OCME · Bilingual mathematics & derivations

Wave4 Closure: Countability, Bipartite Graphs, Distributional Convergence, Quotient Rings, Coloring and Two Depth Extensions

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Cardinality comparison of sets
Natural number
Injectivity and distinct inputs
Surjectivity and codomain coverage
Finite simple undirected graphs
Subset
Function mapping
Cumulative distribution function
Function limits on punctured real domains
Sequence
Finite joint distribution
Finite unital ring
Finite two-sided ideal
Equivalence class
Quotient set
Usual order on the real numbers
Finite integral domain
Finite abelian group
Finite filtration
Finite conditional expectation
Discrete-time stochastic process
Expectation in finite models
Measurable function
At-most-countable set
Bipartite graph
Convergence in distribution
Quotient ring of a finite ring
Graph coloring
Finite field
Finite martingale

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.

Stage 1 · required
Cardinality comparison of sets
Existing canonical prerequisite for the new objects in this pulse.
Related questions Concept guides
Stage 1 · required
Natural number
Existing canonical prerequisite for the new objects in this pulse.
Related questions Concept guides
Stage 1 · required
Injectivity and distinct inputs
Existing canonical prerequisite for the new objects in this pulse.
Related questions Concept guides
Stage 1 · required
Surjectivity and codomain coverage
Existing canonical prerequisite for the new objects in this pulse.
Related questions Concept guides
Stage 1 · required
Finite simple undirected graphs
Existing canonical prerequisite for the new objects in this pulse.
Related questions Concept guides
Stage 1 · required
Subset
Existing canonical prerequisite for the new objects in this pulse.
Related questions Concept guides
Stage 1 · required
Function mapping
Existing canonical prerequisite for the new objects in this pulse.
Related questions Concept guides
Stage 1 · required
Cumulative distribution function
Existing canonical prerequisite for the new objects in this pulse.
Related questions Concept guides
Stage 1 · required
Function limits on punctured real domains
Existing canonical prerequisite for the new objects in this pulse.
Related questions Concept guides
Stage 1 · required
Sequence
Existing canonical prerequisite for the new objects in this pulse.
Related questions Concept guides
Stage 1 · required
Finite joint distribution
Existing canonical prerequisite for the new objects in this pulse.
Related questions Concept guides
Stage 1 · required
Finite unital ring
Existing canonical prerequisite for the new objects in this pulse.
Related questions Concept guides
Stage 1 · required
Finite two-sided ideal
Existing canonical prerequisite for the new objects in this pulse.
Related questions Concept guides
Stage 1 · required
Equivalence class
Existing canonical prerequisite for the new objects in this pulse.
Related questions Concept guides
Stage 1 · required
Quotient set
Existing canonical prerequisite for the new objects in this pulse.
Related questions Concept guides
Stage 1 · required
Usual order on the real numbers
Existing canonical prerequisite for the new objects in this pulse.
Related questions Concept guides
Stage 1 · required
Finite integral domain
Existing canonical prerequisite for the new objects in this pulse.
Related questions Concept guides
Stage 1 · required
Finite abelian group
Existing canonical prerequisite for the new objects in this pulse.
Related questions Concept guides
Stage 1 · required
Finite filtration
Existing canonical prerequisite for the new objects in this pulse.
Related questions Concept guides
Stage 1 · required
Finite conditional expectation
Existing canonical prerequisite for the new objects in this pulse.
Related questions Concept guides
Stage 1 · required
Discrete-time stochastic process
Existing canonical prerequisite for the new objects in this pulse.
Related questions Concept guides
Stage 1 · required
Expectation in finite models
Existing canonical prerequisite for the new objects in this pulse.
Related questions Concept guides
Stage 1 · required
Measurable function
Existing canonical prerequisite for the new objects in this pulse.
Related questions Concept guides
Stage 2 · optional
At-most-countable set
A set is at most countable if it is finite or has an explicit injection into the natural numbers; this is not the same as countably infinite.
Related questions Concept guides
Stage 2 · optional
Bipartite graph
A finite undirected simple graph is bipartite when all vertices split into two disjoint parts and every edge crosses between the parts.
Related questions Concept guides
Stage 2 · optional
Convergence in distribution
Convergence in distribution means that F_n tends pointwise to F at every continuity point of the limiting CDF.
Related questions Concept guides
Stage 2 · optional
Quotient ring of a finite ring
Additive cosets of a proper two-sided ideal in a finite ring form a quotient ring when its operations are independent of representative choice.
Related questions Concept guides
Stage 2 · optional
Graph coloring
A proper k-coloring uses at most k colors and gives different colors to the endpoints of every edge.
Related questions Concept guides
Stage 2 · optional
Finite field
Deepen characteristic, nonzero multiplicative groups, affine equations and Frobenius identities in the complete finite-field model.
Related questions Concept guides
Stage 2 · optional
Finite martingale
Deepen martingale differences, affine stability, fair partial sums and constant-expectation counterexamples under finite filtrations.
Related questions Concept guides

Path source & status

path-pulse-foundations-20261031 · candidate

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