Imbalance Prime Sieving: Every Prime Gap Is a Result of a M\"obius Imbalance Obstruction

TL;DR

This paper presents a novel prime sieving method based on topological obstructions in a M"obius-transformed space, offering an exact prime enumeration algorithm and a geometric perspective on prime gaps.

Contribution

It introduces a new sieve that detects topological obstructions to identify primes and explains prime gaps through geometric collisions in a transformed space.

Findings

Accurately filters primes up to a bound

Provides a geometric interpretation of prime gaps

Suggests potential for new number-theoretic models

Abstract

We introduce a novel sieve for prime numbers based on detecting topological obstructions in a M\"obius-transformed rational metric space. Unlike traditional sieves which rely on divisibility, our method identifies primes as those numbers which contribute new, non-colliding imbalance conjugates. This provides both an exact algorithm for prime enumeration and a new geometric interpretation of prime gaps. This sieve constructs a topological obstruction theory over rational pairs (p, q), from which we observe that every prime gap is a consequence of a collision in this transformed imbalance space. Our empirical results demonstrate that this method precisely filters the prime numbers up to a specified bound, with potential implications for new number-theoretic models and sieving algorithms.