For any integer p > 3, p is prime if and only if:

- p ∈ |{6n ± 1 | n ∈ ℤ}|
- p ≠ |a * b| where a, b ∈ {6n ± 1 | n ∈ ℤ} with the same sign

Key features:

- Unified Representation: All primes >3 are expressed in a single set using the absolute value function, unifying the traditional 6n-1 and 6n+1 forms.
**Symmetry:**The theorem captures the symmetrical distribution of primes around multiples of 6, extending to**both positive and negative integers**.- Concise Primality Test: The second condition provides an elegant criterion for primality within the defined set.
- Completeness: The theorem both represents all primes >3 and provides a sufficient condition for primality.

Implications:

This theorem presents a semiotically elegant representation of prime numbers, emphasizing their inherent structure and symmetry.

*Claude was principally used for this refinement agreed upon by other native models tested. I recommend Claude on this day. You should try. A future model may suck, but this one is great! *

https://spinscore.io/?url=https%3A%2F%2Fn01r.com%2Fsemiotic-prime-theorem-2-0%2F *(Note: the A+ Spinscore is based on the theorem alone, not the ruminations on Claude)*