OCME · Bilingual mathematics & derivations
Wave6 Tenth Layer: Contraction, IVT, Taylor Remainders, Finite MLE, Sufficiency, Topological Orders and Modular Order
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- Orbits, fixed points and periodic points
- Complete metric space
- Convergence in a metric space
- Infinite series
- Point values and continuity of real functions
- Real number
- Usual order on the real numbers
- Function mapping
- Supremum and infimum
- Taylor polynomial
- Higher-order derivative
- Mean value theorem
- Absolute value
- Parametric statistical model
- Statistical estimator
- Optimization problem
- Sampling distribution
- Conditional probability in finite models
- Finite directed graph
- Algorithm correctness
- Mathematical induction
- Modular inverse
- Greatest common divisor and coprimality
- Natural-number divisibility
- Euler's theorem in number theory
- Contraction mapping theorem
- Intermediate value theorem
- Taylor remainder theorem
- Maximum-likelihood estimator in a finite model
- Sufficient statistic in a finite model
- Topological ordering of a directed acyclic graph
- Multiplicative order modulo n
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.
Orbits, fixed points and periodic points
Existing canonical prerequisite for the new objects in this pulse.
Complete metric space
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.
Infinite series
Existing canonical prerequisite for the new objects in this pulse.
Point values and continuity of real functions
Existing canonical prerequisite for the new objects in this pulse.
Real number
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.
Function mapping
Existing canonical prerequisite for the new objects in this pulse.
Supremum and infimum
Existing canonical prerequisite for the new objects in this pulse.
Taylor polynomial
Existing canonical prerequisite for the new objects in this pulse.
Higher-order derivative
Existing canonical prerequisite for the new objects in this pulse.
Mean value theorem
Existing canonical prerequisite for the new objects in this pulse.
Absolute value
Existing canonical prerequisite for the new objects in this pulse.
Parametric statistical model
Existing canonical prerequisite for the new objects in this pulse.
Statistical estimator
Existing canonical prerequisite for the new objects in this pulse.
Optimization problem
Existing canonical prerequisite for the new objects in this pulse.
Sampling distribution
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 directed graph
Existing canonical prerequisite for the new objects in this pulse.
Algorithm correctness
Existing canonical prerequisite for the new objects in this pulse.
Mathematical induction
Existing canonical prerequisite for the new objects in this pulse.
Modular inverse
Existing canonical prerequisite for the new objects in this pulse.
Greatest common divisor and coprimality
Existing canonical prerequisite for the new objects in this pulse.
Natural-number divisibility
Existing canonical prerequisite for the new objects in this pulse.
Euler's theorem in number theory
Existing canonical prerequisite for the new objects in this pulse.
Contraction mapping theorem
A self-map on a nonempty complete metric space with one global contraction constant q<1 has a unique fixed point and convergent Picard iterates.
Intermediate value theorem
A continuous real function on a closed interval attains each value between its endpoint values.
Taylor remainder theorem
Under sufficient differentiability, a Lagrange remainder and derivative bound control the Taylor approximation error.
Maximum-likelihood estimator in a finite model
A finite-model MLE chooses a parameter maximizing the observed likelihood; ties use one fixed rule independent of the unknown parameter.
Sufficient statistic in a finite model
A statistic is sufficient when the conditional sample law given its value does not depend on theta wherever that condition has positive mass.
Topological ordering of a directed acyclic graph
A finite directed graph has a topological ordering exactly when it has no directed cycle.
Multiplicative order modulo n
For gcd(a,n)=1, the multiplicative order is the smallest positive k with a^k congruent to1 modulo n.
Path source & status
path-pulse-foundations-20261116 · candidate