# Wave6 Fourth Layer: Orbit-Stabilizer, MVT, Compact Images, Total Variance, Recurrences, Mobius Inversion and PDAs

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



## [Orbit of a finite group action](/en/mko/mko-finite-group-orbit/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261110%2F)

Existing canonical prerequisite for the new objects in this pulse.

[Related questions](/en/explore/?mko=mko-finite-group-orbit) [Concept guides](/en/derivations/?q=mko-finite-group-orbit)


- Stage 1 · required



## [Stabilizer of a finite group action](/en/mko/mko-finite-group-stabilizer/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261110%2F)

Existing canonical prerequisite for the new objects in this pulse.

[Related questions](/en/explore/?mko=mko-finite-group-stabilizer) [Concept guides](/en/derivations/?q=mko-finite-group-stabilizer)


- Stage 1 · required



## [Cardinality comparison of sets](/en/mko/mko-cardinality-comparison/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261110%2F)

Existing canonical prerequisite for the new objects in this pulse.

[Related questions](/en/explore/?mko=mko-cardinality-comparison) [Concept guides](/en/derivations/?q=mko-cardinality-comparison)


- Stage 1 · required



## [Real derivatives at a point and finite difference quotients](/en/mko/mko-derivative/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261110%2F)

Existing canonical prerequisite for the new objects in this pulse.

[Related questions](/en/explore/?mko=mko-derivative) [Concept guides](/en/derivations/?q=mko-derivative)


- Stage 1 · required



## [Point values and continuity of real functions](/en/mko/mko-continuity/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261110%2F)

Existing canonical prerequisite for the new objects in this pulse.

[Related questions](/en/explore/?mko=mko-continuity) [Concept guides](/en/derivations/?q=mko-continuity)


- Stage 1 · required



## [Average rate of change between distinct inputs](/en/mko/mko-rate-of-change/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261110%2F)

Existing canonical prerequisite for the new objects in this pulse.

[Related questions](/en/explore/?mko=mko-rate-of-change) [Concept guides](/en/derivations/?q=mko-rate-of-change)


- Stage 1 · required



## [Usual order on the real numbers](/en/mko/mko-order-relation/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261110%2F)

Existing canonical prerequisite for the new objects in this pulse.

[Related questions](/en/explore/?mko=mko-order-relation) [Concept guides](/en/derivations/?q=mko-order-relation)


- Stage 1 · required



## [Compact subset](/en/mko/mko-compact-subset/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261110%2F)

Existing canonical prerequisite for the new objects in this pulse.

[Related questions](/en/explore/?mko=mko-compact-subset) [Concept guides](/en/derivations/?q=mko-compact-subset)


- Stage 1 · required



## [Continuous maps between topological spaces](/en/mko/mko-topological-continuity/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261110%2F)

Existing canonical prerequisite for the new objects in this pulse.

[Related questions](/en/explore/?mko=mko-topological-continuity) [Concept guides](/en/derivations/?q=mko-topological-continuity)


- Stage 1 · required



## [Function mapping](/en/mko/mko-function-mapping/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261110%2F)

Existing canonical prerequisite for the new objects in this pulse.

[Related questions](/en/explore/?mko=mko-function-mapping) [Concept guides](/en/derivations/?q=mko-function-mapping)


- Stage 1 · required



## [Law of total expectation](/en/mko/mko-law-of-total-expectation/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261110%2F)

Existing canonical prerequisite for the new objects in this pulse.

[Related questions](/en/explore/?mko=mko-law-of-total-expectation) [Concept guides](/en/derivations/?q=mko-law-of-total-expectation)


- Stage 1 · required



## [Variance in finite probability models](/en/mko/mko-variance/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261110%2F)

Existing canonical prerequisite for the new objects in this pulse.

[Related questions](/en/explore/?mko=mko-variance) [Concept guides](/en/derivations/?q=mko-variance)


- Stage 1 · required



## [Finite conditional expectation](/en/mko/mko-conditional-expectation/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261110%2F)

Existing canonical prerequisite for the new objects in this pulse.

[Related questions](/en/explore/?mko=mko-conditional-expectation) [Concept guides](/en/derivations/?q=mko-conditional-expectation)


- Stage 1 · required



## [Sequence](/en/mko/mko-sequence/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261110%2F)

Existing canonical prerequisite for the new objects in this pulse.

[Related questions](/en/explore/?mko=mko-sequence) [Concept guides](/en/derivations/?q=mko-sequence)


- Stage 1 · required



## [Equation](/en/mko/mko-equation/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261110%2F)

Existing canonical prerequisite for the new objects in this pulse.

[Related questions](/en/explore/?mko=mko-equation) [Concept guides](/en/derivations/?q=mko-equation)


- Stage 1 · required



## [Mobius function](/en/mko/mko-mobius-function/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261110%2F)

Existing canonical prerequisite for the new objects in this pulse.

[Related questions](/en/explore/?mko=mko-mobius-function) [Concept guides](/en/derivations/?q=mko-mobius-function)


- Stage 1 · required



## [Natural-number divisibility](/en/mko/mko-natural-divisibility/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261110%2F)

Existing canonical prerequisite for the new objects in this pulse.

[Related questions](/en/explore/?mko=mko-natural-divisibility) [Concept guides](/en/derivations/?q=mko-natural-divisibility)


- Stage 1 · required



## [Basic counting principles](/en/mko/mko-counting-principle/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261110%2F)

Existing canonical prerequisite for the new objects in this pulse.

[Related questions](/en/explore/?mko=mko-counting-principle) [Concept guides](/en/derivations/?q=mko-counting-principle)


- Stage 1 · required



## [Abstract computational model](/en/mko/mko-computational-model/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261110%2F)

Existing canonical prerequisite for the new objects in this pulse.

[Related questions](/en/explore/?mko=mko-computational-model) [Concept guides](/en/derivations/?q=mko-computational-model)


- Stage 1 · required



## [Deterministic finite automaton](/en/mko/mko-finite-automaton/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261110%2F)

Existing canonical prerequisite for the new objects in this pulse.

[Related questions](/en/explore/?mko=mko-finite-automaton) [Concept guides](/en/derivations/?q=mko-finite-automaton)


- Stage 1 · required



## [Context-free grammar](/en/mko/mko-context-free-grammar/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261110%2F)

Existing canonical prerequisite for the new objects in this pulse.

[Related questions](/en/explore/?mko=mko-context-free-grammar) [Concept guides](/en/derivations/?q=mko-context-free-grammar)


- Stage 2 · optional



## [Orbit-stabilizer theorem](/en/mko/mko-orbit-stabilizer-theorem/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261110%2F)

For a complete finite group action, the orbit size and stabilizer size of the same point multiply to the group order.

[Related questions](/en/explore/?mko=mko-orbit-stabilizer-theorem) [Concept guides](/en/derivations/?q=mko-orbit-stabilizer-theorem)


- Stage 3 · optional



## [Mean value theorem](/en/mko/mko-mean-value-theorem/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261110%2F)

If a<b, f is continuous on [a,b] and differentiable on (a,b), some interior c has derivative equal to the endpoint secant slope.

[Related questions](/en/explore/?mko=mko-mean-value-theorem) [Concept guides](/en/derivations/?q=mko-mean-value-theorem)


- Stage 4 · optional



## [Continuous image of a compact set](/en/mko/mko-continuous-image-of-compact/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261110%2F)

If K is compact in X and f:X→Y is continuous, then f(K) is compact in Y.

[Related questions](/en/explore/?mko=mko-continuous-image-of-compact) [Concept guides](/en/derivations/?q=mko-continuous-image-of-compact)


- Stage 5 · optional



## [Law of total variance](/en/mko/mko-law-of-total-variance/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261110%2F)

For the same finite law and partition-generated G, total variance is the expected within-cell variance plus the variance of the cell means.

[Related questions](/en/explore/?mko=mko-law-of-total-variance) [Concept guides](/en/derivations/?q=mko-law-of-total-variance)


- Stage 6 · optional



## [Recurrence relation](/en/mko/mko-recurrence-relation/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261110%2F)

An order-r recurrence uses explicit initial values and a total single-valued state update to determine successive sequence terms.

[Related questions](/en/explore/?mko=mko-recurrence-relation) [Concept guides](/en/derivations/?q=mko-recurrence-relation)


- Stage 7 · optional



## [Mobius inversion on finite divisor sets](/en/mko/mko-mobius-inversion-finite-divisors/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261110%2F)

If F(n) sums f(d) over every positive divisor of n, Mobius weights exactly recover f(n).

[Related questions](/en/explore/?mko=mko-mobius-inversion-finite-divisors) [Concept guides](/en/derivations/?q=mko-mobius-inversion-finite-divisors)


- Stage 8 · optional



## [Pushdown automaton](/en/mko/mko-pushdown-automaton/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261110%2F)

A nondeterministic PDA uses finite control, input and stack transitions; this slice accepts only after reading the whole input in a final state.

[Related questions](/en/explore/?mko=mko-pushdown-automaton) [Concept guides](/en/derivations/?q=mko-pushdown-automaton)



Path source & statuspath-pulse-foundations-20261110 · candidate

[Original paths JSON](/data/architecture/learning-paths.json)