OCME · Bilingual mathematics & derivations
Structural Depth Wave4 Second Layer: Commutative Rings, Ideals, Banach and Inner-Product Spaces, Operators, Independence and Components
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- Finite unital ring
- Quantifiers and explicit domains
- Subset
- Finite abelian group
- Real normed vector space
- Norm-induced metric
- Complete metric space
- Real vector space
- Real bilinear form
- Usual order on the real numbers
- Linear map between real vector spaces
- Event independence
- Finite joint distribution
- Real random variables on finite sample spaces
- Measurable function
- Graph walks and simple paths
- Equivalence relation
- Finite simple undirected graphs
- Finite commutative ring
- Finite two-sided ideal
- Real Banach space
- Real inner-product space
- Bounded linear operator
- Independence of random variables
- Connected component
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 unital ring
Existing canonical prerequisite for the new objects in this pulse.
Quantifiers and explicit domains
Existing canonical prerequisite for the new objects in this pulse.
Subset
Existing canonical prerequisite for the new objects in this pulse.
Finite abelian group
Existing canonical prerequisite for the new objects in this pulse.
Real normed vector space
Existing canonical prerequisite for the new objects in this pulse.
Norm-induced metric
Existing canonical prerequisite for the new objects in this pulse.
Complete metric space
Existing canonical prerequisite for the new objects in this pulse.
Real vector space
Existing canonical prerequisite for the new objects in this pulse.
Real bilinear form
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.
Linear map between real vector spaces
Existing canonical prerequisite for the new objects in this pulse.
Event independence
Existing canonical prerequisite for the new objects in this pulse.
Finite joint 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.
Measurable function
Existing canonical prerequisite for the new objects in this pulse.
Graph walks and simple paths
Existing canonical prerequisite for the new objects in this pulse.
Equivalence relation
Existing canonical prerequisite for the new objects in this pulse.
Finite simple undirected graphs
Existing canonical prerequisite for the new objects in this pulse.
Finite commutative ring
A finite unital ring is commutative when multiplication commutes for every pair of elements; zero divisors may still exist.
Finite two-sided ideal
A subset of a finite ring is a two-sided ideal when it is an additive subgroup and absorbs multiplication by parent-ring elements on both sides.
Real Banach space
A real normed vector space is a Banach space when it is complete in the metric induced by its own norm.
Real inner-product space
A symmetric, positive-definite real bilinear form is a real inner product; positive semidefiniteness or positive sampled values do not suffice.
Bounded linear operator
A linear map between normed spaces is bounded when one nonnegative constant uniformly controls all output norms relative to input norms.
Independence of random variables
Under one finite joint law, independence requires every value-pair mass to factor as the product of its marginal masses.
Connected component
In a finite undirected simple graph, connected components are exactly the vertex equivalence classes defined by the existence of a path.
Path source & status
path-pulse-foundations-20261028 · candidate