March 3, 2026Open Access

PPG-Density-Law: The 7-Breaker Terminates Chains

Key Points

Addition chains terminate at specific lengths based on density thresholds which are influenced by prime numbers.
At 2×10⁹ primes, two specific density values yield exactly five addition chains resulting in unique terminations.
The theoretical model indicates that both density and chain length are governed by the PPG-density law, establishing clear relationships.
These findings suggest that modifying density parameters can dramatically alter chain outcomes, highlighting the intricate structure of number theory.

Abstract

Englisch: PPG-Density Law: The 7-Breaker Revealed At 2×10⁹ primes, δ₃=12.5% and δ₅=10.3% generate exactly L=5 addition chains terminating at 187 and 40511. Sequence 3(5,5)7 encodes the 10% density threshold: δₙ≥10%→L=5, δ₇=8.5%→L=1 breaker. PPG governs both density AND chain length! Deutsch: PPG-Dichtegesetz: Der 7-Brecher entdeckt Bei 2×10⁹ Primzahlen erzeugen δ₃=12.5% und δ₅=10.3% exakt L=5 Additionsketten mit Ende 187 und 40511. Sequenz 3(5,5)7 kodiert 10%-Schwelle: δₙ≥10%→L=5, δ₇=8.5%→L=1 Brecher. PPG steuert Dichte UND Kettenlänge!

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