OCME · Bilingual mathematics & derivations
Wave6 Third Layer: Stabilizers, Bolzano-Weierstrass, Compact Subsets, Total Expectation, Binomial Theorem, Mobius Function and CFGs
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- Finite group action
- Subgroups and group homomorphisms
- Subsequence
- Sequence limits and exact tail cutoffs
- Real number
- Supremum and infimum
- Compact topological space
- Subspace topology
- Finite conditional expectation
- Expectation in finite models
- Event
- Permutations and combinations
- Mathematical induction
- Univariate polynomials over the rationals
- Prime natural number
- Unique prime factorization of integers
- Function mapping
- Abstract computational model
- Set
- Sequence
- Stabilizer of a finite group action
- Bolzano-Weierstrass theorem for real sequences
- Compact subset
- Law of total expectation
- Binomial theorem
- Mobius function
- Context-free grammar
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 action
Existing canonical prerequisite for the new objects in this pulse.
Subgroups and group homomorphisms
Existing canonical prerequisite for the new objects in this pulse.
Subsequence
Existing canonical prerequisite for the new objects in this pulse.
Sequence limits and exact tail cutoffs
Existing canonical prerequisite for the new objects in this pulse.
Real number
Existing canonical prerequisite for the new objects in this pulse.
Supremum and infimum
Existing canonical prerequisite for the new objects in this pulse.
Compact topological space
Existing canonical prerequisite for the new objects in this pulse.
Subspace topology
Existing canonical prerequisite for the new objects in this pulse.
Finite conditional expectation
Existing canonical prerequisite for the new objects in this pulse.
Expectation in finite models
Existing canonical prerequisite for the new objects in this pulse.
Event
Existing canonical prerequisite for the new objects in this pulse.
Permutations and combinations
Existing canonical prerequisite for the new objects in this pulse.
Mathematical induction
Existing canonical prerequisite for the new objects in this pulse.
Univariate polynomials over the rationals
Existing canonical prerequisite for the new objects in this pulse.
Prime natural number
Existing canonical prerequisite for the new objects in this pulse.
Unique prime factorization of integers
Existing canonical prerequisite for the new objects in this pulse.
Function mapping
Existing canonical prerequisite for the new objects in this pulse.
Abstract computational model
Existing canonical prerequisite for the new objects in this pulse.
Set
Existing canonical prerequisite for the new objects in this pulse.
Sequence
Existing canonical prerequisite for the new objects in this pulse.
Stabilizer of a finite group action
The stabilizer of a specified point consists of every group element fixing that point and forms a subgroup.
Bolzano-Weierstrass theorem for real sequences
Every whole bounded real sequence has a convergent subsequence selected by an infinite strictly increasing index map.
Compact subset
A subset K of X is compact in its subspace topology exactly when every relative-open cover has a finite subcover.
Law of total expectation
On the same finite probability space, weighting every partition-cell conditional mean by its cell probability gives E[X].
Binomial theorem
With commutative multiplication, a binomial power of natural exponent n has a complete finite expansion with binomial coefficients.
Mobius function
On positive integers, complete prime multiplicities determine the Mobius function: mu(1)=1, a repeated prime gives0, and a squarefree integer has the parity sign.
Context-free grammar
A CFG uses finite disjoint nonterminal and terminal alphabets, a start symbol and finite productions with a single nonterminal on each left side.
Path source & status
path-pulse-foundations-20261109 · candidate