real-valued norm satisfying the triangle inequality.
Cauchy sequence
A sequence in a metric space is Cauchy if, for every positive epsilon, all pairs of sufficiently late terms are less than epsilon apart.
Finite joint distribution
Value tuples of random variables on one common finite probability space, together with exact masses, specify a finite joint distribution.
Covariance
Under one finite joint distribution, covariance is E[XY]−E[X]E[Y]; zero covariance does not in general imply independence.
Graph isomorphism
An isomorphism of finite simple graphs is a vertex bijection preserving both adjacency and nonadjacency for every pair.
Equivalence relation
A relation on a fixed carrier is an equivalence relation when it fully satisfies reflexivity, symmetry and transitivity.
Path source & status
path-pulse-foundations-20261026 · candidate