OCME · Bilingual mathematics & derivations
Structural Depth Wave4 Fourth Layer: Finite Fields, Hilbert Spaces, Martingales, Quotients, Cardinality, CDFs and Convergence in Probability
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- Finite integral domain
- Finite unital ring
- Finite abelian group
- Real inner-product space
- Inner-product-induced norm
- Norm-induced metric
- Complete metric space
- Finite filtration
- Finite conditional expectation
- Discrete-time stochastic process
- Expectation in finite models
- Measurable function
- Equivalence class
- Set
- Function mapping
- Equivalence relation
- Injectivity and distinct inputs
- Surjectivity and codomain coverage
- Bijectivity and unique preimages
- Finite discrete probability distribution
- Usual order on the real numbers
- Probability on finite sample spaces
- Finite joint distribution
- Real random variables on finite sample spaces
- Sequence
- Convergence in a metric space
- Finite field
- Real Hilbert space
- Finite martingale
- Quotient set
- Cardinality comparison of sets
- Cumulative distribution function
- Convergence in probability
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 integral domain
Existing canonical prerequisite for the new objects in this pulse.
Finite unital ring
Existing canonical prerequisite for the new objects in this pulse.
Finite abelian group
Existing canonical prerequisite for the new objects in this pulse.
Real inner-product space
Existing canonical prerequisite for the new objects in this pulse.
Inner-product-induced norm
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.
Finite filtration
Existing canonical prerequisite for the new objects in this pulse.
Finite conditional expectation
Existing canonical prerequisite for the new objects in this pulse.
Discrete-time stochastic process
Existing canonical prerequisite for the new objects in this pulse.
Expectation in finite models
Existing canonical prerequisite for the new objects in this pulse.
Measurable function
Existing canonical prerequisite for the new objects in this pulse.
Equivalence class
Existing canonical prerequisite for the new objects in this pulse.
Set
Existing canonical prerequisite for the new objects in this pulse.
Function mapping
Existing canonical prerequisite for the new objects in this pulse.
Equivalence relation
Existing canonical prerequisite for the new objects in this pulse.
Injectivity and distinct inputs
Existing canonical prerequisite for the new objects in this pulse.
Surjectivity and codomain coverage
Existing canonical prerequisite for the new objects in this pulse.
Bijectivity and unique preimages
Existing canonical prerequisite for the new objects in this pulse.
Finite discrete probability distribution
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.
Probability on finite sample spaces
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.
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.
Finite field
A finite commutative unital ring is a finite field when every nonzero element has a multiplicative inverse.
Real Hilbert space
A real inner-product space is Hilbert when it is complete in the norm and metric induced by that same inner product.
Finite martingale
An integrable finite process adapted to the same filtration is a martingale when the next value's conditional expectation equals the current value almost surely at every step.
Quotient set
The distinct equivalence classes of a complete equivalence relation form its quotient set and partition the original carrier.
Cardinality comparison of sets
An injection, surjection or bijection between explicit finite source and target sets supplies a corresponding finite cardinality-comparison certificate.
Cumulative distribution function
For a real-valued random variable, F(t)=P(X≤t) is nondecreasing and right-continuous; in a finite discrete law, its jump is the point mass.
Convergence in probability
X_n converges in probability to X when, for every epsilon>0, the probability of deviation greater than epsilon tends to zero.
Path source & status
path-pulse-foundations-20261030 · candidate