Why - x - = + : The Wave Phase Proof from Superposed Zero Theory

Abstract: For 1,400 years, mathematics has used the rule - x - = + without a physical reason. Brahmagupta stated the rule in 628 CE as "the product of two debts is a fortune" but gave no proof. Later mathematicians proved it with algebra using the distributive law, showing that mathematics would contradict itself if the rule were false. This paper gives the first physical reason: - is a wave trough, + is a wave crest, x is the collision of phases. When two troughs collide, their phases cancel to 0, and 0 splits into a new crest. Therefore - x - = + is not only a mathematical necessity, but a physical necessity. If - x - were not +, waves would not bounce back, time would not flow, and nothing would exist.