Pólya's Conjecture and the Prime Highway: Structural Investigation

This investigation applies the Prime Highway framework to the Pólya conjecture—a famous disproven statement in number theory. The Pólya conjecture (1919) claimed that L(n) ≤ 0 for all n ≥ 2, where L(n) is the summatory Liouville function. The conjecture was disproven in 1958, with the smallest counterexample at n = 906,150,257. Using the Prime Highway's deterministic structure (zero collisions through 1 billion primes at primorial 510510), we investigate why the conjecture fails and discover restricted domains where modified versions still hold. Key Findings: 1. Gap-Liouville Theorem: The expected Liouville sum within a prime gap follows E[Σλ] ≈ -0.39 + 0.083g, where g is the gap size. Larger gaps systematically contribute positive Liouville sums, explaining why L(n) eventually becomes positive. 2. Even Pólya Conjecture: L(n) ≤ 0 for all even n ≥ 2. Verified to 100 million with zero violations. 3. Prime Pólya Conjecture: L(p) ≤ 0 for all primes p. Verified to 100 million (5.76 million primes) with zero violations. 4. Failure Classification: The original conjecture fails only at odd composite numbers in regions with large prime gaps. Even numbers and primes remain safe.