Lesson 5: Winding Lift
Winding keeps full-turn information that ordinary residues forget.
Lesson Goal
Learn how a lifted coordinate records both the number of full turns and the final residue. After this lesson, you should be able to decompose a step count into quotient plus remainder.
The Idea
The earlier S1 lessons deliberately threw away full turns. That is perfect for studying circular closure, but it is not enough when the number of turns matters.
For example, on a circle with base 5, the values 2, 7, 12, and 17 all have residue 2. The lift separates them by recording how many full turns occurred before landing at residue 2.
Definition In Words
A lifted node records a winding count q and residue r so the represented value is q*n + r.
t = q*n + r, with 0 <= r < n
Diagram
Use the widget to see the quotient and residue change separately.
Do not read residue-only diagrams as natural-number proofs. The lift layer is what stores full turns. The widget is an explanation, not a proof.
Worked Example
With base 5, lift t = 17.
17 = 3*5 + 2
q = 3
r = 2The residue 2 tells you where you landed. The winding count 3 tells you how many full turns happened first.
Common Mistake
Do not try to recover the original natural number from the residue alone. Residue 2 could have come from 2, 7, 12, 17, or many later values.
Try It
Set base n = 5 and step count t = 17. The lift should show three full turns and residue 2. Then try t = 2, 7, and 12; the residue stays the same while the winding count changes.
Checkpoint
Why is the residue 2 alone not enough to recover whether the original step count was 2, 7, 12, or 17?
Source Trail
Python reference model:
from circle_math.winding import lift
lift(5, 17) # winding=3, residue=2Paper source: Circle Calculus II