OCME · Bilingual mathematics & derivations
Structural Depth Wave4 Roots: Rings, Metric Completeness, Linear Structures, Independence and Graph Paths
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- Finite abelian group
- Arithmetic operations
- Function mapping
- Real normed vector space
- Metric space
- Real vector space
- Cauchy sequence
- Convergence in a metric space
- Linear combinations and span
- Event
- Probability on finite sample spaces
- Conditional probability in finite models
- Finite simple undirected graphs
- Sequence
- Binary relations and set projections
- Finite unital ring
- Norm-induced metric
- Complete metric space
- Real bilinear form
- Linear map between real vector spaces
- Event independence
- Graph walks and simple paths
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 abelian group
Existing canonical prerequisite for the new objects in this pulse.
Arithmetic operations
Existing canonical prerequisite for the new objects in this pulse.
Function mapping
Existing canonical prerequisite for the new objects in this pulse.
Real normed vector space
Existing canonical prerequisite for the new objects in this pulse.
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.
Cauchy sequence
Existing canonical prerequisite for the new objects in this pulse.
Convergence in a metric space
Existing canonical prerequisite for the new objects in this pulse.
Linear combinations and span
Existing canonical prerequisite for the new objects in this pulse.
Event
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.
Conditional probability in finite 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.
Sequence
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.
Finite unital ring
On one complete finite carrier, this unital-ring protocol requires an additive abelian group, associative multiplication, both distributive laws and a multiplicative identity different from zero.
Norm-induced metric
On the same real normed vector space, taking the norm of a vector difference defines a metric whose axioms follow universally from the norm axioms.
Complete metric space
A metric space is complete when every Cauchy sequence converges to a point still belonging to that space.
Real bilinear form
A bilinear form on a real vector space preserves addition and real scalar multiplication separately in each input.
Linear map between real vector spaces
A map between real vector spaces is linear when it preserves linear combinations for all vectors and real scalars.
Event independence
Two events are independent when their intersection probability equals the product of their marginal probabilities; this definition does not rely on conditioning on a zero-probability event.
Graph walks and simple paths
In a finite undirected simple graph, a walk requires each consecutive pair of vertices to be adjacent; a simple path additionally requires all vertices to be distinct.
Path source & status
path-pulse-foundations-20261027 · candidate