Irreducible prime channels
Shows that residues modulo 30 factor into the 2, 3, and 5 channels.
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Mathematical atlas
PT mathematics / irreducibility
theorem
Why prime numbers?
Why primes appear as irreducible channels of persistence.
Plain
The idea
Prime numbers are the multiplicative building blocks of the integers. In PT, this is not only arithmetic: it is a channel property.
A composite number mixes several constraints. A prime carries an irreducible constraint: it opens its own cycle, which nothing smaller can factor.
prime channels
Standard
Standard reading
The sieve removes multiples of a channel $p$. If $p$ is composite, that channel is already explained by smaller channels. Only a prime adds a new independent obstruction.
PT therefore reads primes as elementary directions of multiplicative persistence: each $p$ introduces an irreducible cyclic torus $\mathbb{Z}/p\mathbb{Z}$.
Takeaways
- A prime adds an independent constraint.
- A composite inherits constraints already present.
- PT turns primes into persistence channels.
Technical
Technical formulation
The Chinese Remainder Theorem factorizes $\mathbb{Z}/m\mathbb{Z}$ into primary components when the factors are coprime. This makes primes structural, not decorative.
In PT, primes are the minimal cyclic-phase channels compatible with CRT decomposition and GFT conservation.
Monograph: ch01_sieve, ch05_geometry, ch06_holonomy.
Formulas
$\mathbb{Z}/(pq)\mathbb{Z}\cong\mathbb{Z}/p\mathbb{Z}\oplus\mathbb{Z}/q\mathbb{Z}\quad(p,q\ \text{coprime})$
$p\ \text{prime}\Rightarrow \mathbb{Z}/p\mathbb{Z}\ \text{irreducible channel}$
public code
Code and scripts
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