Lesson 0: Mathematical Building Blocks

This lesson gives the minimum toolkit for reading the rest of the book. It starts before circles: with objects, names, rules, repetition, and proof.

Lesson Goal

Learn the basic moves that make later Circle Calculus chapters feel natural. After this lesson, you should be able to tell whether a page is naming an object, applying a rule, tracing repeated motion, or claiming a theorem.

1. Objects And Names

Mathematics begins by deciding what kind of thing is being discussed. A thing can be a number, a point, a state, a node, a function, a proof object, or a whole structured space.

Circle Calculus often separates a raw name from the object it names inside a context.

Definition In Words

An address is a raw name for a position. A node is the position after the address has been interpreted inside a finite circle.

address -> rule/context -> node

In ordinary arithmetic, 14 and 2 are different integers. In the finite circle C_12, they can name the same node because the circle keeps only the remainder after division by 12.

2. Rules And Functions

A rule says how to transform one thing into another. When the rule is stable enough to use again and again, mathematics usually treats it as a function.

In C_12, the rule “reduce by 12” sends many addresses to the same node:

2  -> 2
14 -> 2
26 -> 2

The inputs are different. The output node is the same because the rule ignores completed turns.

The first S1 rule is reduction. The second is rotation. A rotation says: start at a node, add a stride, and reduce again.

3. Composition

Composition means doing one rule after another. This is one of the central ideas in algebra, geometry, programming, and proof assistants.

Definition In Words

Two rules compose when the output of the first rule becomes the input of the second rule.

first x -> y, then y -> z

In S1, two rotations compose into one rotation because adding stride b and then stride a lands where adding a + b lands.

4. Iteration

Iteration means applying the same rule repeatedly. Iteration turns a single move into a path.

Start choose a node Step apply the rule once Repeat apply the same rule again Watch record what is visited Close notice first return

This is why coils matter. A coil is not a new mystery object. It is the path created when one rotation is iterated.

5. Invariants

An invariant is something that stays controlled while a process moves. Invariants are what make long calculations understandable.

In S1, the important invariant is often a remainder, a gcd, or a period. You can move many times, but the arithmetic structure keeps constraining what can happen.

In C_18 with stride 6, the orbit from 0 is:

0 -> 6 -> 12 -> 0

The stride shares gcd(18,6) = 6 with the circle size. That shared divisor explains why the path closes after three nodes instead of visiting all eighteen.

6. Evidence Layers

The Living Book separates four kinds of evidence.

Layer	What It Does	What It Does Not Do
Picture	Builds intuition quickly.	It does not prove the theorem.
Rule	Gives the precise calculation.	It may still need proof.
Example	Checks concrete values.	It does not cover every case.
Proof	Covers the stated general claim.	It must match the manifest and Lean source.

Do not let the picture or widget do the job of a proof. The book uses diagrams to teach the idea, then theorem cards and Lean declarations to audit proved claims.

Common Mistake

The most common early mistake is to collapse every layer into one sentence. Instead of saying “14 equals 2,” say “14 and 2 are different integers that reduce to the same node in C_12.”

That wording keeps the object, context, rule, and result separate.

Try It

For each sentence, identify whether it is naming an object, applying a rule, giving an example, or making a theorem-like general claim.

“C_7 is a finite circle with seven nodes.”
“15 lands on node 1 in C_7.”
“In C_12, stride 5 from node 8 lands on node 1.”
“For every n, two rotations compose by adding their strides.”

Checkpoint

Why is “14 and 2 name the same node in C_12” more precise than “14 and 2 are the same”?

Source Trail

These dictionary cards are the stable vocabulary that later lessons reuse. They are reference support, not a substitute for reading the lesson.