Lecture 38 · Mathematics
Read an orbit as a permutation
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Meanwhile, read the outlineWhat you will learn
Construct inverse orbit-rank permutations and state the precise real critical-cycle/Gilbreath correspondence.
Before this lecture
In this lecture
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Separate two orders on an orbit
- Chronological ranks: 2, 1, 3
Compute forward and inverse maps
- F = [3,1,2]
- G = F⁻¹ = [2,3,1]
Realize the inverse as a shuffle
- Real orbit → ranks → inverse map → deck list
Work the first period-four example
- Ranks 3, 1, 4, 2 → G = [3,4,2,1]
Compare the second period-four example
- Ranks 2, 1, 4, 3 → G = [2,3,4,1]
State the precise deeper theorem
- Exact real critical period n ↔ Gilbreath n-cycles
Check small cycle counts
- All Gilbreath permutations ≠ single-cycle members
Play the correspondence
- Use predecessors, not successors
Companion interactive
Planned
Rank an exact real critical orbit, invert its forward map, test prefixes and reconstruct a shuffle.
The interactive for this lecture is being built and will appear here.
Notes
Written notes for this lecture will appear here.
Sources
- Magical Mathematics, Chapter 5, pp. 74–78, theorem on p. 77
How these sources were used
- Use right-to-left orbit reading as the inverse forward-rank map. Correctly distinguish chronological rank word, forward map, inverse map and deck-list convention.