Does constant market cap imply reversion to the mean

If the market cap of a market is constant, do stock prices necessarily follow the principle of reversion to the mean? How would one prove this? (I'm pretty sure this is trivial, but I don't know how to express the necessary ideas).

Identically: $$Market\_Cap \equiv Number\_of\_Shares \cdot Price\_per\_Share$$ If a market cap is constant than stock prices only fluctuate because of changes in the number of shares. Therefore, this would imply mean reversion in share prices only if the fluctuations in the number of shares were also mean reverting.
As a general matter in the United States over the last hundred or so years, the average price per share has mostly been \$20-\$40 while market capitalization has increased substantially, implying a large increase in the number of shares. This has mostly been achieved by stock splits, stock based acquisitions, and secondary equity offerings.