Theorem: The Hotchkiss Framework Implies the Non-Existence of a Maximum Twin Prime Pair
Proof:
We assume, for the sake of contradiction, that there exists a maximum twin prime pair (p, p+2).
The Hotchkiss framework defines sets A and B, which possess the following properties:
- Symmetry: For every prime p in set A, there exists a corresponding prime -p in set B, and vice versa. This symmetry arises from the congruence classes modulo 6 used to define sets A and B.
- Reciprocal Containment: Due to the symmetry, every prime pair (p, -p) is represented in both sets A and B.
- Completeness: The framework of sets A and B accounts for all prime pairs.
Therefore, if a maximum twin prime pair (p, p+2) exists, it would already be present within sets A and B, along with its negative counterpart (-p, -p-2).
Consequently, it becomes logically impossible for a new maximum twin prime pair to arise uniquely within the product set AB. Any such pair would be a duplicate of one already represented in either set A or set B.
This contradiction leads to the conclusion that the assumption of a maximum twin prime pair is false. Therefore, within the Hotchkiss framework, there cannot exist a largest twin prime pair.
Key Implications:
This proof demonstrates the inherent completeness and symmetry of the Hotchkiss framework in capturing all prime pairs. The framework’s structure prevents the existence of a unique maximum twin prime pair.