Lesson 6: S1 Review And Exercises

This review checks whether the finite-circle unit is usable as a single story. Work through it after the first five S1 lessons and before moving to S2.

Unit Readiness Goal

You are ready for higher dimensions when you can solve finite-circle problems by hand, explain what the widgets are showing, and identify which facts are theorem-backed.

What You Should Be Able To Do

  • Reduce an integer address to a node in C_n.
  • Compose two rotations by adding their strides.
  • Trace an orbit from a start node and stride.
  • Predict period from n / gcd(n,k).
  • Recognize why prime circles make every nonzero stride a full coil.
  • Separate residue from winding in a lifted coordinate.

These exercises are learning checks. They are not formal proofs. Use the theorem cards below when you want the proof-status trail.

Exercises

  1. In C_10, reduce the addresses 3, 13, 23, and -7. What do they have in common?

  2. In C_12, start at 8. Apply stride 5, then stride 7. What is the final node? What single stride gives the same result?

  3. In C_12, trace the orbit from start 0 with stride 4. How many nodes are visited before closure?

  4. In C_18, use stride 6. Compute gcd(18,6), the number of disjoint cycles, and the length of each cycle.

  5. Compare C_11 and C_12. Which nonzero strides are full coils in each circle?

  6. With base 5, lift t = 17 into quotient-plus-residue form. Then explain why residue alone loses information.

Answer Sketches

  1. They all reduce to node 3 in C_10.

  2. The two-step stride is 5 + 7 = 12, so the final node is the same as start 8 in C_12.

  3. The orbit is 0, 4, 8, then it returns to 0; the period is 3.

  4. gcd(18,6) = 6; there are six disjoint cycles, each of length 3.

  5. In C_11, every stride 1 through 10 is a full coil. In C_12, only strides coprime to 12 are full coils: 1, 5, 7, and 11.

  6. 17 = 3*5 + 2, so the winding count is 3 and the residue is 2. The residue alone cannot distinguish 2, 7, 12, and 17.

Readiness Check

If someone gives you a new C_n, a start, and a stride, you should now know how to draw the finite circle, reduce addresses, trace repeated motion, predict closure, and decide whether the output is an intuition diagram, a Python example, or a theorem-backed claim.

Source Trail

These theorem cards cover the main S1 proof spine used in the exercises.

Finite-circle paper: Circle Calculus I

Winding paper: Circle Calculus II

Previous: Winding Lift Next Unit: S2 Sphere Scaffolding