Definition: An algebraic rule stating that the modulation cycle of a composite base (b₁×b₂) is the element-wise product of the weight sequences of its component base cycles.
Chapter 1: The Team-Up Rule for Codes (Elementary School Understanding)
Imagine every number "base" is a special code-making machine.
The Base-2 Machine turns the numbers 1, 2, 3, 4... into a repeating code 1, 0, 1, 0....
The Base-3 Machine turns the numbers 1, 2, 3, 4... into a different repeating code 1, 2, 0, 1, 2, 0....
Now, what if we want to know the code for a Base-6 Machine? The number 6 is a "composite base" because it's made of 2 × 3.
The Law of Composite Bases is a magical shortcut. It says you don't need a new machine for Base-6! You can figure out its code just by looking at the codes from the Base-2 and Base-3 machines.
The Rule: To get the Base-6 code, you just multiply the numbers from the other two codes in order.
Base-2 Code: 1, 0, 1, 0, 1, 0...
Base-3 Code: 1, 2, 0, 1, 2, 0...
Base-6 Code: 1×1, 0×2, 1×0, 0×1, 1×2, 0×0... which is 1, 0, 0, 0, 2, 0...
This law tells us that the behavior of a big, composite number language is just the "teamed-up" behavior of its smaller, prime number language parts.
Chapter 2: Predicting Cycles by Multiplication (Middle School Understanding)
When we study a number N in a specific base b, we often care about its remainder when divided by a prime p. The sequence of remainders for b⁰, b¹, b², b³, ... when divided by p is called the modulation cycle or weight sequence.
Example: Modulo p=5
Weight Sequence for Base-2:
2⁰=1 ≡ 1 (mod 5)
2¹=2 ≡ 2 (mod 5)
2²=4 ≡ 4 (mod 5)
2³=8 ≡ 3 (mod 5)
2⁴=16 ≡ 1 (mod 5) → The cycle repeats.
Sequence is (1, 2, 4, 3)
Weight Sequence for Base-3:
3⁰=1 ≡ 1 (mod 5)
3¹=3 ≡ 3 (mod 5)
3²=9 ≡ 4 (mod 5)
3³=27 ≡ 2 (mod 5)
3⁴=81 ≡ 1 (mod 5) → The cycle repeats.
Sequence is (1, 3, 4, 2)
The Law of Composite Bases gives us a shortcut to find the weight sequence for a composite base, like Base-6 (2 × 3), without doing all the calculations.
The Law: The weight sequence for b = b₁ × b₂ is the element-wise product of the individual sequences (modulo p).
Weight Sequence for Base-6 (mod 5):
W₆ = (W₂[i] × W₃[i]) mod 5
W₆(0) = (1 × 1) mod 5 = 1
W₆(1) = (2 × 3) mod 5 = 6 mod 5 = 1
W₆(2) = (4 × 4) mod 5 = 16 mod 5 = 1
W₆(3) = (3 × 2) mod 5 = 6 mod 5 = 1
The predicted sequence for Base-6 is (1, 1, 1, 1). Let's check: 6⁰=1, 6¹≡1, 6²=36≡1, 6³≡1. It's correct!
Chapter 3: A Consequence of Modular Arithmetic Laws (High School Understanding)
The Law of Composite Bases is a theorem that describes the structure of the Modulation Group G(b mod p). This group consists of the weight sequence W_b = (b⁰, b¹, b², ...) modulo a prime p.
The Theorem: Let b = b₁ × b₂. The weight sequence for the composite base b modulo p, W_b, is the element-wise product of the weight sequences W_{b₁} and W_{b₂} modulo p.
W_b[i] = (W_{b₁}[i] × W_{b₂}[i]) mod p
Proof:
The i-th term of the weight sequence W_b is, by definition, bⁱ mod p.
Substitute b = b₁ × b₂:
W_b[i] = (b₁ × b₂)ⁱ mod p
By the laws of exponents, (xy)ⁱ = xⁱyⁱ.
W_b[i] = (b₁ⁱ × b₂ⁱ) mod p
By the laws of modular arithmetic, (x × y) mod p = ((x mod p) × (y mod p)) mod p.
W_b[i] = ((b₁ⁱ mod p) × (b₂ⁱ mod p)) mod p
By definition, b₁ⁱ mod p is the i-th term of the weight sequence W_{b₁}. The same is true for b₂.
W_b[i] = (W_{b₁}[i] × W_{b₂}[i]) mod p
The law is proven.
This theorem is powerful because it allows us to decompose the analysis of a complex, composite base into the analysis of its simpler, prime component bases. It shows that the structural interactions are perfectly compositional.
Chapter 4: A Homomorphism on the Ring of Cyclical Sequences (College Level)
The Law of Composite Bases describes a homomorphism related to the structure of (ℤ/pℤ)ˣ, the multiplicative group of integers modulo p.
The Mathematical Objects:
Let S_p be the set of all periodic sequences of elements from (ℤ/pℤ). We can define an element-wise multiplication on this set, turning it into a ring.
Let W: ℤ⁺ → S_p be a map that takes an integer base b and maps it to its weight sequence, W(b) = (b⁰, b¹, b², ...) mod p.
The Law as a Homomorphism:
The Law of Composite Bases is the statement that this map W is a semigroup homomorphism from the multiplicative semigroup of positive integers (ℤ⁺, ×) to the ring of sequences (S_p, ×).
W(b₁ × b₂) = W(b₁) × W(b₂)
This provides a powerful analytical tool. To understand the structural properties of a composite base b (like its cycle length, |G(b mod p)|), we can first analyze the properties of its prime factors pᵢ and then combine the results.
Application to Cycle Length:
The length of the cycle W(b) must be a divisor of the least common multiple (lcm) of the lengths of the component cycles W(pᵢ).
Example from Chapter 2: p=5.
|G(2 mod 5)| has length 4. W₂ = (1,2,4,3).
|G(3 mod 5)| has length 4. W₃ = (1,3,4,2).
lcm(4, 4) = 4.
The cycle length for base 6 (2×3) must be a divisor of 4. We calculated it to be 1 ((1,1,1,1) repeated), which is a divisor of 4.
This theorem allows us to decompose the study of Frame Incompatibility. The total dissonance generated by a composite base b can be understood as the superposition of the dissonances generated by its prime factors.
Chapter 5: Worksheet - The Team-Up Rule
Part 1: The Team-Up Rule for Codes (Elementary Level)
The Base-2 machine has a code (1,0,1,0...). The Base-5 machine has a code (1,2,3,4,0,1...) modulo 5. No, that's not right. 5⁰≡1, 5¹≡0. The code is (1,0,0,0...) mod 5.
Let's use a better example. Modulo 7:
Base-2 code is (1,2,4,1,2,4...)
Base-3 code is (1,3,2,6,4,5...)
What is the start of the code for the composite Base-6 (2×3) machine?
Part 2: Predicting Cycles (Middle School Understanding)
Let's work modulo p=7.
Find the first 3 terms of the weight sequence for Base-2.
Find the first 3 terms of the weight sequence for Base-4.
Using the law, predict the first 3 terms of the weight sequence for Base-8 (2×4).
Part 3: Modular Arithmetic (High School Understanding)
Provide the step-by-step proof of the Law of Composite Bases using the laws of exponents and modular arithmetic.
The cycle length for Base-2 mod 11 is 10. The cycle length for Base-3 mod 11 is 5.
What can you say about the maximum possible cycle length for Base-6 mod 11?
Part 4: The Homomorphism (College Level)
What is a semigroup homomorphism?
The Law of Composite Bases states that the map W(b) is a homomorphism between which two algebraic structures?
How does this law allow us to "decompose" the problem of analyzing the structural interaction between a composite base b and a prime modulus p?